Stokes problem - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Stokes problem - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"222360b6-e1cf-4613-b047-e2d2df1479ed","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Stokes_problem","wgTitle":"Stokes problem","wgCurRevisionId":1249345868,"wgRevisionId":1249345868,"wgArticleId":53946357,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Fluid dynamics"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Stokes_problem","wgRelevantArticleId":53946357,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true, "wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q30715867","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready", "user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns", "ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/b/be/Stokes_boundary_layer.gif"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1684"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/b/be/Stokes_boundary_layer.gif"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="1123"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="898"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Stokes problem - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Stokes_problem"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Stokes_problem&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Stokes_problem"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Stokes_problem rootpage-Stokes_problem skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Stokes+problem" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Stokes+problem" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Stokes+problem" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Stokes+problem" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Flow_description[3][4]" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Flow_description[3][4]"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Flow description<sup><span>[</span>3<span>]</span></sup><sup><span>[</span>4<span>]</span></sup></span> </div> </a> <button aria-controls="toc-Flow_description[3][4]-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Flow description<sup><span>[</span>3<span>]</span></sup><sup><span>[</span>4<span>]</span></sup> subsection</span> </button> <ul id="toc-Flow_description[3][4]-sublist" class="vector-toc-list"> <li id="toc-Solution[5][6]" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solution[5][6]"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Solution<sup><span>[</span>5<span>]</span></sup><sup><span>[</span>6<span>]</span></sup></span> </div> </a> <ul id="toc-Solution[5][6]-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vorticity_oscillations_near_the_boundary" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vorticity_oscillations_near_the_boundary"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Vorticity oscillations near the boundary</span> </div> </a> <ul id="toc-Vorticity_oscillations_near_the_boundary-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fluid_bounded_by_an_upper_wall" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fluid_bounded_by_an_upper_wall"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Fluid bounded by an upper wall</span> </div> </a> <ul id="toc-Fluid_bounded_by_an_upper_wall-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fluid_bounded_by_a_free_surface" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fluid_bounded_by_a_free_surface"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Fluid bounded by a free surface</span> </div> </a> <ul id="toc-Fluid_bounded_by_a_free_surface-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flow_due_to_an_oscillating_pressure_gradient_near_a_plane_rigid_plate" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Flow_due_to_an_oscillating_pressure_gradient_near_a_plane_rigid_plate"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Flow due to an oscillating pressure gradient near a plane rigid plate</span> </div> </a> <ul id="toc-Flow_due_to_an_oscillating_pressure_gradient_near_a_plane_rigid_plate-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Stokes_problem_in_cylindrical_geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stokes_problem_in_cylindrical_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Stokes problem in cylindrical geometry</span> </div> </a> <button aria-controls="toc-Stokes_problem_in_cylindrical_geometry-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Stokes problem in cylindrical geometry subsection</span> </button> <ul id="toc-Stokes_problem_in_cylindrical_geometry-sublist" class="vector-toc-list"> <li id="toc-Torsional_oscillation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Torsional_oscillation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Torsional oscillation</span> </div> </a> <ul id="toc-Torsional_oscillation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Axial_oscillation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Axial_oscillation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Axial oscillation</span> </div> </a> <ul id="toc-Axial_oscillation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Stokes–Couette_flow[11]" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stokes–Couette_flow[11]"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Stokes–Couette flow<sup><span>[</span>11<span>]</span></sup></span> </div> </a> <ul id="toc-Stokes–Couette_flow[11]-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Stokes problem</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="This article exist only in this language. Add the article for other languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-0" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">Add languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> <div class="after-portlet after-portlet-lang"><span class="uls-after-portlet-link"></span><span class="wb-langlinks-add wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q30715867#sitelinks-wikipedia" title="Add interlanguage links" class="wbc-editpage">Add links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Stokes_problem" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Stokes_problem" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Stokes_problem"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Stokes_problem&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Stokes_problem&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Stokes_problem"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Stokes_problem&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Stokes_problem&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Stokes_problem" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Stokes_problem" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Stokes_problem&oldid=1249345868" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Stokes_problem&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Stokes_problem&id=1249345868&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FStokes_problem"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FStokes_problem"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Stokes_problem&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Stokes_problem&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q30715867" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Stokes_boundary_layer.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/be/Stokes_boundary_layer.gif" decoding="async" width="233" height="327" class="mw-file-element" data-file-width="233" data-file-height="327" /></a><figcaption>Stokes problem in a viscous fluid due to the harmonic oscillation of a plane rigid plate (bottom black edge). Velocity (blue line) and particle excursion (red dots) as a function of the distance to the wall.</figcaption></figure> <p>In fluid dynamics, <b>Stokes problem</b> also known as <b>Stokes second problem</b> or sometimes referred to as <b>Stokes boundary layer</b> or <b>Oscillating boundary layer</b> is a problem of determining the flow created by an oscillating solid surface, named after <a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st Baronet">Sir George Stokes</a>. This is considered one of the simplest unsteady problems that has an exact solution for the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Turbulent" class="mw-redirect" title="Turbulent">turbulent</a> flow, this is still named a Stokes boundary layer, but now one has to rely on <a href="/wiki/Flow_measurement" title="Flow measurement">experiments</a>, <a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">numerical simulations</a> or <a href="/wiki/Approximation" title="Approximation">approximate methods</a> in order to obtain useful information on the flow. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Flow_description[3][4]"><span id="Flow_description.5B3.5D.5B4.5D"></span>Flow description<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=1" title="Edit section: Flow description[3][4]"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider an infinitely long plate which is oscillating with a velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cos \omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cos \omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf58e3d09ffe334eb30f97f12c5bbb7af383f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.953ex; height:2.176ex;" alt="{\displaystyle U\cos \omega t}"></span> in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> direction, which is located at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=0}"></span> in an infinite domain of fluid, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is the frequency of the oscillations. The incompressible <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> reduce to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial u}{\partial t}}=\nu {\frac {\partial ^{2}u}{\partial y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial u}{\partial t}}=\nu {\frac {\partial ^{2}u}{\partial y^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a88fb6264da28cd5bc263f656bd1bc121e3e3a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.378ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial u}{\partial t}}=\nu {\frac {\partial ^{2}u}{\partial y^{2}}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> is the <a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a>. The pressure gradient does not enter into the problem. The initial, <a href="/wiki/No-slip_condition" title="No-slip condition">no-slip condition</a> on the wall is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(0,t)=U\cos \omega t,\quad u(\infty ,t)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>,</mo> <mspace width="1em" /> <mi>u</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(0,t)=U\cos \omega t,\quad u(\infty ,t)=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0dfabfc487a3d72d1dcfe61d0f973487677b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.827ex; height:2.843ex;" alt="{\displaystyle u(0,t)=U\cos \omega t,\quad u(\infty ,t)=0,}"></span></dd></dl> <p>and the second boundary condition is due to the fact that the motion at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=0}"></span> is not felt at infinity. The flow is only due to the motion of the plate, there is no imposed pressure gradient. </p> <div class="mw-heading mw-heading3"><h3 id="Solution[5][6]"><span id="Solution.5B5.5D.5B6.5D"></span>Solution<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=2" title="Edit section: Solution[5][6]"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The initial condition is not required because of periodicity. Since both the equation and the boundary conditions are linear, the velocity can be written as the real part of some complex function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=U\Re \left[e^{i\omega t}f(y)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>U</mi> <mi mathvariant="normal">ℜ</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=U\Re \left[e^{i\omega t}f(y)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb2857823969b2e69b39aa0d3c3d4f544da6a550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.204ex; height:3.343ex;" alt="{\displaystyle u=U\Re \left[e^{i\omega t}f(y)\right]}"></span></dd></dl> <p>because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \omega t=\Re e^{i\omega t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>=</mo> <mi mathvariant="normal">ℜ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \omega t=\Re e^{i\omega t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10f8002585886388745733a66bcc77f9046c5902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.306ex; height:2.676ex;" alt="{\displaystyle \cos \omega t=\Re e^{i\omega t}}"></span>. </p><p>Substituting this into the partial differential equation reduces it to ordinary differential equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''-{\frac {i\omega }{\nu }}f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi>ω</mi> </mrow> <mi>ν</mi> </mfrac> </mrow> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''-{\frac {i\omega }{\nu }}f=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/009ddedfd71cfadd5d6c6483acf3341c99920006" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.922ex; height:5.176ex;" alt="{\displaystyle f''-{\frac {i\omega }{\nu }}f=0}"></span></dd></dl> <p>with boundary conditions </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(0)=1,\quad f(\infty )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=1,\quad f(\infty )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e395ddc37ef88c2ae1b13c36fa4208e2d22e72f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.54ex; height:2.843ex;" alt="{\displaystyle f(0)=1,\quad f(\infty )=0}"></span></dd></dl> <p>The solution to the above problem is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(y)=\exp \left[-{\frac {1+i}{\sqrt {2}}}{\sqrt {\frac {\omega }{\nu }}}y\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>ω</mi> <mi>ν</mi> </mfrac> </msqrt> </mrow> <mi>y</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(y)=\exp \left[-{\frac {1+i}{\sqrt {2}}}{\sqrt {\frac {\omega }{\nu }}}y\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f9a77d8ddd35344056f57db7802abab3ab7a8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.56ex; height:6.509ex;" alt="{\displaystyle f(y)=\exp \left[-{\frac {1+i}{\sqrt {2}}}{\sqrt {\frac {\omega }{\nu }}}y\right]}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(y,t)=Ue^{-{\sqrt {\frac {\omega }{2\nu }}}y}\cos \left(\omega t-{\sqrt {\frac {\omega }{2\nu }}}y\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>ν</mi> </mrow> </mfrac> </msqrt> </mrow> <mi>y</mi> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>ν</mi> </mrow> </mfrac> </msqrt> </mrow> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(y,t)=Ue^{-{\sqrt {\frac {\omega }{2\nu }}}y}\cos \left(\omega t-{\sqrt {\frac {\omega }{2\nu }}}y\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0123bd0ecd3dbd03e5551a27210e4564dd6458d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.07ex; height:6.343ex;" alt="{\displaystyle u(y,t)=Ue^{-{\sqrt {\frac {\omega }{2\nu }}}y}\cos \left(\omega t-{\sqrt {\frac {\omega }{2\nu }}}y\right)}"></span></dd></dl> <p>The disturbance created by the oscillating plate travels as the transverse wave through the fluid, but it is highly damped by the exponential factor. The depth of penetration <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ={\sqrt {2\nu /\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ω</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ={\sqrt {2\nu /\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6c91662ac9ecf9d9bbdefd48f2080157e815b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.474ex; height:4.843ex;" alt="{\displaystyle \delta ={\sqrt {2\nu /\omega }}}"></span> of this wave decreases with the frequency of the oscillation, but increases with the kinematic viscosity of the fluid. </p><p>The force per unit area exerted on the plate by the fluid is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\mu \left({\frac {\partial u}{\partial y}}\right)_{y=0}={\sqrt {\rho \omega \mu }}U\cos \left(\omega t-{\frac {\pi }{4}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>μ</mi> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ρ</mi> <mi>ω</mi> <mi>μ</mi> </msqrt> </mrow> <mi>U</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\mu \left({\frac {\partial u}{\partial y}}\right)_{y=0}={\sqrt {\rho \omega \mu }}U\cos \left(\omega t-{\frac {\pi }{4}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67ded3498adb5a2ca73d3c879abb5d3370a80523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:40.73ex; height:6.509ex;" alt="{\displaystyle F=\mu \left({\frac {\partial u}{\partial y}}\right)_{y=0}={\sqrt {\rho \omega \mu }}U\cos \left(\omega t-{\frac {\pi }{4}}\right)}"></span></dd></dl> <p>There is a phase shift between the oscillation of the plate and the force created. </p> <div class="mw-heading mw-heading3"><h3 id="Vorticity_oscillations_near_the_boundary">Vorticity oscillations near the boundary</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=3" title="Edit section: Vorticity oscillations near the boundary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An important observation from Stokes' solution for the oscillating Stokes flow is that <a href="/wiki/Vorticity" title="Vorticity">vorticity</a> oscillations are confined to a thin boundary layer and damp <a href="/wiki/Exponential_decay" title="Exponential decay">exponentially</a> when moving away from the wall.<sup id="cite_ref-Phil46_7-0" class="reference"><a href="#cite_note-Phil46-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This observation is also valid for the case of a turbulent boundary layer. Outside the Stokes boundary layer – which is often the bulk of the fluid volume – the vorticity oscillations may be neglected. To good approximation, the flow velocity oscillations are <a href="/wiki/Irrotational" class="mw-redirect" title="Irrotational">irrotational</a> outside the boundary layer, and <a href="/wiki/Potential_flow" title="Potential flow">potential flow</a> theory can be applied to the oscillatory part of the motion. This significantly simplifies the solution of these flow problems, and is often applied in the irrotational flow regions of <a href="/wiki/Sound_wave" class="mw-redirect" title="Sound wave">sound waves</a> and <a href="/wiki/Water_wave" class="mw-redirect" title="Water wave">water waves</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Fluid_bounded_by_an_upper_wall">Fluid bounded by an upper wall</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=4" title="Edit section: Fluid bounded by an upper wall"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the fluid domain is bounded by an upper, stationary wall, located at a height <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadff51da4a316d5ddd2165ebdce91dd563c187f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle y=h}"></span>, the flow velocity is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(y,t)={\frac {U}{2(\cosh 2\lambda h-\cos 2\lambda h)}}[e^{-\lambda (y-2h)}\cos(\omega t-\lambda y)+e^{\lambda (y-2h)}\cos(\omega t+\lambda y)-e^{-\lambda y}\cos(\omega t-\lambda y+2\lambda h)-e^{\lambda y}\cos(\omega t+\lambda y-2\lambda h)]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>U</mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>cosh</mi> <mo>⁡</mo> <mn>2</mn> <mi>λ</mi> <mi>h</mi> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>λ</mi> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <mi>λ</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>λ</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>λ</mi> <mi>y</mi> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <mi>λ</mi> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>λ</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>y</mi> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>λ</mi> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>λ</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(y,t)={\frac {U}{2(\cosh 2\lambda h-\cos 2\lambda h)}}[e^{-\lambda (y-2h)}\cos(\omega t-\lambda y)+e^{\lambda (y-2h)}\cos(\omega t+\lambda y)-e^{-\lambda y}\cos(\omega t-\lambda y+2\lambda h)-e^{\lambda y}\cos(\omega t+\lambda y-2\lambda h)]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e44cc1792c5c6d6c2169f1e826ed40076de12f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:130.477ex; height:6.009ex;" alt="{\displaystyle u(y,t)={\frac {U}{2(\cosh 2\lambda h-\cos 2\lambda h)}}[e^{-\lambda (y-2h)}\cos(\omega t-\lambda y)+e^{\lambda (y-2h)}\cos(\omega t+\lambda y)-e^{-\lambda y}\cos(\omega t-\lambda y+2\lambda h)-e^{\lambda y}\cos(\omega t+\lambda y-2\lambda h)]}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\sqrt {\omega /(2\nu )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>ν</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\sqrt {\omega /(2\nu )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6c01d1811b977fbc6bc26ab87608b045d13956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.59ex; height:4.843ex;" alt="{\displaystyle \lambda ={\sqrt {\omega /(2\nu )}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Fluid_bounded_by_a_free_surface">Fluid bounded by a free surface</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=5" title="Edit section: Fluid bounded by a free surface"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose the extent of the fluid domain be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<y<h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <mi>y</mi> <mo><</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<y<h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7acbb7ff96bb5b55eae6130407448a098a91b528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.854ex; height:2.509ex;" alt="{\displaystyle 0<y<h}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadff51da4a316d5ddd2165ebdce91dd563c187f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle y=h}"></span> representing a free surface. Then the solution as shown by <a href="/wiki/Chia-Shun_Yih" title="Chia-Shun Yih">Chia-Shun Yih</a> in 1968<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(y,t)={\frac {U\cos h/\delta \,\mathrm {cosh} \,h/\delta }{2(\cos ^{2}h/\delta +\mathrm {sinh} ^{2}h/\delta )}}\Re \left\{W+W^{*}-i\mathrm {tanh} \,h/\delta \,\tan h/\delta \,(W-W^{*})\right\},\qquad W=\mathrm {cosh} [(1+i)(h-y)/\delta ]e^{i\omega t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>U</mi> <mi>cos</mi> <mo>⁡</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>δ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">h</mi> </mrow> <mspace width="thinmathspace" /> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>δ</mi> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>δ</mi> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi mathvariant="normal">ℜ</mi> <mrow> <mo>{</mo> <mrow> <mi>W</mi> <mo>+</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">h</mi> </mrow> <mspace width="thinmathspace" /> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>δ</mi> <mspace width="thinmathspace" /> <mi>tan</mi> <mo>⁡</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>δ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>W</mi> <mo>−</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">h</mi> </mrow> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>δ</mi> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(y,t)={\frac {U\cos h/\delta \,\mathrm {cosh} \,h/\delta }{2(\cos ^{2}h/\delta +\mathrm {sinh} ^{2}h/\delta )}}\Re \left\{W+W^{*}-i\mathrm {tanh} \,h/\delta \,\tan h/\delta \,(W-W^{*})\right\},\qquad W=\mathrm {cosh} [(1+i)(h-y)/\delta ]e^{i\omega t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccaf389a956570e4c65e429bbb3a0ad46bacd3bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:114.347ex; height:6.843ex;" alt="{\displaystyle u(y,t)={\frac {U\cos h/\delta \,\mathrm {cosh} \,h/\delta }{2(\cos ^{2}h/\delta +\mathrm {sinh} ^{2}h/\delta )}}\Re \left\{W+W^{*}-i\mathrm {tanh} \,h/\delta \,\tan h/\delta \,(W-W^{*})\right\},\qquad W=\mathrm {cosh} [(1+i)(h-y)/\delta ]e^{i\omega t}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ={\sqrt {2\nu /\omega }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ω</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ={\sqrt {2\nu /\omega }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8d71437e9bb4687974a973e61317777512ee86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.121ex; height:4.843ex;" alt="{\displaystyle \delta ={\sqrt {2\nu /\omega }}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Flow_due_to_an_oscillating_pressure_gradient_near_a_plane_rigid_plate">Flow due to an oscillating pressure gradient near a plane rigid plate</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=6" title="Edit section: Flow due to an oscillating pressure gradient near a plane rigid plate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Stokes_boundary_layer_oscillating_flow.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/5c/Stokes_boundary_layer_oscillating_flow.gif" decoding="async" width="233" height="327" class="mw-file-element" data-file-width="233" data-file-height="327" /></a><figcaption>Stokes boundary layer due to the <a href="/wiki/Sinusoidal" class="mw-redirect" title="Sinusoidal">sinusoidal</a> oscillation of the far-field flow velocity. The horizontal velocity is the blue line, and the corresponding horizontal particle excursions are the red dots.</figcaption></figure> <p>The case for an oscillating <a href="/wiki/Far-field" class="mw-redirect" title="Far-field">far-field</a> flow, with the plate held at rest, can easily be constructed from the previous solution for an oscillating plate by using <a href="/wiki/Linear_superposition" class="mw-redirect" title="Linear superposition">linear superposition</a> of solutions. Consider a uniform velocity oscillation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(\infty ,t)=U_{\infty }\cos \omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(\infty ,t)=U_{\infty }\cos \omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3624db4c6959b8b241350ff4378383544d89b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.068ex; height:2.843ex;" alt="{\displaystyle u(\infty ,t)=U_{\infty }\cos \omega t}"></span> far away from the plate and a vanishing velocity at the plate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(0,t)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(0,t)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdec7eea77ad89b5a2beab682c733e80ef648c86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.436ex; height:2.843ex;" alt="{\displaystyle u(0,t)=0}"></span>. Unlike the stationary fluid in the original problem, the pressure gradient here at infinity must be a harmonic function of time. The solution is then given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(y,t)=U_{\infty }\left[\,\cos \omega t-{\text{e}}^{-{\sqrt {\frac {\omega }{2\nu }}}y}\,\cos \left(\omega t-{\sqrt {\frac {\omega }{2\nu }}}y\right)\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mrow> <mo>[</mo> <mrow> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>ν</mi> </mrow> </mfrac> </msqrt> </mrow> <mi>y</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>ω</mi> <mrow> <mn>2</mn> <mi>ν</mi> </mrow> </mfrac> </msqrt> </mrow> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(y,t)=U_{\infty }\left[\,\cos \omega t-{\text{e}}^{-{\sqrt {\frac {\omega }{2\nu }}}y}\,\cos \left(\omega t-{\sqrt {\frac {\omega }{2\nu }}}y\right)\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b383f8e5e9e17b34a35daec90d3c55693d85ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.973ex; height:6.343ex;" alt="{\displaystyle u(y,t)=U_{\infty }\left[\,\cos \omega t-{\text{e}}^{-{\sqrt {\frac {\omega }{2\nu }}}y}\,\cos \left(\omega t-{\sqrt {\frac {\omega }{2\nu }}}y\right)\right],}"></span></dd></dl> <p>which is zero at the wall <i>y = 0</i>, corresponding with the <a href="/wiki/No-slip_condition" title="No-slip condition">no-slip condition</a> for a wall at rest. This situation is often encountered in <a href="/wiki/Sound_waves" class="mw-redirect" title="Sound waves">sound waves</a> near a solid wall, or for the fluid motion near the sea bed in <a href="/wiki/Water_waves" class="mw-redirect" title="Water waves">water waves</a>. The vorticity, for the oscillating flow near a wall at rest, is equal to the vorticity in case of an oscillating plate but of opposite sign. </p> <div class="mw-heading mw-heading2"><h2 id="Stokes_problem_in_cylindrical_geometry">Stokes problem in cylindrical geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=7" title="Edit section: Stokes problem in cylindrical geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Torsional_oscillation">Torsional oscillation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=8" title="Edit section: Torsional oscillation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider an infinitely long cylinder of radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> exhibiting torsional oscillation with angular velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \cos \omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \cos \omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7eb2fa5ea090a88b5cf8e8c2c6df5faa97189ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.849ex; height:2.176ex;" alt="{\displaystyle \Omega \cos \omega t}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is the frequency. Then the velocity approaches after the initial transient phase to<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\theta }=a\Omega \ \Re \left[{\frac {K_{1}(r{\sqrt {i\omega /\nu }})}{K_{1}(a{\sqrt {i\omega /\nu }})}}e^{i\omega t}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mi mathvariant="normal">Ω</mi> <mtext> </mtext> <mi mathvariant="normal">ℜ</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>i</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ν</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>i</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ν</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\theta }=a\Omega \ \Re \left[{\frac {K_{1}(r{\sqrt {i\omega /\nu }})}{K_{1}(a{\sqrt {i\omega /\nu }})}}e^{i\omega t}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a127c1d08bcafa93beb50c24266bc83b6903a923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.108ex; height:7.509ex;" alt="{\displaystyle v_{\theta }=a\Omega \ \Re \left[{\frac {K_{1}(r{\sqrt {i\omega /\nu }})}{K_{1}(a{\sqrt {i\omega /\nu }})}}e^{i\omega t}\right]}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8520077dbcf03c2aabefd98d41a2269ed41a54fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{1}}"></span> is the modified Bessel function of the second kind. This solution can be expressed with real argument<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}v_{\theta }\left(r,t\right)&=\Psi \left\lbrace \left[{\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}r\right)+{\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}r\right)\right]\cos \left(t\right)\right.\\&+\left.\left[{\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}r\right)-{\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}r\right)\right]\sin \left(t\right)\right\rbrace \\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">Ψ</mi> <mrow> <mo>{</mo> <mrow> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>kei</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>kei</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ker</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ker</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mi>r</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>kei</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ker</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ker</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mo>)</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>kei</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mi>r</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v_{\theta }\left(r,t\right)&=\Psi \left\lbrace \left[{\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}r\right)+{\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}r\right)\right]\cos \left(t\right)\right.\\&+\left.\left[{\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}r\right)-{\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}r\right)\right]\sin \left(t\right)\right\rbrace \\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e14063663f03df2b3655d35c7c627840f687e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:76.259ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}v_{\theta }\left(r,t\right)&=\Psi \left\lbrace \left[{\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}r\right)+{\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}r\right)\right]\cos \left(t\right)\right.\\&+\left.\left[{\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}r\right)-{\textrm {ker}}_{1}\left({\sqrt {R_{\omega }}}\right){\textrm {kei}}_{1}\left({\sqrt {R_{\omega }}}r\right)\right]\sin \left(t\right)\right\rbrace \\\end{aligned}}}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi =\left[{\textrm {kei}}_{1}^{2}\left({\sqrt {R_{\omega }}}\right)+{\textrm {ker}}_{1}^{2}\left({\sqrt {R_{\omega }}}\right)\right]^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ</mi> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>kei</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>ker</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </msqrt> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi =\left[{\textrm {kei}}_{1}^{2}\left({\sqrt {R_{\omega }}}\right)+{\textrm {ker}}_{1}^{2}\left({\sqrt {R_{\omega }}}\right)\right]^{-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c63f12ee316c74e344c73199f3926cd4cabcb37a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:36.571ex; height:4.009ex;" alt="{\displaystyle \Psi =\left[{\textrm {kei}}_{1}^{2}\left({\sqrt {R_{\omega }}}\right)+{\textrm {ker}}_{1}^{2}\left({\sqrt {R_{\omega }}}\right)\right]^{-1},}"></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {kei} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">i</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {kei} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1109c476a460388e5cd3beb5b0f204c72467cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.907ex; height:2.176ex;" alt="{\displaystyle \mathrm {kei} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ker} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ker} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e112c3a6e281bb27fe64c7579aa9c2aafffb567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.171ex; height:2.176ex;" alt="{\displaystyle \mathrm {ker} }"></span> are <a href="/wiki/Kelvin_functions" title="Kelvin functions">Kelvin functions</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a16af1ecd1cedd66afc2dae8f72bfd78edb6bbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.019ex; height:2.509ex;" alt="{\displaystyle R_{\omega }}"></span> is to the dimensionless oscillatory Reynolds number defined as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\omega }=\omega a^{2}/\nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\omega }=\omega a^{2}/\nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3c5bc9e773ad233c760cc7b9385f5cec13cf924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.242ex; height:3.176ex;" alt="{\displaystyle R_{\omega }=\omega a^{2}/\nu }"></span>, being <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> the kinematic viscosity. </p> <div class="mw-heading mw-heading3"><h3 id="Axial_oscillation">Axial oscillation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=9" title="Edit section: Axial oscillation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the cylinder oscillates in the axial direction with velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cos \omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cos \omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf58e3d09ffe334eb30f97f12c5bbb7af383f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.953ex; height:2.176ex;" alt="{\displaystyle U\cos \omega t}"></span>, then the velocity field is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=U\ \Re \left[{\frac {K_{0}(r{\sqrt {i\omega /\nu }})}{K_{0}(a{\sqrt {i\omega /\nu }})}}e^{i\omega t}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>U</mi> <mtext> </mtext> <mi mathvariant="normal">ℜ</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>i</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ν</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>i</mi> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ν</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ω</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=U\ \Re \left[{\frac {K_{0}(r{\sqrt {i\omega /\nu }})}{K_{0}(a{\sqrt {i\omega /\nu }})}}e^{i\omega t}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/275fc087cd44651ace11fe1f8428f6fc6318c7d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.182ex; height:7.509ex;" alt="{\displaystyle u=U\ \Re \left[{\frac {K_{0}(r{\sqrt {i\omega /\nu }})}{K_{0}(a{\sqrt {i\omega /\nu }})}}e^{i\omega t}\right]}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b0af6cafb690d3dbb0f3f30a032631338dc476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="{\displaystyle K_{0}}"></span> is the modified Bessel function of the second kind. </p> <div class="mw-heading mw-heading2"><h2 id="Stokes–Couette_flow[11]"><span id="Stokes.E2.80.93Couette_flow.5B11.5D"></span>Stokes–Couette flow<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=10" title="Edit section: Stokes–Couette flow[11]"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the <a href="/wiki/Couette_flow" title="Couette flow">Couette flow</a>, instead of the translational motion of one of the plate, an oscillation of one plane will be executed. If we have a bottom wall at rest at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=0}"></span> and the upper wall at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aadff51da4a316d5ddd2165ebdce91dd563c187f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle y=h}"></span> is executing an oscillatory motion with velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cos \omega t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mi>cos</mi> <mo>⁡</mo> <mi>ω</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cos \omega t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bf58e3d09ffe334eb30f97f12c5bbb7af383f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.953ex; height:2.176ex;" alt="{\displaystyle U\cos \omega t}"></span>, then the velocity field is given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=U\ \Re \left\{{\frac {\sin ky}{\sin kh}}\right\},\quad {\text{where}}\quad k={\frac {1+i}{\sqrt {2}}}{\sqrt {\frac {\omega }{\nu }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mi>U</mi> <mtext> </mtext> <mi mathvariant="normal">ℜ</mi> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>k</mi> <mi>y</mi> </mrow> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>k</mi> <mi>h</mi> </mrow> </mfrac> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>where</mtext> </mrow> <mspace width="1em" /> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>ω</mi> <mi>ν</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=U\ \Re \left\{{\frac {\sin ky}{\sin kh}}\right\},\quad {\text{where}}\quad k={\frac {1+i}{\sqrt {2}}}{\sqrt {\frac {\omega }{\nu }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/783bee8cc5c511a14d0bf0f580676323e069275d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:46.435ex; height:6.509ex;" alt="{\displaystyle u=U\ \Re \left\{{\frac {\sin ky}{\sin kh}}\right\},\quad {\text{where}}\quad k={\frac {1+i}{\sqrt {2}}}{\sqrt {\frac {\omega }{\nu }}}.}"></span></dd></dl> <p>The frictional force per unit area on the moving plane is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\mu U\Re \{k\cot kh\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>μ</mi> <mi>U</mi> <mi mathvariant="normal">ℜ</mi> <mo fence="false" stretchy="false">{</mo> <mi>k</mi> <mi>cot</mi> <mo>⁡</mo> <mi>k</mi> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mu U\Re \{k\cot kh\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35977ab022242146470440cbb9540e420cf3b5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.877ex; height:2.843ex;" alt="{\displaystyle -\mu U\Re \{k\cot kh\}}"></span> and on the fixed plane is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu U\Re \{k\csc kh\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mi>U</mi> <mi mathvariant="normal">ℜ</mi> <mo fence="false" stretchy="false">{</mo> <mi>k</mi> <mi>csc</mi> <mo>⁡</mo> <mi>k</mi> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu U\Re \{k\csc kh\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcc0759e49d36848f874b35422fc14bf3373381d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.95ex; height:2.843ex;" alt="{\displaystyle \mu U\Re \{k\csc kh\}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Rayleigh_problem" title="Rayleigh problem">Rayleigh problem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Stokes_problem&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWang1991" class="citation journal cs1">Wang, C. Y. (1991). "Exact solutions of the steady-state Navier-Stokes equations". <i>Annual Review of Fluid Mechanics</i>. <b>23</b>: 159–177. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991AnRFM..23..159W">1991AnRFM..23..159W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.fl.23.010191.001111">10.1146/annurev.fl.23.010191.001111</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Fluid+Mechanics&rft.atitle=Exact+solutions+of+the+steady-state+Navier-Stokes+equations&rft.volume=23&rft.pages=159-177&rft.date=1991&rft_id=info%3Adoi%2F10.1146%2Fannurev.fl.23.010191.001111&rft_id=info%3Abibcode%2F1991AnRFM..23..159W&rft.aulast=Wang&rft.aufirst=C.+Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+problem" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Landau & Lifshitz (1987), pp. 83–85.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Lagerstrom, Paco Axel. Laminar flow theory. Princeton University Press, 1996.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Acheson, David J. Elementary fluid dynamics. Oxford University Press, 1990.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. "Fluid mechanics." (1987).</span> </li> <li id="cite_note-Phil46-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Phil46_7-0">^</a></b></span> <span class="reference-text">Phillips (1977), p. 46.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Yih, C. S. (1968). Instability of unsteady flows or configurations Part 1. Instability of a horizontal liquid layer on an oscillating plane. Journal of Fluid Mechanics, 31(4), 737-751.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="/wiki/Philip_Drazin" title="Philip Drazin">Drazin, Philip G.</a>, and <a href="/wiki/Norman_Riley_(professor)" title="Norman Riley (professor)">Norman Riley</a>. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiveroGarzónNúñezFigueroa2019" class="citation journal cs1">Rivero, M.; Garzón, F.; Núñez, J.; Figueroa, A. (2019). "Study of the flow induced by circular cylinder performing torsional oscillation". <i>European Journal of Mechanics - B/Fluids</i>. <b>78</b>: 245–251. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.euromechflu.2019.08.002">10.1016/j.euromechflu.2019.08.002</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:201253195">201253195</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Mechanics+-+B%2FFluids&rft.atitle=Study+of+the+flow+induced+by+circular+cylinder+performing+torsional+oscillation&rft.volume=78&rft.pages=245-251&rft.date=2019&rft_id=info%3Adoi%2F10.1016%2Fj.euromechflu.2019.08.002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A201253195%23id-name%3DS2CID&rft.aulast=Rivero&rft.aufirst=M.&rft.au=Garz%C3%B3n%2C+F.&rft.au=N%C3%BA%C3%B1ez%2C+J.&rft.au=Figueroa%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+problem" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Landau, L. D., & Sykes, J. B. (1987). Fluid Mechanics: Vol 6. pp. 88</span> </li> </ol></div><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Physical_oceanography" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Physical_oceanography" title="Template:Physical oceanography"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Physical_oceanography" title="Template talk:Physical oceanography"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Physical_oceanography" title="Special:EditPage/Template:Physical oceanography"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Physical_oceanography" style="font-size:114%;margin:0 4em"><a href="/wiki/Physical_oceanography" title="Physical oceanography">Physical oceanography</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Wind_wave" title="Wind wave">Waves</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Airy_wave_theory" title="Airy wave theory">Airy wave theory</a></li> <li><a href="/wiki/Ballantine_scale" title="Ballantine scale">Ballantine scale</a></li> <li><a href="/wiki/Modulational_instability" title="Modulational instability">Benjamin–Feir instability</a></li> <li><a href="/wiki/Boussinesq_approximation_(water_waves)" title="Boussinesq approximation (water waves)">Boussinesq approximation</a></li> <li><a href="/wiki/Breaking_wave" title="Breaking wave">Breaking wave</a></li> <li><a href="/wiki/Clapotis" title="Clapotis">Clapotis</a></li> <li><a href="/wiki/Cnoidal_wave" title="Cnoidal wave">Cnoidal wave</a></li> <li><a href="/wiki/Cross_sea" title="Cross sea">Cross sea</a></li> <li><a href="/wiki/Dispersion_(water_waves)" title="Dispersion (water waves)">Dispersion</a></li> <li><a href="/wiki/Edge_wave" title="Edge wave">Edge wave</a></li> <li><a href="/wiki/Equatorial_wave" title="Equatorial wave">Equatorial waves</a></li> <li><a href="/wiki/Gravity_wave" title="Gravity wave">Gravity wave</a></li> <li><a href="/wiki/Green%27s_law" title="Green's law">Green's law</a></li> <li><a href="/wiki/Infragravity_wave" title="Infragravity wave">Infragravity wave</a></li> <li><a href="/wiki/Internal_wave" title="Internal wave">Internal wave</a></li> <li><a href="/wiki/Iribarren_number" title="Iribarren number">Iribarren number</a></li> <li><a href="/wiki/Kelvin_wave" title="Kelvin wave">Kelvin wave</a></li> <li><a href="/wiki/Kinematic_wave" title="Kinematic wave">Kinematic wave</a></li> <li><a href="/wiki/Longshore_drift" title="Longshore drift">Longshore drift</a></li> <li><a href="/wiki/Luke%27s_variational_principle" title="Luke's variational principle">Luke's variational principle</a></li> <li><a href="/wiki/Mild-slope_equation" title="Mild-slope equation">Mild-slope equation</a></li> <li><a href="/wiki/Radiation_stress" title="Radiation stress">Radiation stress</a></li> <li><a href="/wiki/Rogue_wave" title="Rogue wave">Rogue wave</a></li> <li><a href="/wiki/Rossby_wave" title="Rossby wave">Rossby wave</a></li> <li><a href="/wiki/Rossby-gravity_waves" title="Rossby-gravity waves">Rossby-gravity waves</a></li> <li><a href="/wiki/Sea_state" title="Sea state">Sea state</a></li> <li><a href="/wiki/Seiche" title="Seiche">Seiche</a></li> <li><a href="/wiki/Significant_wave_height" title="Significant wave height">Significant wave height</a></li> <li><a href="/wiki/Soliton" title="Soliton">Soliton</a></li> <li><a href="/wiki/Stokes_drift" title="Stokes drift">Stokes drift</a></li> <li><a class="mw-selflink selflink">Stokes problem</a></li> <li><a href="/wiki/Stokes_wave" title="Stokes wave">Stokes wave</a></li> <li><a href="/wiki/Swell_(ocean)" title="Swell (ocean)">Swell</a></li> <li><a href="/wiki/Trochoidal_wave" title="Trochoidal wave">Trochoidal wave</a></li> <li><a href="/wiki/Tsunami" title="Tsunami">Tsunami</a> <ul><li><a href="/wiki/Megatsunami" title="Megatsunami">megatsunami</a></li></ul></li> <li><a href="/wiki/Undertow_(water_waves)" title="Undertow (water waves)">Undertow</a></li> <li><a href="/wiki/Ursell_number" title="Ursell number">Ursell number</a></li> <li><a href="/wiki/Wave_action_(continuum_mechanics)" title="Wave action (continuum mechanics)">Wave action</a></li> <li><a href="/wiki/Wave_base" title="Wave base">Wave base</a></li> <li><a href="/wiki/Wave_height" title="Wave height">Wave height</a></li> <li><a href="/wiki/Wave_nonlinearity" title="Wave nonlinearity">Wave nonlinearity</a></li> <li><a href="/wiki/Wave_power" title="Wave power">Wave power</a></li> <li><a href="/wiki/Wave_radar" title="Wave radar">Wave radar</a></li> <li><a href="/wiki/Wave_setup" title="Wave setup">Wave setup</a></li> <li><a href="/wiki/Wave_shoaling" title="Wave shoaling">Wave shoaling</a></li> <li><a href="/wiki/Wave_turbulence" title="Wave turbulence">Wave turbulence</a></li> <li><a href="/wiki/Wave%E2%80%93current_interaction" title="Wave–current interaction">Wave–current interaction</a></li> <li><a href="/wiki/Waves_and_shallow_water" title="Waves and shallow water">Waves and shallow water</a> <ul><li><a href="/wiki/One-dimensional_Saint-Venant_equations" class="mw-redirect" title="One-dimensional Saint-Venant equations">one-dimensional Saint-Venant equations</a></li> <li><a href="/wiki/Shallow_water_equations" title="Shallow water equations">shallow water equations</a></li></ul></li> <li><a href="/wiki/Wind_fetch" title="Wind fetch">Wind fetch</a></li> <li><a href="/wiki/Wind_setup" title="Wind setup">Wind setup</a></li> <li><a href="/wiki/Wind_wave" title="Wind wave">Wind wave</a> <ul><li><a href="/wiki/Wind_wave_model" title="Wind wave model">model</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="10" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Upwelling.svg" class="mw-file-description" title="Upwelling"><img alt="Upwelling" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/120px-Upwelling.svg.png" decoding="async" width="120" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/180px-Upwelling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/240px-Upwelling.svg.png 2x" data-file-width="365" data-file-height="242" /></a></span><br /><br /><br /><br /><br /><br /><span typeof="mw:File"><a href="/wiki/File:Antarctic_bottom_water.svg" class="mw-file-description" title="Antarctic bottom water"><img alt="Antarctic bottom water" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Antarctic_bottom_water.svg/120px-Antarctic_bottom_water.svg.png" decoding="async" width="120" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Antarctic_bottom_water.svg/180px-Antarctic_bottom_water.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Antarctic_bottom_water.svg/240px-Antarctic_bottom_water.svg.png 2x" data-file-width="745" data-file-height="470" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ocean_current" title="Ocean current">Circulation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atmospheric_circulation" title="Atmospheric circulation">Atmospheric circulation</a></li> <li><a href="/wiki/Baroclinity" title="Baroclinity">Baroclinity</a></li> <li><a href="/wiki/Boundary_current" title="Boundary current">Boundary current</a></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Coriolis%E2%80%93Stokes_force" title="Coriolis–Stokes force">Coriolis–Stokes force</a></li> <li><a href="/wiki/Craik%E2%80%93Leibovich_vortex_force" title="Craik–Leibovich vortex force">Craik–Leibovich vortex force</a></li> <li><a href="/wiki/Downwelling" title="Downwelling">Downwelling</a></li> <li><a href="/wiki/Eddy_(fluid_dynamics)" title="Eddy (fluid dynamics)">Eddy</a></li> <li><a href="/wiki/Ekman_layer" title="Ekman layer">Ekman layer</a></li> <li><a href="/wiki/Ekman_spiral" title="Ekman spiral">Ekman spiral</a></li> <li><a href="/wiki/Ekman_transport" title="Ekman transport">Ekman transport</a></li> <li><a href="/wiki/El_Ni%C3%B1o%E2%80%93Southern_Oscillation" title="El Niño–Southern Oscillation">El Niño–Southern Oscillation</a></li> <li><a href="/wiki/General_circulation_model" title="General circulation model">General circulation model</a></li> <li><a href="/wiki/Geochemical_Ocean_Sections_Study" title="Geochemical Ocean Sections Study">Geochemical Ocean Sections Study</a></li> <li><a href="/wiki/Geostrophic_current" title="Geostrophic current">Geostrophic current</a></li> <li><a href="/wiki/Global_Ocean_Data_Analysis_Project" title="Global Ocean Data Analysis Project">Global Ocean Data Analysis Project</a></li> <li><a href="/wiki/Gulf_Stream" title="Gulf Stream">Gulf Stream</a></li> <li><a href="/wiki/Humboldt_Current" title="Humboldt Current">Humboldt Current</a></li> <li><a href="/wiki/Hydrothermal_circulation" title="Hydrothermal circulation">Hydrothermal circulation</a></li> <li><a href="/wiki/Langmuir_circulation" title="Langmuir circulation">Langmuir circulation</a></li> <li><a href="/wiki/Longshore_drift" title="Longshore drift">Longshore drift</a></li> <li><a href="/wiki/Loop_Current" title="Loop Current">Loop Current</a></li> <li><a href="/wiki/Modular_Ocean_Model" title="Modular Ocean Model">Modular Ocean Model</a></li> <li><a href="/wiki/Ocean_current" title="Ocean current">Ocean current</a></li> <li><a href="/wiki/Ocean_dynamical_thermostat" title="Ocean dynamical thermostat">Ocean dynamical thermostat</a></li> <li><a href="/wiki/Ocean_dynamics" title="Ocean dynamics">Ocean dynamics</a></li> <li><a href="/wiki/Ocean_gyre" title="Ocean gyre">Ocean gyre</a></li> <li><a href="/wiki/Overflow_(oceanography)" title="Overflow (oceanography)">Overflow</a></li> <li><a href="/wiki/Princeton_Ocean_Model" title="Princeton Ocean Model">Princeton Ocean Model</a></li> <li><a href="/wiki/Rip_current" title="Rip current">Rip current</a></li> <li><a href="/wiki/Subsurface_ocean_current" title="Subsurface ocean current">Subsurface ocean current</a></li> <li><a href="/wiki/Sverdrup_balance" title="Sverdrup balance">Sverdrup balance</a></li> <li><a href="/wiki/Thermohaline_circulation" title="Thermohaline circulation">Thermohaline circulation</a> <ul><li><a href="/wiki/Shutdown_of_thermohaline_circulation" class="mw-redirect" title="Shutdown of thermohaline circulation">shutdown</a></li></ul></li> <li><a href="/wiki/Upwelling" title="Upwelling">Upwelling</a></li> <li><a href="/wiki/Whirlpool" title="Whirlpool">Whirlpool</a></li> <li><a href="/wiki/Wind_generated_current" title="Wind generated current">Wind generated current</a></li> <li><a href="/wiki/World_Ocean_Circulation_Experiment" title="World Ocean Circulation Experiment">World Ocean Circulation Experiment</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tide" title="Tide">Tides</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Amphidromic_point" title="Amphidromic point">Amphidromic point</a></li> <li><a href="/wiki/Earth_tide" title="Earth tide">Earth tide</a></li> <li><a href="/wiki/Head_of_tide" title="Head of tide">Head of tide</a></li> <li><a href="/wiki/Internal_tide" title="Internal tide">Internal tide</a></li> <li><a href="/wiki/Lunitidal_interval" title="Lunitidal interval">Lunitidal interval</a></li> <li><a href="/wiki/Perigean_spring_tide" title="Perigean spring tide">Perigean spring tide</a></li> <li><a href="/wiki/Rip_tide" title="Rip tide">Rip tide</a></li> <li><a href="/wiki/Rule_of_twelfths" title="Rule of twelfths">Rule of twelfths</a></li> <li><a href="/wiki/Slack_tide" title="Slack tide">Slack tide</a></li> <li><a href="/wiki/Theory_of_tides" title="Theory of tides">Theory of tides</a></li> <li><a href="/wiki/Tidal_bore" title="Tidal bore">Tidal bore</a></li> <li><a href="/wiki/Tidal_force" title="Tidal force">Tidal force</a></li> <li><a href="/wiki/Tidal_power" title="Tidal power">Tidal power</a></li> <li><a href="/wiki/Tidal_race" title="Tidal race">Tidal race</a></li> <li><a href="/wiki/Tidal_range" title="Tidal range">Tidal range</a></li> <li><a href="/wiki/Tidal_resonance" title="Tidal resonance">Tidal resonance</a></li> <li><a href="/wiki/Tide_gauge" title="Tide gauge">Tide gauge</a></li> <li><a href="/wiki/Tideline" title="Tideline">Tideline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Landform" title="Landform">Landforms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abyssal_fan" title="Abyssal fan">Abyssal fan</a></li> <li><a href="/wiki/Abyssal_plain" title="Abyssal plain">Abyssal plain</a></li> <li><a href="/wiki/Atoll" title="Atoll">Atoll</a></li> <li><a href="/wiki/Bathymetric_chart" title="Bathymetric chart">Bathymetric chart</a></li> <li><a href="/wiki/Carbonate_platform" title="Carbonate platform">Carbonate platform</a></li> <li><a href="/wiki/Coastal_geography" title="Coastal geography">Coastal geography</a></li> <li><a href="/wiki/Cold_seep" title="Cold seep">Cold seep</a></li> <li><a href="/wiki/Continental_margin" title="Continental margin">Continental margin</a></li> <li><a href="/wiki/Continental_rise" title="Continental rise">Continental rise</a></li> <li><a href="/wiki/Continental_shelf" title="Continental shelf">Continental shelf</a></li> <li><a href="/wiki/Contourite" title="Contourite">Contourite</a></li> <li><a href="/wiki/Guyot" title="Guyot">Guyot</a></li> <li><a href="/wiki/Hydrography" title="Hydrography">Hydrography</a></li> <li><a href="/wiki/Knoll_(oceanography)" title="Knoll (oceanography)">Knoll</a></li> <li><a href="/wiki/Ocean_bank" title="Ocean bank">Ocean bank</a></li> <li><a href="/wiki/Oceanic_basin" title="Oceanic basin">Oceanic basin</a></li> <li><a href="/wiki/Oceanic_plateau" title="Oceanic plateau">Oceanic plateau</a></li> <li><a href="/wiki/Oceanic_trench" title="Oceanic trench">Oceanic trench</a></li> <li><a href="/wiki/Passive_margin" title="Passive margin">Passive margin</a></li> <li><a href="/wiki/Seabed" title="Seabed">Seabed</a></li> <li><a href="/wiki/Seamount" title="Seamount">Seamount</a></li> <li><a href="/wiki/Submarine_canyon" title="Submarine canyon">Submarine canyon</a></li> <li><a href="/wiki/Submarine_volcano" title="Submarine volcano">Submarine volcano</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plate_tectonics" title="Plate tectonics">Plate<br />tectonics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convergent_boundary" title="Convergent boundary">Convergent boundary</a></li> <li><a href="/wiki/Divergent_boundary" title="Divergent boundary">Divergent boundary</a></li> <li><a href="/wiki/Fracture_zone" title="Fracture zone">Fracture zone</a></li> <li><a href="/wiki/Hydrothermal_vent" title="Hydrothermal vent">Hydrothermal vent</a></li> <li><a href="/wiki/Marine_geology" title="Marine geology">Marine geology</a></li> <li><a href="/wiki/Mid-ocean_ridge" title="Mid-ocean ridge">Mid-ocean ridge</a></li> <li><a href="/wiki/Mohorovi%C4%8Di%C4%87_discontinuity" title="Mohorovičić discontinuity">Mohorovičić discontinuity</a></li> <li><a href="/wiki/Oceanic_crust" title="Oceanic crust">Oceanic crust</a></li> <li><a href="/wiki/Outer_trench_swell" title="Outer trench swell">Outer trench swell</a></li> <li><a href="/wiki/Ridge_push" title="Ridge push">Ridge push</a></li> <li><a href="/wiki/Seafloor_spreading" title="Seafloor spreading">Seafloor spreading</a></li> <li><a href="/wiki/Slab_pull" title="Slab pull">Slab pull</a></li> <li><a href="/wiki/Slab_suction" title="Slab suction">Slab suction</a></li> <li><a href="/wiki/Slab_window" title="Slab window">Slab window</a></li> <li><a href="/wiki/Subduction" title="Subduction">Subduction</a></li> <li><a href="/wiki/Transform_fault" title="Transform fault">Transform fault</a></li> <li><a href="/wiki/Vine%E2%80%93Matthews%E2%80%93Morley_hypothesis" title="Vine–Matthews–Morley hypothesis">Vine–Matthews–Morley hypothesis</a></li> <li><a href="/wiki/Volcanic_arc" title="Volcanic arc">Volcanic arc</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ocean zones</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benthic_zone" title="Benthic zone">Benthic</a></li> <li><a href="/wiki/Deep_ocean_water" title="Deep ocean water">Deep ocean water</a></li> <li><a href="/wiki/Deep_sea" title="Deep sea">Deep sea</a></li> <li><a href="/wiki/Littoral_zone" title="Littoral zone">Littoral</a></li> <li><a href="/wiki/Mesopelagic_zone" title="Mesopelagic zone">Mesopelagic</a></li> <li><a href="/wiki/Oceanic_zone" title="Oceanic zone">Oceanic</a></li> <li><a href="/wiki/Pelagic_zone" title="Pelagic zone">Pelagic</a></li> <li><a href="/wiki/Photic_zone" title="Photic zone">Photic</a></li> <li><a href="/wiki/Surf_zone" title="Surf zone">Surf</a></li> <li><a href="/wiki/Swash" title="Swash">Swash</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Sea_level" title="Sea level">Sea level</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Deep-ocean_Assessment_and_Reporting_of_Tsunamis" title="Deep-ocean Assessment and Reporting of Tsunamis">Deep-ocean Assessment and Reporting of Tsunamis</a></li> <li><a href="/wiki/Global_Sea_Level_Observing_System" title="Global Sea Level Observing System">Global Sea Level Observing System</a></li> <li><a href="/wiki/North_West_Shelf_Operational_Oceanographic_System" title="North West Shelf Operational Oceanographic System">North West Shelf Operational Oceanographic System</a></li> <li><a href="/wiki/Sea-level_curve" title="Sea-level curve">Sea-level curve</a></li> <li><a href="/wiki/Sea_level_drop" title="Sea level drop">Sea level drop</a></li> <li><a href="/wiki/Sea_level_rise" title="Sea level rise">Sea level rise</a></li> <li><a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Acoustical_oceanography" class="mw-redirect" title="Acoustical oceanography">Acoustics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Deep_scattering_layer" title="Deep scattering layer">Deep scattering layer</a></li> <li><a href="/wiki/Ocean_acoustic_tomography" title="Ocean acoustic tomography">Ocean acoustic tomography</a></li> <li><a href="/wiki/Sofar_bomb" title="Sofar bomb">Sofar bomb</a></li> <li><a href="/wiki/SOFAR_channel" title="SOFAR channel">SOFAR channel</a></li> <li><a href="/wiki/Underwater_acoustics" title="Underwater acoustics">Underwater acoustics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satellites</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jason-1" title="Jason-1">Jason-1</a></li> <li><a href="/wiki/OSTM/Jason-2" title="OSTM/Jason-2">OSTM/Jason-2</a></li> <li><a href="/wiki/Jason-3" title="Jason-3">Jason-3</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ocean_acidification" title="Ocean acidification">Acidification</a></li> <li><a href="/wiki/Argo_(oceanography)" title="Argo (oceanography)">Argo</a></li> <li><a href="/wiki/Benthic_lander" title="Benthic lander">Benthic lander</a></li> <li><a href="/wiki/Color_of_water" title="Color of water">Color of water</a></li> <li><a href="/wiki/DSV_Alvin" title="DSV Alvin">DSV <i>Alvin</i></a></li> <li><a href="/wiki/Marginal_sea" class="mw-redirect" title="Marginal sea">Marginal sea</a></li> <li><a href="/wiki/Marine_energy" title="Marine energy">Marine energy</a></li> <li><a href="/wiki/Marine_pollution" title="Marine pollution">Marine pollution</a></li> <li><a href="/wiki/Mooring_(oceanography)" title="Mooring (oceanography)">Mooring</a></li> <li><a href="/wiki/National_Oceanographic_Data_Center" title="National Oceanographic Data Center">National Oceanographic Data Center</a></li> <li><a href="/wiki/Ocean" title="Ocean">Ocean</a></li> <li><a href="/wiki/Ocean_exploration" title="Ocean exploration">Explorations</a></li> <li><a href="/wiki/Ocean_observations" title="Ocean observations">Observations</a></li> <li><a href="/wiki/Ocean_reanalysis" title="Ocean reanalysis">Reanalysis</a></li> <li><a href="/wiki/Ocean_surface_topography" title="Ocean surface topography">Ocean surface topography</a></li> <li><a href="/wiki/Ocean_temperature" title="Ocean temperature">Ocean temperature</a></li> <li><a href="/wiki/Ocean_thermal_energy_conversion" title="Ocean thermal energy conversion">Ocean thermal energy conversion</a></li> <li><a href="/wiki/Oceanography" title="Oceanography">Oceanography</a> <ul><li><a href="/wiki/Outline_of_oceanography" title="Outline of oceanography">Outline of oceanography</a></li></ul></li> <li><a href="/wiki/Pelagic_sediment" title="Pelagic sediment">Pelagic sediment</a></li> <li><a href="/wiki/Sea_surface_microlayer" title="Sea surface microlayer">Sea surface microlayer</a></li> <li><a href="/wiki/Sea_surface_temperature" title="Sea surface temperature">Sea surface temperature</a></li> <li><a href="/wiki/Seawater" title="Seawater">Seawater</a></li> <li><a href="/wiki/Science_On_a_Sphere" title="Science On a Sphere">Science On a Sphere</a></li> <li><a href="/wiki/Ocean_stratification" title="Ocean stratification">Stratification</a></li> <li><a href="/wiki/Thermocline" title="Thermocline">Thermocline</a></li> <li><a href="/wiki/Underwater_glider" title="Underwater glider">Underwater glider</a></li> <li><a href="/wiki/Water_column" title="Water column">Water column</a></li> <li><a href="/wiki/World_Ocean_Atlas" title="World Ocean Atlas">World Ocean Atlas</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Physical_oceanography" title="Category:Physical oceanography">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a href="https://commons.wikimedia.org/wiki/Category:Physical_oceanography" class="extiw" title="commons:Category:Physical oceanography">Commons</a></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Waves_in_pacifica_1.jpg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Waves_in_pacifica_1.jpg/16px-Waves_in_pacifica_1.jpg" decoding="async" width="16" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Waves_in_pacifica_1.jpg/24px-Waves_in_pacifica_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Waves_in_pacifica_1.jpg/32px-Waves_in_pacifica_1.jpg 2x" data-file-width="2000" data-file-height="1358" /></a></span> </span><a href="/wiki/Portal:Oceans" title="Portal:Oceans">Oceans portal</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Stokes_problem&oldid=1249345868">https://en.wikipedia.org/w/index.php?title=Stokes_problem&oldid=1249345868</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Fluid_dynamics" title="Category:Fluid dynamics">Fluid dynamics</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 4 October 2024, at 12:59<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Stokes_problem&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-nt2gf","wgBackendResponseTime":150,"wgPageParseReport":{"limitreport":{"cputime":"0.241","walltime":"0.402","ppvisitednodes":{"value":635,"limit":1000000},"postexpandincludesize":{"value":31463,"limit":2097152},"templateargumentsize":{"value":0,"limit":2097152},"expansiondepth":{"value":6,"limit":100},"expensivefunctioncount":{"value":2,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":21594,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 195.491 1 -total"," 53.26% 104.111 1 Template:Physical_oceanography"," 51.98% 101.623 1 Template:Navbox"," 45.09% 88.141 2 Template:Cite_journal"," 6.58% 12.871 1 Template:Portal-inline"," 6.36% 12.426 2 Template:Icon"]},"scribunto":{"limitreport-timeusage":{"value":"0.126","limit":"10.000"},"limitreport-memusage":{"value":3148746,"limit":52428800},"limitreport-logs":"table#1 {\n [\"size\"] = \"tiny\",\n}\n"},"cachereport":{"origin":"mw-web.eqiad.main-958868bfc-vtczg","timestamp":"20241104180958","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Stokes problem","url":"https:\/\/en.wikipedia.org\/wiki\/Stokes_problem","sameAs":"http:\/\/www.wikidata.org\/entity\/Q30715867","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q30715867","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2017-05-03T06:09:38Z","dateModified":"2024-10-04T12:59:28Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/be\/Stokes_boundary_layer.gif"}</script> </body> </html>

CINXE.COM

Stokes problem - Wikipedia