View source for Circulation (physics)

{{short description|Line integral of the fluid velocity around a closed curve}}
[[File:General circulation-vorticity diagram.svg|300px|thumb|Field lines of a vector field {{math|'''v'''}}, around the boundary of an open curved surface with infinitesimal line element {{math|''d'''''l'''}} along boundary, and through its interior with {{math|''dS''}} the infinitesimal surface element and {{math|'''n'''}} the [[unit vector|unit]] normal to the surface. '''Top:''' Circulation is the line integral of {{math|'''v'''}} around a closed loop {{math|''C''}}. Project {{math|'''v'''}} along {{math|''d'''''l'''}}, then sum. Here {{math|'''v'''}} is split into components perpendicular (⊥) parallel ( ‖ ) to {{math|''d'''''l'''}}, the parallel components are [[tangent]]ial to the closed loop and contribute to circulation, the perpendicular components do not. '''Bottom:''' Circulation is also the [[flux]] of vorticity {{math|1='''ω''' = '''∇''' × '''v'''}} through the surface, and the [[curl (mathematics)|curl]] of {{math|'''v'''}} is ''heuristically'' depicted as a helical arrow (not a literal representation). Note the projection of {{math|'''v'''}} along {{math|''d'''''l'''}} and curl of {{math|'''v'''}} may be in the negative sense, reducing the circulation.]]

In physics, '''circulation''' is the [[line integral]] of a [[vector field]] around a closed curve embedded in the field. In [[fluid dynamics]], the field is the fluid [[velocity field]]. In [[Electromagnetism|electrodynamics]], it can be the electric or the magnetic field.

In [[aerodynamics]], it finds applications in the calculation of [[Lift (force)|lift]], for which circulation was first used independently by [[Frederick Lanchester]]<ref>{{cite book |last1=Lanchester |first1=Frederick. W |title=AERODYNAMICS |date=1907 |publisher=ARCHIBALD CONSTABLE & CO. |location=London}}</ref>,[[Ludwig Prandtl]],<ref>{{cite book |last1=Prandtl |first1=Ludwig |title=APPLICATIONS OF MODERN HYDRODYNAMICS TO AERONAUTICS |date=1922 |publisher=National Advisory Committee for Aeronautics |location=United States |url=https://ntrs.nasa.gov/api/citations/19930091180/downloads/19930091180.pdf}}</ref> [[Martin Kutta]] and [[Nikolay Zhukovsky (scientist)|Nikolay Zhukovsky]].{{Citation needed|date=November 2020}} It is usually denoted {{math|Γ}} (uppercase [[gamma]]).

==Definition and properties==

If {{math|'''V'''}} is a vector field and {{math|d'''l'''}} is a vector representing the [[Differential (infinitesimal)|differential]] length of a small element of a defined curve, the contribution of that differential length to circulation is {{math|dΓ}}:
<math display="block">\mathrm{d}\Gamma = \mathbf{V} \cdot \mathrm{d}\mathbf{l} = \left|\mathbf{V}\right| \left|\mathrm{d}\mathbf{l}\right| \cos \theta.</math>

Here, {{math|''θ''}} is the angle between the vectors {{math|'''V'''}} and {{math|d'''l'''}}.

The '''circulation''' {{math|Γ}} of a vector field {{math|'''V'''}} around a [[closed curve]] {{math|''C''}} is the [[line integral]]:<ref>{{cite book 
  | title = Introduction to Fluid Mechanics 
  | author1 =  Robert W. Fox 
  | author2 =  Alan T. McDonald
  | author3 =  Philip J. Pritchard 
  | edition = 6
  | publisher = [[John Wiley & Sons|Wiley]]
  | year = 2003
  | isbn = 978-0-471-20231-8
  }}</ref><ref name=":0">{{Cite web| title=The Feynman Lectures on Physics Vol. II Ch. 3: Vector Integral Calculus| url=https://feynmanlectures.caltech.edu/II_03.html|access-date=2020-11-02 | website=feynmanlectures.caltech.edu}}</ref>
<math display="block">\Gamma = \oint_{C}\mathbf{V} \cdot \mathrm d \mathbf{l}.</math>

In a [[conservative vector field]] 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 [[gradient]] of a scalar function, which is called a [[Scalar potential|potential]].<ref name=":0" />
<!--
The dimensions of circulation in fluid dynamics are length squared, divided by time; L2⋅T−1, which is equivalent to velocity times length.
-->
== Relation to vorticity and curl ==

Circulation can be related to [[Curl (mathematics)|curl]] of a vector field {{math|'''V'''}} and, more specifically, to [[vorticity]] if the field is a fluid velocity field,
<math display="block">\boldsymbol{\omega} = \nabla\times\mathbf{V}.</math>

By [[Stokes' theorem]], the [[flux]] of curl or vorticity vectors through a surface ''S'' is equal to the circulation around its perimeter,<ref name=":0" />
<math display="block">\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}</math>

Here, the closed integration path {{math|''∂S''}} is the [[boundary (topology)|boundary]] or perimeter of an open surface {{math|''S''}}, whose infinitesimal element [[Normal (geometry)|normal]] {{math|1=d'''S''' = '''n'''dS}} is oriented according to the [[Right-hand rule#Curve orientation and normal vectors|right-hand rule]]. Thus curl and vorticity are the circulation per unit area, taken around a local infinitesimal loop.

In [[potential flow]] of a fluid with a region of [[vorticity]], all closed curves that enclose the vorticity have the same value for circulation.<ref name="JDA">Anderson, John D. (1984), ''Fundamentals of Aerodynamics'', section 3.16. McGraw-Hill. {{ISBN|0-07-001656-9}}</ref>

== Uses ==

=== Kutta–Joukowski theorem in fluid dynamics ===
{{main|Kutta–Joukowski theorem}}

In fluid dynamics, the [[lift (force)|lift]] per unit span (L') acting on a body in a two-dimensional flow field is directly proportional to the circulation, i.e. it can be expressed as the product of the circulation Γ about the body, the fluid density <math>\rho</math>, and the speed of the body relative to the free-stream <math>v_{\infty}</math>:
<math display="block">L' = \rho v_{\infty} \Gamma</math>

This is known as the Kutta–Joukowski theorem.<ref name="K&S">{{cite book | author1=A.M. Kuethe | title=Foundations of Aerodynamics | author2=J.D. Schetzer | publisher=[[John Wiley & Sons]] | year=1959 | isbn=978-0-471-50952-3 | edition=2 | at=§4.11}}</ref>

This equation applies around airfoils, where the circulation is generated by ''airfoil action''; and around spinning objects experiencing the [[Magnus effect]] where the circulation is induced mechanically. In airfoil action, the magnitude of the circulation is determined by the [[Kutta condition]].<ref name="K&S" />

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.<ref name="JDA" />

Circulation is often used in [[computational fluid dynamics]] as an intermediate variable to calculate forces on an [[airfoil]] or other body.

=== Fundamental equations of electromagnetism ===
In electrodynamics, the [[Faraday's law of induction#Maxwell–Faraday equation|Maxwell-Faraday law of induction]] can be stated in two equivalent forms:<ref>{{Cite web | title=The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction | url=https://feynmanlectures.caltech.edu/II_17.html | access-date=2020-11-02 | website=feynmanlectures.caltech.edu}}</ref> that the curl of the electric field is equal to the negative rate of change of the magnetic field,
<math display="block">\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>

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
<math display="block">\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}.</math>

Circulation of a [[static magnetic field]] is, by [[Ampère's law]], proportional to the total current enclosed by the loop
<math display="block">\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}.</math>

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

==See also==
{{Continuum mechanics| cTopic=Fluid mechanics}}
* [[Maxwell's equations]]
* [[Biot–Savart law#Aerodynamics applications|Biot–Savart law in aerodynamics]]
* [[Kelvin's circulation theorem]]
{{Clear}}

==References==
{{reflist}}

[[Category:Fluid dynamics]]
[[Category:Physical quantities]]
[[Category:Electromagnetism]]