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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Computational fluid dynamics tools</div> <p class="mw-empty-elt"> </p> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about continuum mechanics. For the use of generalized coordinates in classical mechanics, see <a href="/wiki/Generalized_coordinates" title="Generalized coordinates">generalized coordinates</a>, <a href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</a> and <a href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/Lagrangian_vs_Eulerian.webm/220px--Lagrangian_vs_Eulerian.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="83" data-durationhint="12" data-mwtitle="Lagrangian_vs_Eulerian.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Lagrangian_vs_Eulerian.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3b/Lagrangian_vs_Eulerian.webm/Lagrangian_vs_Eulerian.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="320" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3b/Lagrangian_vs_Eulerian.webm/Lagrangian_vs_Eulerian.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/3/3b/Lagrangian_vs_Eulerian.webm" type="video/webm; codecs="vp8"" data-width="1280" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3b/Lagrangian_vs_Eulerian.webm/Lagrangian_vs_Eulerian.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="96" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3b/Lagrangian_vs_Eulerian.webm/Lagrangian_vs_Eulerian.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="160" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3b/Lagrangian_vs_Eulerian.webm/Lagrangian_vs_Eulerian.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/3b/Lagrangian_vs_Eulerian.webm/Lagrangian_vs_Eulerian.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="240" /></video></span><figcaption>File:Lagrangian vs Eulerian<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="How does this image show the difference between the Lagrangian and the Eulerian specification of the flow field? (August 2024)">further explanation needed</span></a></i>]</sup></figcaption></figure><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Eulerian_versus_Lagrangian_perspectives_illustrated.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Eulerian_versus_Lagrangian_perspectives_illustrated.svg/220px-Eulerian_versus_Lagrangian_perspectives_illustrated.svg.png" decoding="async" width="220" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Eulerian_versus_Lagrangian_perspectives_illustrated.svg/330px-Eulerian_versus_Lagrangian_perspectives_illustrated.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Eulerian_versus_Lagrangian_perspectives_illustrated.svg/440px-Eulerian_versus_Lagrangian_perspectives_illustrated.svg.png 2x" data-file-width="701" data-file-height="367" /></a><figcaption>Eulerian perspective of <a href="/wiki/Flow_velocity" title="Flow velocity">fluid velocity</a> versus Lagrangian depiction of <a href="/wiki/Finite_strain_theory" title="Finite strain theory">strain.</a></figcaption></figure> <p>In <a href="/wiki/Classical_field_theory" title="Classical field theory">classical field theories</a>, the <b>Lagrangian specification of the flow field</b> is a way of looking at fluid motion where the observer follows an individual <a href="/wiki/Fluid_parcel" title="Fluid parcel">fluid parcel</a> as it moves through space and time.<sup id="cite_ref-Batchelor_1-0" class="reference"><a href="#cite_note-Batchelor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lamb_2-0" class="reference"><a href="#cite_note-Lamb-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Plotting the position of an individual parcel through time gives the <a href="/wiki/Streamlines,_streaklines,_and_pathlines" title="Streamlines, streaklines, and pathlines">pathline</a> of the parcel. This can be visualized as sitting in a boat and drifting down a river. </p><p>The <b>Eulerian specification of the flow field</b> is a way of looking at fluid motion that focuses on specific locations in the space through which the fluid flows as time passes.<sup id="cite_ref-Batchelor_1-1" class="reference"><a href="#cite_note-Batchelor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lamb_2-1" class="reference"><a href="#cite_note-Lamb-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> This can be visualized by sitting on the bank of a river and watching the water pass the fixed location. </p><p>The Lagrangian and Eulerian specifications of the flow field are sometimes loosely denoted as the <b>Lagrangian and Eulerian frame of reference</b>. However, in general both the Lagrangian and Eulerian specification of the flow field can be applied in any observer's <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a>, and in any <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> used within the chosen frame of reference. The Lagrangian and Eulerian specifications are named after <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, respectively. </p><p>These specifications are reflected in <a href="/wiki/Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a>, where "Eulerian" simulations employ a fixed <a href="/wiki/Types_of_mesh" title="Types of mesh">mesh</a> while "Lagrangian" ones (such as <a href="/wiki/Meshfree_methods" title="Meshfree methods">meshfree simulations</a>) feature simulation nodes that may move following the <a href="/wiki/Velocity_field" class="mw-redirect" title="Velocity field">velocity field</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrangian_and_Eulerian_specification_of_the_flow_field&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> is credited of introducing both specifications in two publications written in 1755<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> and 1759.<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><sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> studied the equations of motion in connection to the <a href="/wiki/Action_principles" title="Action principles">principle of least action</a> in 1760, later in a treaty of fluid mechanics in 1781,<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> and thirdly in his book <i><a href="/wiki/M%C3%A9canique_analytique" title="Mécanique analytique">Mécanique analytique</a></i>.<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In this book Lagrange starts with the Lagrangian specification but later converts them into the Eulerian specification.<sup id="cite_ref-:0_5-2" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrangian_and_Eulerian_specification_of_the_flow_field&action=edit&section=2" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the <i>Eulerian specification</i> of a <a href="/wiki/Field_(physics)" title="Field (physics)">field</a>, the field is represented as a function of position <b>x</b> and time <i>t</i>. For example, the <a href="/wiki/Flow_velocity" title="Flow velocity">flow velocity</a> is represented by a function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \left(\mathbf {x} ,t\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \left(\mathbf {x} ,t\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f6606810b20f4fba086816847f00a4121f6064e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8ex; height:2.843ex;" alt="{\displaystyle \mathbf {u} \left(\mathbf {x} ,t\right).}"></span> </p><p>On the other hand, in the <i>Lagrangian specification</i>, individual fluid parcels are followed through time. The fluid parcels are labelled by some (time-independent) vector field <b>x</b><sub>0</sub>. (Often, <b>x</b><sub>0</sub> is chosen to be the position of the center of mass of the parcels at some initial time <i>t</i><sub>0</sub>. It is chosen in this particular manner to account for the possible changes of the shape over time. Therefore, the center of mass is a good parameterization of the flow velocity <b>u</b> of the parcel.)<sup id="cite_ref-Batchelor_1-2" class="reference"><a href="#cite_note-Batchelor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In the Lagrangian description, the flow is described by a function <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {X} \left(\mathbf {x} _{0},t\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} \left(\mathbf {x} _{0},t\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325c42def30a6edd2cebea1ea5cc1616acd6f8af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.589ex; height:2.843ex;" alt="{\displaystyle \mathbf {X} \left(\mathbf {x} _{0},t\right),}"></span> giving the position of the particle labeled <b>x</b><sub>0</sub> at time <i>t</i>. </p><p>The two specifications are related as follows:<sup id="cite_ref-Lamb_2-2" class="reference"><a href="#cite_note-Lamb-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \left(\mathbf {X} (\mathbf {x} _{0},t),t\right)={\frac {\partial \mathbf {X} }{\partial t}}\left(\mathbf {x} _{0},t\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \left(\mathbf {X} (\mathbf {x} _{0},t),t\right)={\frac {\partial \mathbf {X} }{\partial t}}\left(\mathbf {x} _{0},t\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcfdb730a2fe441106316719f3027540290467a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.564ex; height:5.509ex;" alt="{\displaystyle \mathbf {u} \left(\mathbf {X} (\mathbf {x} _{0},t),t\right)={\frac {\partial \mathbf {X} }{\partial t}}\left(\mathbf {x} _{0},t\right),}"></span> because both sides describe the velocity of the particle labeled <b>x</b><sub>0</sub> at time <i>t</i>. </p><p>Within a chosen coordinate system, <b>x</b><sub>0</sub> and <b>x</b> are referred to as the <b>Lagrangian coordinates</b> and <b>Eulerian coordinates</b> of the flow respectively. </p> <div class="mw-heading mw-heading2"><h2 id="Material_derivative">Material derivative</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrangian_and_Eulerian_specification_of_the_flow_field&action=edit&section=3" title="Edit section: Material derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Material_derivative" title="Material derivative">Material derivative</a></div> <p>The Lagrangian and Eulerian specifications of the <a href="/wiki/Kinematics" title="Kinematics">kinematics</a> and <a href="/wiki/Dynamics_(physics)" class="mw-redirect" title="Dynamics (physics)">dynamics</a> of the flow field are related by the <a href="/wiki/Material_derivative" title="Material derivative">material derivative</a> (also called the Lagrangian derivative, convective derivative, substantial derivative, or particle derivative).<sup id="cite_ref-Batchelor_1-3" class="reference"><a href="#cite_note-Batchelor-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Suppose we have a flow field <b>u</b>, and we are also given a generic field with Eulerian specification <b>F</b>(<b>x</b>, <i>t</i>). Now one might ask about the total rate of change of <b>F</b> experienced by a specific flow parcel. This can be computed as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {D} \mathbf {F} }{\mathrm {D} t}}={\frac {\partial \mathbf {F} }{\partial t}}+\left(\mathbf {u} \cdot \nabla \right)\mathbf {F} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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">F</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {D} \mathbf {F} }{\mathrm {D} t}}={\frac {\partial \mathbf {F} }{\partial t}}+\left(\mathbf {u} \cdot \nabla \right)\mathbf {F} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bbeef01eca0ca66a9ba9cc08f76d19c6faf3db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.696ex; height:5.509ex;" alt="{\displaystyle {\frac {\mathrm {D} \mathbf {F} }{\mathrm {D} t}}={\frac {\partial \mathbf {F} }{\partial t}}+\left(\mathbf {u} \cdot \nabla \right)\mathbf {F} ,}"></span> where ∇ denotes the <a href="/wiki/Del" title="Del">nabla</a> operator with respect to <b>x</b>, and the operator <b>u</b>⋅∇ is to be applied to each component of <b>F</b>. This tells us that the total rate of change of the function <b>F</b> as the fluid parcels moves through a flow field described by its Eulerian specification <b>u</b> is equal to the sum of the local rate of change and the convective rate of change of <b>F</b>. This is a consequence of the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> since we are differentiating the function <b>F</b>(<b>X</b>(<b>x</b><sub>0</sub>, <i>t</i>), <i>t</i>) with respect to <i>t</i>. </p><p><a href="/wiki/Conservation_law" title="Conservation law">Conservation laws</a> for a unit mass have a Lagrangian form, which together with mass conservation produce Eulerian conservation; on the contrary, when fluid particles can exchange a quantity (like energy or momentum), only Eulerian conservation laws exist.<sup id="cite_ref-Falkovich_7-0" class="reference"><a href="#cite_note-Falkovich-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </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=Lagrangian_and_Eulerian_specification_of_the_flow_field&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Brewer%E2%80%93Dobson_circulation" title="Brewer–Dobson circulation">Brewer-Dobson Circulation</a></li> <li><a href="/wiki/Conservation_form" title="Conservation form">Conservation form</a></li> <li><a href="/wiki/Contour_advection" title="Contour advection">Contour advection</a></li> <li><a href="/wiki/Displacement_field_(mechanics)" title="Displacement field (mechanics)">Displacement field (mechanics)</a></li> <li><a href="/wiki/Equivalent_latitude" title="Equivalent latitude">Equivalent latitude</a></li> <li><a href="/wiki/Generalized_Lagrangian_mean" title="Generalized Lagrangian mean">Generalized Lagrangian mean</a></li> <li><a href="/wiki/Trajectory_(fluid_mechanics)" title="Trajectory (fluid mechanics)">Trajectory (fluid mechanics)</a></li> <li><a href="/wiki/Liouville%27s_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville's theorem (Hamiltonian)</a></li> <li><a href="/wiki/Lagrangian_particle_tracking" title="Lagrangian particle tracking">Lagrangian particle tracking</a></li> <li><a href="/wiki/Rolling#Deformable_bodies" title="Rolling">Rolling</a></li> <li><a href="/wiki/Streamlines,_streaklines,_and_pathlines" title="Streamlines, streaklines, and pathlines">Streamlines, streaklines, and pathlines</a></li> <li><a href="/wiki/Immersed_Boundary_Method" class="mw-redirect" title="Immersed Boundary Method">Immersed Boundary Method</a></li> <li><a href="/wiki/Semi-Lagrangian_scheme" title="Semi-Lagrangian scheme">Semi-Lagrangian scheme</a></li> <li><a href="/wiki/Stochastic_Eulerian_Lagrangian_method" title="Stochastic Eulerian Lagrangian method">Stochastic Eulerian Lagrangian methods</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrangian_and_Eulerian_specification_of_the_flow_field&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Batchelor-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Batchelor_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Batchelor_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Batchelor_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Batchelor_1-3"><sup><i><b>d</b></i></sup></a></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="CITEREFBatchelor1973" class="citation book cs1"><a href="/wiki/George_Batchelor" title="George Batchelor">Batchelor, G. K.</a> (1973). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/847527173"><i>An Introduction to Fluid dynamics</i></a>. Cambridge, U.K.: Cambridge University Press. pp. 71–73. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-09817-5" title="Special:BookSources/978-0-521-09817-5"><bdi>978-0-521-09817-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/847527173">847527173</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Fluid+dynamics&rft.place=Cambridge%2C+U.K.&rft.pages=71-73&rft.pub=Cambridge+University+Press&rft.date=1973&rft_id=info%3Aoclcnum%2F847527173&rft.isbn=978-0-521-09817-5&rft.aulast=Batchelor&rft.aufirst=G.+K.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F847527173&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+and+Eulerian+specification+of+the+flow+field" class="Z3988"></span></span> </li> <li id="cite_note-Lamb-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lamb_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lamb_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lamb_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLamb1994" class="citation book cs1"><a href="/wiki/Horace_Lamb" title="Horace Lamb">Lamb, H.</a> (1994) [1932]. <i>Hydrodynamics</i> (6th ed.). Cambridge University Press. §3–§7 and §13–§16. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-45868-9" title="Special:BookSources/978-0-521-45868-9"><bdi>978-0-521-45868-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hydrodynamics&rft.pages=%C2%A73-%C2%A77+and+%C2%A713-%C2%A716&rft.edition=6th&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=978-0-521-45868-9&rft.aulast=Lamb&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+and+Eulerian+specification+of+the+flow+field" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler1757" class="citation journal cs1">Euler, Leonhard (1757-01-01). <a rel="nofollow" class="external text" href="https://scholarlycommons.pacific.edu/euler-works/226/">"Principes généraux du mouvement des fluides"</a>. <i>Mémoires de l'académie des sciences de Berlin</i>: 274–315.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=M%C3%A9moires+de+l%27acad%C3%A9mie+des+sciences+de+Berlin&rft.atitle=Principes+g%C3%A9n%C3%A9raux+du+mouvement+des+fluides&rft.pages=274-315&rft.date=1757-01-01&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=https%3A%2F%2Fscholarlycommons.pacific.edu%2Feuler-works%2F226%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+and+Eulerian+specification+of+the+flow+field" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler1761" class="citation journal cs1">Euler, Leonhard (1761-01-01). <a rel="nofollow" class="external text" href="https://scholarlycommons.pacific.edu/euler-works/258/">"Principia motus fluidorum"</a>. <i>Novi Commentarii Academiae Scientiarum Petropolitanae</i>: 271–311.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Novi+Commentarii+Academiae+Scientiarum+Petropolitanae&rft.atitle=Principia+motus+fluidorum&rft.pages=271-311&rft.date=1761-01-01&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=https%3A%2F%2Fscholarlycommons.pacific.edu%2Feuler-works%2F258%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+and+Eulerian+specification+of+the+flow+field" class="Z3988"></span></span> </li> <li id="cite_note-:0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLamb1945" class="citation book cs1">Lamb, Sir Horace (1945-01-01). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=237xDg7T0RkC&q=;+Lamb,+Hydrodynamics,+6th+ed.,+(Cambridge+University.+Press,+1937)+c"><i>Hydrodynamics</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60256-1" title="Special:BookSources/978-0-486-60256-1"><bdi>978-0-486-60256-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hydrodynamics&rft.pub=Courier+Corporation&rft.date=1945-01-01&rft.isbn=978-0-486-60256-1&rft.aulast=Lamb&rft.aufirst=Sir+Horace&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D237xDg7T0RkC%26q%3D%3B%2BLamb%2C%2BHydrodynamics%2C%2B6th%2Bed.%2C%2B%28Cambridge%2BUniversity.%2BPress%2C%2B1937%29%2Bc&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+and+Eulerian+specification+of+the+flow+field" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_LAGRANGE__4_695_0">"Joseph Louis de Lagrange: Mémoire sur la théorie du mouvement des fluides"</a>. <i>sites.mathdoc.fr</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=sites.mathdoc.fr&rft.atitle=Joseph+Louis+de+Lagrange%3A+M%C3%A9moire+sur+la+th%C3%A9orie+du+mouvement+des+fluides&rft_id=http%3A%2F%2Fsites.mathdoc.fr%2Fcgi-bin%2Foeitem%3Fid%3DOE_LAGRANGE__4_695_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+and+Eulerian+specification+of+the+flow+field" class="Z3988"></span></span> </li> <li id="cite_note-Falkovich-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Falkovich_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFalkovich2011" class="citation book cs1">Falkovich, Gregory (2011). <i>Fluid Mechanics (A short course for physicists)</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-00575-4" title="Special:BookSources/978-1-107-00575-4"><bdi>978-1-107-00575-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics+%28A+short+course+for+physicists%29&rft.pub=Cambridge+University+Press&rft.date=2011&rft.isbn=978-1-107-00575-4&rft.aulast=Falkovich&rft.aufirst=Gregory&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+and+Eulerian+specification+of+the+flow+field" class="Z3988"></span></span> </li> </ol></div></div> <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=Lagrangian_and_Eulerian_specification_of_the_flow_field&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBadinCrisciani2018" class="citation book cs1">Badin, G.; Crisciani, F. (2018). <i>Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws</i>. Springer. p. 218. <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/2018vffg.book.....B">2018vffg.book.....B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-59695-2">10.1007/978-3-319-59695-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-59694-5" title="Special:BookSources/978-3-319-59694-5"><bdi>978-3-319-59694-5</bdi></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:125902566">125902566</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Variational+Formulation+of+Fluid+and+Geophysical+Fluid+Dynamics+-+Mechanics%2C+Symmetries+and+Conservation+Laws&rft.pages=218&rft.pub=Springer&rft.date=2018&rft_id=info%3Adoi%2F10.1007%2F978-3-319-59695-2&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A125902566%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2018vffg.book.....B&rft.isbn=978-3-319-59694-5&rft.aulast=Badin&rft.aufirst=G.&rft.au=Crisciani%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+and+Eulerian+specification+of+the+flow+field" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1987" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, Lev</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, E.M.</a> (1987). <i>Fluid Mechanics</i>. Course of Theoretical Physics, Volume 6 (2nd ed.). Butterworth-Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0750627672" title="Special:BookSources/978-0750627672"><bdi>978-0750627672</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics&rft.series=Course+of+Theoretical+Physics%2C+Volume+6&rft.edition=2nd&rft.pub=Butterworth-Heinemann&rft.date=1987&rft.isbn=978-0750627672&rft.aulast=Landau&rft.aufirst=Lev&rft.au=Lifshitz%2C+E.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrangian+and+Eulerian+specification+of+the+flow+field" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrangian_and_Eulerian_specification_of_the_flow_field&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a rel="nofollow" class="external autonumber" href="http://www.isima.fr/~leborgne/IsimathMeca/Lageul.pdf">[1]</a> Objectivity in classical continuum mechanics: Motions, Eulerian and Lagrangian functions; Deformation gradient; Lie derivatives; Velocity-addition formula, Coriolis; Objectivity. </p>    </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="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Lagrangian_and_Eulerian_specification_of_the_flow_field&oldid=1256151236">https://en.wikipedia.org/w/index.php?title=Lagrangian_and_Eulerian_specification_of_the_flow_field&oldid=1256151236</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">Categories</a>: <ul><li><a href="/wiki/Category:Fluid_dynamics" title="Category:Fluid dynamics">Fluid dynamics</a></li><li><a href="/wiki/Category:Aerodynamics" title="Category:Aerodynamics">Aerodynamics</a></li><li><a href="/wiki/Category:Computational_fluid_dynamics" title="Category:Computational fluid dynamics">Computational fluid dynamics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_American_English_from_February_2019" title="Category:Use American English from February 2019">Use American English from February 2019</a></li><li><a href="/wiki/Category:All_Wikipedia_articles_written_in_American_English" title="Category:All Wikipedia articles written in American English">All Wikipedia articles written in American English</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_clarification_from_August_2024" title="Category:Wikipedia articles needing clarification from August 2024">Wikipedia articles needing clarification from August 2024</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 8 November 2024, at 14:01<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. 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