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class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>De vergelijkingen</span> </div> </a> <ul id="toc-De_vergelijkingen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuïteitsvergelijking" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Continuïteitsvergelijking"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Continuïteitsvergelijking</span> </div> </a> <ul id="toc-Continuïteitsvergelijking-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vereenvoudigingen" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vereenvoudigingen"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Vereenvoudigingen</span> </div> </a> <ul id="toc-Vereenvoudigingen-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Toepassingen" class="vector-toc-list-item vector-toc-level-1 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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">Navier-Stokesvergelijkingen</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="Ga naar een artikel in een andere taal. Beschikbaar in 45 talen" > <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-45" 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">45 talen</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D9%86%D8%A7%D9%81%D9%8A%D9%8A%D9%87-%D8%B3%D8%AA%D9%88%D9%83%D8%B3" title="معادلات نافييه-ستوكس – Arabisch" lang="ar" hreflang="ar" data-title="معادلات نافييه-ستوكس" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D9%86%D8%A7%DA%A4%D9%8A%D9%8A_%D8%B3%D8%AA%D9%88%D9%83" title="معادلات ناڤيي ستوك – Marokkaans Arabisch" lang="ary" hreflang="ary" data-title="معادلات ناڤيي ستوك" data-language-autonym="الدارجة" data-language-local-name="Marokkaans Arabisch" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Ecuaciones_de_Navier-Stokes" title="Ecuaciones de Navier-Stokes – Asturisch" lang="ast" hreflang="ast" data-title="Ecuaciones de Navier-Stokes" data-language-autonym="Asturianu" data-language-local-name="Asturisch" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%9D%D0%B0%D0%B2%D0%B8%D0%B5-%D0%A1%D1%82%D0%BE%D0%BA%D1%81" title="Уравнения на Навие-Стокс – Bulgaars" lang="bg" hreflang="bg" data-title="Уравнения на Навие-Стокс" data-language-autonym="Български" data-language-local-name="Bulgaars" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Navier%E2%80%93Stokesove_jedna%C4%8Dine" title="Navier–Stokesove jednačine – Bosnisch" lang="bs" hreflang="bs" data-title="Navier–Stokesove jednačine" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equacions_de_Navier-Stokes" title="Equacions de Navier-Stokes – Catalaans" lang="ca" hreflang="ca" data-title="Equacions de Navier-Stokes" data-language-autonym="Català" data-language-local-name="Catalaans" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Navierovy%E2%80%93Stokesovy_rovnice" title="Navierovy–Stokesovy rovnice – Tsjechisch" lang="cs" hreflang="cs" data-title="Navierovy–Stokesovy rovnice" data-language-autonym="Čeština" data-language-local-name="Tsjechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Navier-Stokes%27_ligning" title="Navier-Stokes' ligning – Deens" lang="da" hreflang="da" data-title="Navier-Stokes' ligning" data-language-autonym="Dansk" data-language-local-name="Deens" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Navier-Stokes-Gleichungen" title="Navier-Stokes-Gleichungen – Duits" lang="de" hreflang="de" data-title="Navier-Stokes-Gleichungen" data-language-autonym="Deutsch" data-language-local-name="Duits" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations – Engels" lang="en" hreflang="en" data-title="Navier–Stokes equations" data-language-autonym="English" data-language-local-name="Engels" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaciones_de_Navier-Stokes" title="Ecuaciones de Navier-Stokes – Spaans" lang="es" hreflang="es" data-title="Ecuaciones de Navier-Stokes" data-language-autonym="Español" data-language-local-name="Spaans" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Navier-Stokes_ekuazioak" title="Navier-Stokes ekuazioak – Baskisch" lang="eu" hreflang="eu" data-title="Navier-Stokes ekuazioak" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D8%A7%D8%AA_%D9%86%D8%A7%D9%88%DB%8C%D9%87%E2%80%93%D8%A7%D8%B3%D8%AA%D9%88%DA%A9%D8%B3" title="معادلات ناویه–استوکس – Perzisch" lang="fa" hreflang="fa" data-title="معادلات ناویه–استوکس" data-language-autonym="فارسی" data-language-local-name="Perzisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Navierin%E2%80%93Stokesin_yht%C3%A4l%C3%B6t" title="Navierin–Stokesin yhtälöt – Fins" lang="fi" hreflang="fi" data-title="Navierin–Stokesin yhtälöt" data-language-autonym="Suomi" data-language-local-name="Fins" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quations_de_Navier-Stokes" title="Équations de Navier-Stokes – Frans" lang="fr" hreflang="fr" data-title="Équations de Navier-Stokes" data-language-autonym="Français" data-language-local-name="Frans" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Cothrom%C3%B3id_Navier-Stokes" title="Cothromóid Navier-Stokes – Iers" lang="ga" hreflang="ga" data-title="Cothromóid Navier-Stokes" data-language-autonym="Gaeilge" data-language-local-name="Iers" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ecuaci%C3%B3ns_de_Navier-Stokes" title="Ecuacións de Navier-Stokes – Galicisch" lang="gl" hreflang="gl" data-title="Ecuacións de Navier-Stokes" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%A0%D7%90%D7%95%D7%95%D7%99%D7%94-%D7%A1%D7%98%D7%95%D7%A7%D7%A1" title="משוואות נאוויה-סטוקס – Hebreeuws" lang="he" hreflang="he" data-title="משוואות נאוויה-סטוקס" data-language-autonym="עברית" data-language-local-name="Hebreeuws" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A8%E0%A5%87%E0%A4%B5%E0%A4%BF%E0%A4%AF%E0%A4%B0-%E0%A4%B8%E0%A5%8D%E0%A4%9F%E0%A5%8B%E0%A4%95%E0%A5%8D%E0%A4%B8_%E0%A4%B8%E0%A4%AE%E0%A5%80%E0%A4%95%E0%A4%B0%E0%A4%A3" title="नेवियर-स्टोक्स समीकरण – Hindi" lang="hi" hreflang="hi" data-title="नेवियर-स्टोक्स समीकरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Navier-Stokesove_jednad%C5%BEbe" title="Navier-Stokesove jednadžbe – Kroatisch" lang="hr" hreflang="hr" data-title="Navier-Stokesove jednadžbe" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Navier%E2%80%93Stokes-egyenletek" title="Navier–Stokes-egyenletek – Hongaars" lang="hu" hreflang="hu" data-title="Navier–Stokes-egyenletek" data-language-autonym="Magyar" data-language-local-name="Hongaars" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%86%D5%A1%D5%BE%D5%AB%D5%A5-%D5%8D%D5%BF%D5%B8%D6%84%D5%BD%D5%AB_%D5%B0%D5%A1%D5%BE%D5%A1%D5%BD%D5%A1%D6%80%D5%B8%D6%82%D5%B4" title="Նավիե-Ստոքսի հավասարում – Armeens" lang="hy" hreflang="hy" data-title="Նավիե-Ստոքսի հավասարում" data-language-autonym="Հայերեն" data-language-local-name="Armeens" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazioni_di_Navier-Stokes" title="Equazioni di Navier-Stokes – Italiaans" lang="it" hreflang="it" data-title="Equazioni di Navier-Stokes" data-language-autonym="Italiano" data-language-local-name="Italiaans" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8A%E3%83%93%E3%82%A8%E2%80%93%E3%82%B9%E3%83%88%E3%83%BC%E3%82%AF%E3%82%B9%E6%96%B9%E7%A8%8B%E5%BC%8F" title="ナビエ–ストークス方程式 – Japans" lang="ja" hreflang="ja" data-title="ナビエ–ストークス方程式" data-language-autonym="日本語" data-language-local-name="Japans" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%82%98%EB%B9%84%EC%97%90-%EC%8A%A4%ED%86%A0%ED%81%AC%EC%8A%A4_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="나비에-스토크스 방정식 – Koreaans" lang="ko" hreflang="ko" data-title="나비에-스토크스 방정식" data-language-autonym="한국어" data-language-local-name="Koreaans" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Navier-Stokes-Equatiounen" title="Navier-Stokes-Equatiounen – Luxemburgs" lang="lb" hreflang="lb" data-title="Navier-Stokes-Equatiounen" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxemburgs" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A8%E0%B5%87%E0%B4%B5%E0%B4%BF%E0%B4%AF%E0%B5%BC-%E0%B4%B8%E0%B5%8D%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B5%8B%E0%B4%95%E0%B5%8D%E0%B4%B8%E0%B5%8D_%E0%B4%B8%E0%B4%AE%E0%B4%B5%E0%B4%BE%E0%B4%95%E0%B5%8D%E0%B4%AF%E0%B4%99%E0%B5%8D%E0%B4%99%E0%B5%BE" title="നേവിയർ-സ്റ്റോക്സ് സമവാക്യങ്ങൾ – Malayalam" lang="ml" hreflang="ml" data-title="നേവിയർ-സ്റ്റോക്സ് സമവാക്യങ്ങൾ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Navier-Stokes-likningane" title="Navier-Stokes-likningane – Noors - Nynorsk" lang="nn" hreflang="nn" data-title="Navier-Stokes-likningane" data-language-autonym="Norsk nynorsk" data-language-local-name="Noors - Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Navier-Stokes-ligningene" title="Navier-Stokes-ligningene – Noors - Bokmål" lang="nb" hreflang="nb" data-title="Navier-Stokes-ligningene" data-language-autonym="Norsk bokmål" data-language-local-name="Noors - Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnania_Naviera-Stokesa" title="Równania Naviera-Stokesa – Pools" lang="pl" hreflang="pl" data-title="Równania Naviera-Stokesa" data-language-autonym="Polski" data-language-local-name="Pools" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Equassion_%C3%ABd_Navier-Stokes" title="Equassion ëd Navier-Stokes – Piëmontees" lang="pms" hreflang="pms" data-title="Equassion ëd Navier-Stokes" data-language-autonym="Piemontèis" data-language-local-name="Piëmontees" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%B5es_de_Navier-Stokes" title="Equações de Navier-Stokes – Portugees" lang="pt" hreflang="pt" data-title="Equações de Navier-Stokes" data-language-autonym="Português" data-language-local-name="Portugees" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ecua%C8%9Biile_Navier-Stokes" title="Ecuațiile Navier-Stokes – Roemeens" lang="ro" hreflang="ro" data-title="Ecuațiile Navier-Stokes" data-language-autonym="Română" data-language-local-name="Roemeens" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D1%8F_%D0%9D%D0%B0%D0%B2%D1%8C%D0%B5_%E2%80%94_%D0%A1%D1%82%D0%BE%D0%BA%D1%81%D0%B0" title="Уравнения Навье — Стокса – Russisch" lang="ru" hreflang="ru" data-title="Уравнения Навье — Стокса" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations – Simple English" lang="en-simple" hreflang="en-simple" data-title="Navier–Stokes equations" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Navier-Stokesove_ena%C4%8Dbe" title="Navier-Stokesove enačbe – Sloveens" lang="sl" hreflang="sl" data-title="Navier-Stokesove enačbe" data-language-autonym="Slovenščina" data-language-local-name="Sloveens" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Navje%E2%80%93Stoksove_jedna%C4%8Dine" title="Navje–Stoksove jednačine – Servisch" lang="sr" hreflang="sr" data-title="Navje–Stoksove jednačine" data-language-autonym="Српски / srpski" data-language-local-name="Servisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Navier%E2%80%93Stokes_ekvationer" title="Navier–Stokes ekvationer – Zweeds" lang="sv" hreflang="sv" data-title="Navier–Stokes ekvationer" data-language-autonym="Svenska" data-language-local-name="Zweeds" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B0%E0%AF%8D-%E0%AE%B8%E0%AF%8D%E0%AE%9F%E0%AF%8B%E0%AE%95%E0%AF%8D%E0%AE%9A%E0%AF%81_%E0%AE%9A%E0%AE%AE%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AF%8D" title="நேவியர்-ஸ்டோக்சு சமன்பாடுகள் – Tamil" lang="ta" hreflang="ta" data-title="நேவியர்-ஸ்டோக்சு சமன்பாடுகள்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A1%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B8%99%E0%B8%B2%E0%B9%80%E0%B8%A7%E0%B8%B5%E0%B8%A2%E0%B8%A3%E0%B9%8C%E2%80%93%E0%B8%AA%E0%B9%82%E0%B8%95%E0%B8%81%E0%B8%AA%E0%B9%8C" title="สมการนาเวียร์–สโตกส์ – Thai" lang="th" hreflang="th" data-title="สมการนาเวียร์–สโตกส์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Navier-Stokes_denklemleri" title="Navier-Stokes denklemleri – Turks" lang="tr" hreflang="tr" data-title="Navier-Stokes denklemleri" data-language-autonym="Türkçe" data-language-local-name="Turks" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D1%96%D0%B2%D0%BD%D1%8F%D0%BD%D0%BD%D1%8F_%D0%9D%D0%B0%D0%B2%27%D1%94_%E2%80%94_%D0%A1%D1%82%D0%BE%D0%BA%D1%81%D0%B0" title="Рівняння Нав'є — Стокса – Oekraïens" lang="uk" hreflang="uk" data-title="Рівняння Нав'є — Стокса" data-language-autonym="Українська" data-language-local-name="Oekraïens" 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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">Uiterlijk</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">naar zijbalk verplaatsen</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">verbergen</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">Uit Wikipedia, de vrije encyclopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Doorverwezen vanaf <a href="/w/index.php?title=Navier-Stokes-vergelijkingen&redirect=no" class="mw-redirect" title="Navier-Stokes-vergelijkingen">Navier-Stokes-vergelijkingen</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="nl" dir="ltr"><p>De <b>Navier-Stokesvergelijkingen</b>, genoemd naar <a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Claude-Louis Navier</a> en <a href="/wiki/George_Stokes" title="George Stokes">George Stokes</a>, zijn <a href="/wiki/Parti%C3%ABle_differentiaalvergelijking" title="Partiële differentiaalvergelijking">partiële differentiaalvergelijkingen</a> die de stroming van <a href="/wiki/Flu%C3%AFdum" title="Fluïdum">fluïda</a> beschrijven. Die vergelijkingen zeggen, dat een verandering in <a href="/wiki/Impuls_(natuurkunde)" title="Impuls (natuurkunde)">impuls</a> van een fluïdumdeel - bijvoorbeeld als een vloeistof versnelt - altijd in evenwicht is met <a href="/wiki/Drukgradi%C3%ABnt" title="Drukgradiënt">drukgradiënten</a> die er zijn en met de dissipatieve <a href="/wiki/Viscositeit" title="Viscositeit">viskeuze</a> <a href="/wiki/Kracht" title="Kracht">kracht</a> die inwerkt op het fluïdum. Die viskeuze kracht ontstaat door <a href="/wiki/Molecuul" title="Molecuul">moleculaire</a> interactie en bepaalt hoe "stroperig" (of hoe viskeus) een fluïdum is. De Navier-Stokesvergelijkingen zijn dus een dynamische uitdrukking van het krachtenevenwicht inwerkend op een willekeurig deel van een fluïdum. In feite drukken de vergelijkingen dus de <a href="/wiki/Wetten_van_Newton" title="Wetten van Newton">wetten van Newton</a> d(m<b>v</b>)/dt = <b>F</b> uit voor een eenheidsvolume. De vergelijkingen gelden algemeen, zolang de snelheid veel kleiner blijft dan de <a href="/wiki/Lichtsnelheid" title="Lichtsnelheid">lichtsnelheid</a> en zolang geen <a href="/wiki/Kwantumeffect" title="Kwantumeffect">kwantumeffecten</a> zoals <a href="/wiki/Superflu%C3%AFditeit" title="Superfluïditeit">supervloeibaarheid</a> meespelen. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="De_vergelijkingen">De vergelijkingen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&veaction=edit&section=1" title="Bewerk dit kopje: De vergelijkingen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&action=edit&section=1" title="De broncode bewerken van de sectie: De vergelijkingen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De meest algemene vorm van de Navier-Stokesvergelijkingen (eigenlijk zijn er drie, voor alle drie dimensies, maar in <a href="/wiki/Vector_(wiskunde)" title="Vector (wiskunde)">vectornotatie</a> worden die in één vergelijking bijeengenomen) - afgeleid van de <a href="/wiki/Wet_van_behoud_van_impuls" class="mw-redirect" title="Wet van behoud van impuls">wet van behoud van impuls</a> 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 \rho {\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=-\nabla \cdot \mathbb {P} +\mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho {\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=-\nabla \cdot \mathbb {P} +\mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/041a8d9d01513d8295e098cf6fac7afcb1617634" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.577ex; height:5.509ex;" alt="{\displaystyle \rho {\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=-\nabla \cdot \mathbb {P} +\mathbf {f} }" /></span></dd></dl> <ul><li><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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" /></span> is de (massa)dichtheid;</li> <li><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 \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }" /></span> is de vectoriële snelheid van het fluïdum;</li> <li><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 \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" /></span> is de <a href="/wiki/Nabla" title="Nabla">nabla-operator</a> (en <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 \nabla \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df6024211b717870f07844116e116b2eb314d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.583ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot }" /></span> is dus de <a href="/wiki/Divergentie_(vectorveld)" title="Divergentie (vectorveld)">divergentie</a>);</li> <li><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 \mathbb {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1053af9e662ceaf56c4455f90e0f67273422eded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" /></span> is een <a href="/wiki/Tensor" title="Tensor">tensor</a> voor de veralgemeende druk (of <i><a href="/wiki/Mechanische_spanning" title="Mechanische spanning">stress</a></i>). Deze tensor omvat onder meer een deel dat de <a href="/wiki/Viscositeit" title="Viscositeit">viscositeit</a> voorstelt, en is een maat voor hoeveel <a href="/wiki/Schuifspanning" title="Schuifspanning">afschuifkracht</a> een laag bewegende stof uitoefent op een daarnaast bewegende andere laag;</li> <li><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 \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6194e680a4e7c521f2178c50eea302843a852d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }" /></span> is de krachtvector per volume die op het fluïdum werkt, bijvoorbeeld de zwaartekracht.</li></ul> <p>Deze vergelijking doet geen aannamen over de aard of oorzaak van de gegenereerde afschuifkrachten, en is geldig voor zowel <a href="/wiki/Newtoniaans_flu%C3%AFdum" title="Newtoniaans fluïdum">newtoniaanse vloeistoffen</a> als niet-newtoniaanse vloeistoffen. In de praktijk moet deze <a href="/wiki/Bewegingsvergelijking" title="Bewegingsvergelijking">bewegingsvergelijking</a> daarom worden gecombineerd met een <a href="/w/index.php?title=Constitutieve_vergelijking&action=edit&redlink=1" class="new" title="Constitutieve vergelijking (de pagina bestaat niet)">constitutieve vergelijking</a> die beschrijft hoe de druk en stress afhangen van de snelheidsgradiënten. Wanneer we aannemen dat de vloeistof newtoniaans is, vinden we: </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 \rho {\frac {\partial \mathbf {v} }{\partial t}}+\rho (\mathbf {v} \cdot \nabla )\mathbf {v} =-\nabla p+\eta \nabla ^{2}\mathbf {v} +(\lambda +\eta )\nabla (\nabla \cdot \mathbf {v} )+\mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mi>η</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>+</mo> <mi>η</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho {\frac {\partial \mathbf {v} }{\partial t}}+\rho (\mathbf {v} \cdot \nabla )\mathbf {v} =-\nabla p+\eta \nabla ^{2}\mathbf {v} +(\lambda +\eta )\nabla (\nabla \cdot \mathbf {v} )+\mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9499c7eff6b2336749769a7705bc10a7e28d8cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:56.158ex; height:5.509ex;" alt="{\displaystyle \rho {\frac {\partial \mathbf {v} }{\partial t}}+\rho (\mathbf {v} \cdot \nabla )\mathbf {v} =-\nabla p+\eta \nabla ^{2}\mathbf {v} +(\lambda +\eta )\nabla (\nabla \cdot \mathbf {v} )+\mathbf {f} }" /></span></dd></dl> <p>Hoewel de constitutieve vergelijking voor een newtoniaanse vloeistof lineair is, is deze vorm van de Navier-Stokesvergelijkingen steeds een niet-lineaire <a href="/wiki/Differentiaalvergelijking" title="Differentiaalvergelijking">differentiaalvergelijking</a> vanwege de advectieterm <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 (\mathbf {v} \cdot \nabla )\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f4e7bbac3438d526b551fbd87720fc8bbce7166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.246ex; height:2.843ex;" alt="{\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} }" /></span>, die uitdrukt hoe lokale grootheden veranderen als materiaal wordt meegevoerd door de stroming. </p> <ul><li>De operator <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 }{\partial t}}+(\mathbf {v} \cdot \nabla )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}+(\mathbf {v} \cdot \nabla )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8975f9f476c3e152ee975fe1bcee3297c7b3762b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.669ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}+(\mathbf {v} \cdot \nabla )}" /></span></dd></dl></li></ul> <dl><dd>wordt de <i><a href="/wiki/Materi%C3%ABle_afgeleide" title="Materiële afgeleide">materiële afgeleide</a></i> genoemd. Hij beschrijft de verandering in de tijd van een lokale grootheid onder invloed van zowel expliciete tijdsafhankelijkheid als het meevoeren van materiaal onder invloed van de stroming.</dd></dl> <ul><li>De vector <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 \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6194e680a4e7c521f2178c50eea302843a852d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }" /></span> beschrijft de kracht per volume, dus in SI-eenheden in N/m³. Die wordt bijvoorbeeld veroorzaakt door de <a href="/wiki/Zwaartekracht" title="Zwaartekracht">zwaartekracht</a> of de <a href="/wiki/Corioliskracht" class="mw-redirect" title="Corioliskracht">corioliskracht</a> voor het weer.</li> <li>De scalar <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span> is de druk in Pa.</li> <li>De constanten <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span> en <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" /></span> zijn <a href="/wiki/Materiaalconstante" class="mw-redirect" title="Materiaalconstante">materiaalconstanten</a> met betrekking tot de <a href="/wiki/Viscositeit" title="Viscositeit">viscositeit</a> in Pa·s. Daarbij 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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" /></span> de 'gewone' of <i>dynamische viscositeit</i> voor onsamendrukbare stroming en <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span> de coëfficiënt van de 'tweede viscositeit', die gerelateerd is aan de <i>volumeviscositeit</i> of <i>bulkviscositeit</i>, een viscositeit die boven op de dynamische viscositeit komt bij stromingen waarbij vloeistof wordt samengedrukt of uitgerekt.</li></ul> <p>In veel praktische stromingen van vloeistoffen kan men aannemen dat de vloeistof <a href="/wiki/Compressibiliteit" class="mw-redirect" title="Compressibiliteit">onsamendrukbaar</a> is. Wiskundig kan men dit tot uitdrukking brengen door de voorwaarde dat de divergentie van de stroming wegvalt, dus </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 \nabla \cdot \mathbf {v} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {v} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/760025de0202db29f86a1c2c4bef18d54ccc0f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.287ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {v} =0}" /></span>,</dd></dl> <p>zodat de vergelijking reduceert tot de bekendere vorm </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 \rho {\frac {\partial \mathbf {v} }{\partial t}}+\rho (\mathbf {v} \cdot \nabla )\mathbf {v} =-\nabla p+\eta \nabla ^{2}\mathbf {v} +\mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mi>η</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho {\frac {\partial \mathbf {v} }{\partial t}}+\rho (\mathbf {v} \cdot \nabla )\mathbf {v} =-\nabla p+\eta \nabla ^{2}\mathbf {v} +\mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/933e2b3b148d493c35c5ea4fcff6b24cdecd882b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:37.372ex; height:5.509ex;" alt="{\displaystyle \rho {\frac {\partial \mathbf {v} }{\partial t}}+\rho (\mathbf {v} \cdot \nabla )\mathbf {v} =-\nabla p+\eta \nabla ^{2}\mathbf {v} +\mathbf {f} }" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Continuïteitsvergelijking"><span id="Continu.C3.AFteitsvergelijking"></span>Continuïteitsvergelijking</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&veaction=edit&section=2" title="Bewerk dit kopje: Continuïteitsvergelijking" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&action=edit&section=2" title="De broncode bewerken van de sectie: Continuïteitsvergelijking"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote" style="margin-bottom:0.5em; padding:0.5em 0 0.5em 1.6em; font-size:95%;" role="note"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/20px-1rightarrow_blue.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/1rightarrow_blue.svg/40px-1rightarrow_blue.svg.png 1.5x" data-file-width="480" data-file-height="480" /></span></span> <i>Zie <a href="/wiki/Continu%C3%AFteitsvergelijking" title="Continuïteitsvergelijking">continuïteitsvergelijking</a> voor het hoofdartikel over dit onderwerp.</i></div> <p>De wet van behoud van massa geeft: </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 {\partial \mathbf {\rho } \over \partial t}+\nabla \cdot (\rho {\vec {v}})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial \mathbf {\rho } \over \partial t}+\nabla \cdot (\rho {\vec {v}})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26b20e1fd512f3fcaeb59ff89bd816a78100c483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.259ex; height:5.676ex;" alt="{\displaystyle {\partial \mathbf {\rho } \over \partial t}+\nabla \cdot (\rho {\vec {v}})=0}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Vereenvoudigingen">Vereenvoudigingen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&veaction=edit&section=3" title="Bewerk dit kopje: Vereenvoudigingen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&action=edit&section=3" title="De broncode bewerken van de sectie: Vereenvoudigingen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Voor een onsamendrukbaar fluïdum zoals een vloeistof bij niet al te hoge drukken of een gas ver beneden de <a href="/wiki/Geluidssnelheid" title="Geluidssnelheid">geluidssnelheid</a> is de massadichtheid constant. Dit geeft: </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 \nabla \cdot {\vec {v}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot {\vec {v}}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e8ef68f79faaa1d66b4059743e1d1ac39cd82b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.698ex; height:2.676ex;" alt="{\displaystyle \nabla \cdot {\vec {v}}=0,}" /></span>.</dd></dl> <p>Zo valt dus de factor met <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 +\eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>+</mo> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda +\eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f04325533838b0b3f01ec24979b2991d2da20fdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.365ex; height:2.676ex;" alt="{\displaystyle \lambda +\eta }" /></span> weg. </p><p>De Navier-Stokesvergelijkingen gelden ook voor samendrukbare fluïda zoals gassen. Voor gassen ligt de viscositeit een factor 100 lager dan voor vloeistoffen, zodat in dat geval de viscositeit bij benadering te verwaarlozen is en de term met <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 \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" /></span> wegvalt. </p><p>Gelijktijdige verwaarlozing van samendrukbaarheid en viscositeit geeft de <a href="/wiki/Stromingsvergelijkingen_van_Euler" title="Stromingsvergelijkingen van Euler">stromingsvergelijkingen van Euler</a> die dus gelden voor de stroming van weinig viskeuze vloeistoffen of voor gassen ver onder de geluidssnelheid. </p><p>In stationaire stromingen is de snelheid onafhankelijk van de tijd, oftewel <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 {\partial {\vec {v}} \over \partial t}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial {\vec {v}} \over \partial t}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9ac7b0ca2fe03ec25e90888581701205d54ad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.59ex; height:5.509ex;" alt="{\displaystyle {\partial {\vec {v}} \over \partial t}=0}" /></span>. Dan blijft er dus een differentiaalvergelijking over met alleen afgeleiden naar de plaats en de oplossing is dus een snelheidsvector als functie van de plaats. </p> <div class="mw-heading mw-heading2"><h2 id="Toepassingen">Toepassingen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&veaction=edit&section=4" title="Bewerk dit kopje: Toepassingen" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&action=edit&section=4" title="De broncode bewerken van de sectie: Toepassingen"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In de praktijk lost men de Navier-Stokesvergelijkingen op met <a href="/wiki/Numerieke_stromingsleer" title="Numerieke stromingsleer">numerieke stromingsleer</a>: de zogenaamde <b>CFD</b> (Engels: <i>Computational Fluid Dynamics</i>). Deze wiskunde wordt onder andere gebruikt om het weer te voorspellen, branden te simuleren en de <a href="/wiki/Aerodynamica" title="Aerodynamica">aerodynamica</a> van vliegtuigen en auto's te bestuderen. </p> <div class="mw-heading mw-heading2"><h2 id="Millenniumprijsprobleem">Millenniumprijsprobleem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&veaction=edit&section=5" title="Bewerk dit kopje: Millenniumprijsprobleem" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&action=edit&section=5" title="De broncode bewerken van de sectie: Millenniumprijsprobleem"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Het <a href="/wiki/Clay_Mathematics_Institute" title="Clay Mathematics Institute">Clay Mathematics Institute</a> uit <a href="/wiki/Cambridge_(Massachusetts)" title="Cambridge (Massachusetts)">Cambridge</a> (<a href="/wiki/Massachusetts" title="Massachusetts">Massachusetts</a>) heeft bij het ingaan van het nieuwe millennium in 2000 zeven <a href="/wiki/Millenniumprijsprobleem" title="Millenniumprijsprobleem">millenniumprijsproblemen</a> ingesteld. Een wetenschappelijk panel selecteerde een zevental problemen op wiskundig gebied die al jaren op een oplossing lagen te wachten. De raad van directeuren van CMI heeft een fonds van $7 miljoen beschikbaar voor de oplossing van deze problemen, $1 miljoen voor elk van de problemen. Een van de millenniumprijsproblemen is het vinden van een oplossing van de Navier-Stokesvergelijkingen. Zie voor een exacte beschrijving van het probleem de website van het Clay Mathematics Institute.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Voetnoten">Voetnoten</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&veaction=edit&section=6" title="Bewerk dit kopje: Voetnoten" class="mw-editsection-visualeditor"><span>bewerken</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Navier-Stokesvergelijkingen&action=edit&section=6" title="De broncode bewerken van de sectie: Voetnoten"><span>brontekst bewerken</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="citation"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20151222054637/http://www.claymath.org/millennium-problems/navier%E2%80%93stokes-equation">Clay Mathematics Institute Clay Mathematics Institute</a>. Gearchiveerd op <span class="mw-formatted-date" title="2015-12-22">22 december 2015</span>. Geraadpleegd op <span class="mw-formatted-date" title="2015-11-07">7 november 2015</span>. </span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Overgenomen van "<a dir="ltr" href="https://nl.wikipedia.org/w/index.php?title=Navier-Stokesvergelijkingen&oldid=67921806">https://nl.wikipedia.org/w/index.php?title=Navier-Stokesvergelijkingen&oldid=67921806</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categorie:Alles" title="Categorie:Alles">Categorieën</a>: <ul><li><a href="/wiki/Categorie:Fysische_transportverschijnselen" title="Categorie:Fysische transportverschijnselen">Fysische transportverschijnselen</a></li><li><a href="/wiki/Categorie:Differentiaalvergelijking" title="Categorie:Differentiaalvergelijking">Differentiaalvergelijking</a></li><li><a href="/wiki/Categorie:Vloeistofdynamica" title="Categorie:Vloeistofdynamica">Vloeistofdynamica</a></li><li><a href="/wiki/Categorie:Natuurkundige_wet" title="Categorie:Natuurkundige wet">Natuurkundige wet</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"> Deze pagina is voor het laatst bewerkt op 4 aug 2024 om 17:22.</li> <li id="footer-info-copyright">De tekst is beschikbaar onder de licentie <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.nl">Creative Commons Naamsvermelding/Gelijk delen</a>, er kunnen aanvullende voorwaarden van toepassing zijn. 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