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<title>Circulation (physics)</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Circulation (physics)</span></span>

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<p>In physics, <b>circulation</b> is the <a href="Line_integral" title="Line integral">line integral</a> of a <a href="Vector_field" title="Vector field">vector field</a> around a closed curve embedded in the field. In <a href="Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, the field is the fluid <a href="Velocity_field" class="mw-redirect" title="Velocity field">velocity field</a>. In <a href="Electromagnetism" title="Electromagnetism">electrodynamics</a>, it can be the electric or the magnetic field.

</p><p>In <a href="Aerodynamics" title="Aerodynamics">aerodynamics</a>, it finds applications in the calculation of <a href="Lift_(force)" title="Lift (force)">lift</a>, for which circulation was first used independently by <a href="Frederick_Lanchester" class="mw-redirect" title="Frederick Lanchester">Frederick Lanchester</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="Ludwig_Prandtl" title="Ludwig Prandtl">Ludwig Prandtl</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="Martin_Kutta" title="Martin Kutta">Martin Kutta</a> and <a href="Nikolay_Zhukovsky_(scientist)" title="Nikolay Zhukovsky (scientist)">Nikolay Zhukovsky</a>.<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> It is usually denoted <span class="texhtml">Γ</span> (uppercase <a href="Gamma" title="Gamma">gamma</a>).

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<div class="mw-heading mw-heading2"><h2 id="Definition_and_properties">Definition and properties</h2></div>

<p>If <span class="texhtml"><b>V</b></span> is a vector field and <span class="texhtml">d<b>l</b></span> is a vector representing the <a href="Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">differential</a> length of a small element of a defined curve, the contribution of that differential length to circulation is <span class="texhtml">dΓ</span>:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \Gamma =\mathbf {V} \cdot \mathrm {d} \mathbf {l} =\left|\mathbf {V} \right|\left|\mathrm {d} \mathbf {l} \right|\cos \theta .}</annotation>

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</p><p>Here, <span class="texhtml"><i>θ</i></span> is the angle between the vectors <span class="texhtml"><b>V</b></span> and <span class="texhtml">d<b>l</b></span>.

</p><p>The <b>circulation</b> <span class="texhtml">Γ</span> of a vector field <span class="texhtml"><b>V</b></span> around a <a href="Closed_curve" class="mw-redirect" title="Closed curve">closed curve</a> <span class="texhtml"><i>C</i></span> is the <a href="Line_integral" title="Line integral">line integral</a>:<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>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma =\oint _{C}\mathbf {V} \cdot \mathrm {d} \mathbf {l} .}</annotation>

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</p><p>In a <a href="Conservative_vector_field" title="Conservative vector field">conservative vector field</a> this integral evaluates to zero for every closed curve. That means that a line integral between any two points in the field is independent of the path taken. It also implies that the vector field can be expressed as the <a href="Gradient" title="Gradient">gradient</a> of a scalar function, which is called a <a href="Scalar_potential" title="Scalar potential">potential</a>.<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>

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<div class="mw-heading mw-heading2"><h2 id="Relation_to_vorticity_and_curl">Relation to vorticity and curl</h2></div>

<p>Circulation can be related to <a href="Curl_(mathematics)" title="Curl (mathematics)">curl</a> of a vector field <span class="texhtml"><b>V</b></span> and, more specifically, to <a href="Vorticity" title="Vorticity">vorticity</a> if the field is a fluid velocity field,

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {V} .}</annotation>

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</p><p>By <a href="Stokes'_theorem" title="Stokes' theorem">Stokes' theorem</a>, the <a href="Flux" title="Flux">flux</a> of curl or vorticity vectors through a surface <i>S</i> is equal to the circulation around its perimeter,<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>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma =\oint _{\partial S}\mathbf {V} \cdot \mathrm {d} \mathbf {l} =\iint _{S}\nabla \times \mathbf {V} \cdot \mathrm {d} \mathbf {S} =\iint _{S}{\boldsymbol {\omega }}\cdot \mathrm {d} \mathbf {S} }</annotation>

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</p><p>Here, the closed integration path <span class="texhtml"><i>∂S</i></span> is the <a href="Boundary_(topology)" title="Boundary (topology)">boundary</a> or perimeter of an open surface <span class="texhtml"><i>S</i></span>, whose infinitesimal element <a href="Normal_(geometry)" title="Normal (geometry)">normal</a> <span class="texhtml">d<b>S</b> = <b>n</b>dS</span> is oriented according to the <a href="Right-hand_rule#Curve_orientation_and_normal_vectors" title="Right-hand rule">right-hand rule</a>. Thus curl and vorticity are the circulation per unit area, taken around a local infinitesimal loop.

</p><p>In <a href="Potential_flow" title="Potential flow">potential flow</a> of a fluid with a region of <a href="Vorticity" title="Vorticity">vorticity</a>, all closed curves that enclose the vorticity have the same value for circulation.<sup id="cite_ref-JDA_6-0" class="reference"><a href="#cite_note-JDA-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Kutta–Joukowski_theorem_in_fluid_dynamics">Kutta–Joukowski theorem in fluid dynamics</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Kutta%E2%80%93Joukowski_theorem" title="Kutta–Joukowski theorem">Kutta–Joukowski theorem</a></div>

<p>In fluid dynamics, the <a href="Lift_(force)" title="Lift (force)">lift</a> per unit span (L') acting on a body in a two-dimensional flow field is directly proportional to the circulation. Lift per unit span can be expressed as the product of the circulation Γ about the body, the fluid density <span class="mwe-math-element mwe-math-element-inline"><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 }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math></span><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" 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 }" loading="lazy"></span>, and the speed of the body relative to the free-stream <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\infty }}">

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<annotation encoding="application/x-tex">{\displaystyle v_{\infty }}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle L'=\rho v_{\infty }\Gamma }</annotation>

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</p><p>This is known as the Kutta–Joukowski theorem.<sup id="cite_ref-K&S_7-0" class="reference"><a href="#cite_note-K&S-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>This equation applies around airfoils, where the circulation is generated by <i>airfoil action</i>; and around spinning objects experiencing the <a href="Magnus_effect" title="Magnus effect">Magnus effect</a> where the circulation is induced mechanically. In airfoil action, the magnitude of the circulation is determined by the <a href="Kutta_condition" title="Kutta condition">Kutta condition</a>.<sup id="cite_ref-K&S_7-1" class="reference"><a href="#cite_note-K&S-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>The circulation on every closed curve around the airfoil has the same value, and is related to the lift generated by each unit length of span. Provided the closed curve encloses the airfoil, the choice of curve is arbitrary.<sup id="cite_ref-JDA_6-1" class="reference"><a href="#cite_note-JDA-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>Circulation is often used in <a href="Computational_fluid_dynamics" title="Computational fluid dynamics">computational fluid dynamics</a> as an intermediate variable to calculate forces on an <a href="Airfoil" title="Airfoil">airfoil</a> or other body.

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<div class="mw-heading mw-heading3"><h3 id="Fundamental_equations_of_electromagnetism">Fundamental equations of electromagnetism</h3></div>

<p>In electrodynamics, the <a href="Faraday's_law_of_induction#Maxwell–Faraday_equation" title="Faraday's law of induction">Maxwell-Faraday law of induction</a> can be stated in two equivalent forms:<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> that the curl of the electric field is equal to the negative rate of change of the magnetic field,

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<annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}</annotation>

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</p><p>or that the circulation of the electric field around a loop is equal to the negative rate of change of the magnetic field flux through any surface spanned by the loop, by Stokes' theorem

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<annotation encoding="application/x-tex">{\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\iint _{S}\nabla \times \mathbf {E} \cdot \mathrm {d} \mathbf {S} =-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} .}</annotation>

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</p><p>Circulation of a <a href="Static_magnetic_field" class="mw-redirect" title="Static magnetic field">static magnetic field</a> is, by <a href="Amp%C3%A8re's_law" class="mw-redirect" title="Ampère's law">Ampère's law</a>, proportional to the total current enclosed by the loop

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<annotation encoding="application/x-tex">{\displaystyle \oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}\iint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =\mu _{0}I_{\text{enc}}.}</annotation>

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</p><p>For systems with electric fields that change over time, the law must be modified to include a term known as Maxwell's correction.

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D{\frac {d\varphi }{dx}}}">

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<annotation encoding="application/x-tex">{\displaystyle J=-D{\frac {d\varphi }{dx}}}</annotation>

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</math></span><img src="./1856f88def2056f28ed27c7d31180a6240820ea6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.874ex; height:5.509ex;" alt="{\displaystyle J=-D{\frac {d\varphi }{dx}}}" loading="lazy"></span><div class="sidebar-caption"><a href="Fick's_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></div></td></tr><tr><td class="sidebar-content-with-subgroup">

Conservations</th></tr><tr><td class="sidebar-content hlist">

<li><a href="Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">Momentum</a></li>

<li><a href="Conservation_of_energy" title="Conservation of energy">Energy</a></li></ul></td>

</tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;">

Inequalities</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Clausius–Duhem (entropy)</a></li></ul></td>

</tr></tbody></table></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="Solid_mechanics" title="Solid mechanics">Solid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist">

<ul><li><a href="Deformation_(physics)" title="Deformation (physics)">Deformation</a></li>

<li><a href="Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a>

<ul><li><a href="Linear_elasticity" title="Linear elasticity">linear</a></li></ul></li>

<li><a href="Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a></li>

<li><a href="Hooke's_law" title="Hooke's law">Hooke's law</a></li>

<li><a href="Stress_(mechanics)" title="Stress (mechanics)">Stress</a></li>

<li><a href="Strain_(mechanics)" title="Strain (mechanics)">Strain</a>

<ul><li><a href="Finite_strain_theory" title="Finite strain theory">Finite strain</a></li>

<li><a href="Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain</a></li></ul></li>

<li><a href="Compatibility_(mechanics)" title="Compatibility (mechanics)">Compatibility</a></li>

<li><a href="Bending" title="Bending">Bending</a></li>

<li><a href="Contact_mechanics" title="Contact mechanics">Contact mechanics</a>

<ul><li><a href="Frictional_contact_mechanics" title="Frictional contact mechanics">frictional</a></li></ul></li>

<li><a href="Material_failure_theory" title="Material failure theory">Material failure theory</a></li>

<li><a href="Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a></li></ul>

</div></div></div></td>

</tr><tr><td class="sidebar-content-with-subgroup">

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<a href="Fluid" title="Fluid">Fluids</a></th></tr><tr><td class="sidebar-content">

<ul><li><a href="Hydrostatics" title="Hydrostatics">Statics</a> <b>·</b> <a href="Fluid_dynamics" title="Fluid dynamics">Dynamics</a></li>

<li><a href="Archimedes'_principle" title="Archimedes' principle">Archimedes' principle</a> <b>·</b> <a href="Bernoulli's_principle" title="Bernoulli's principle">Bernoulli's principle</a></li>

<li><a href="Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li>

<li><a href="Hagen%E2%80%93Poiseuille_equation" title="Hagen–Poiseuille equation">Poiseuille equation</a> <b>·</b> <a href="Pascal's_law" title="Pascal's law">Pascal's law</a></li>

<li><a href="Viscosity" title="Viscosity">Viscosity</a>

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<li><a href="Buoyancy" title="Buoyancy">Buoyancy</a> <b>·</b> <a href="Mixing_(process_engineering)" title="Mixing (process engineering)">Mixing</a> <b>·</b> <a href="Pressure" title="Pressure">Pressure</a></li></ul>

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<ul><li><a href="Atmosphere" title="Atmosphere">Atmosphere</a></li>

<li><a href="Boyle's_law" title="Boyle's law">Boyle's law</a></li>

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<a href="Plasma_(physics)" title="Plasma (physics)">Plasma</a></th></tr></tbody></table></div></div></td>

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<ul><li><a href="Electrorheological_fluid" title="Electrorheological fluid">Electrorheological</a></li>

<li><a href="Magnetorheological_fluid" title="Magnetorheological fluid">Magnetorheological</a></li>

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<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist">

<ul><li><a href="Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li>

<li><a href="Robert_Boyle" title="Robert Boyle">Boyle</a></li>

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<li><a href="Jacques_Charles" title="Jacques Charles">Charles</a></li>

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<li><a href="Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Gay-Lussac</a></li>

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<li><a href="Claude-Louis_Navier" title="Claude-Louis Navier">Navier</a></li>

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<li><a href="Sir_George_Stokes%2C_1st_Baronet" title="Sir George Stokes, 1st Baronet">Stokes</a></li>

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<ul><li><a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a></li>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFLanchester1907" class="citation book cs1">Lanchester, Frederick. W (1907). <i>AERODYNAMICS</i>. London: ARCHIBALD CONSTABLE & CO.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFPrandtl1922" class="citation book cs1">Prandtl, Ludwig (1922). <a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/api/citations/19930091180/downloads/19930091180.pdf"><i>APPLICATIONS OF MODERN HYDRODYNAMICS TO AERONAUTICS</i></a> <span class="cs1-format">(PDF)</span>. United States: National Advisory Committee for Aeronautics.</cite></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">Anderson, John D. (1984), <i>Fundamentals of Aerodynamics</i>, Section 2.13, McGraw Hill</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"><cite id="CITEREFRobert_W._FoxAlan_T._McDonaldPhilip_J._Pritchard2003" class="citation book cs1">Robert W. Fox; Alan T. McDonald; Philip J. Pritchard (2003). <i>Introduction to Fluid Mechanics</i> (6 ed.). <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">Wiley</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-20231-8</bdi>.</cite></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"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_03.html">"The Feynman Lectures on Physics Vol. II Ch. 3: Vector Integral Calculus"</a>. <i>feynmanlectures.caltech.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-02</span></span>.</cite></span>

</li>

<li id="cite_note-JDA-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-JDA_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-JDA_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Anderson, John D. (1984), <i>Fundamentals of Aerodynamics</i>, section 3.16. McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-001656-9</bdi></span>

</li>

<li id="cite_note-K&S-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-K&S_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-K&S_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFA.M._KuetheJ.D._Schetzer1959" class="citation book cs1">A.M. Kuethe; J.D. Schetzer (1959). <i>Foundations of Aerodynamics</i> (2 ed.). <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. §4.11. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-50952-3</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_17.html">"The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction"</a>. <i>feynmanlectures.caltech.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-11-02</span></span>.</cite></span>

</li>

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