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 electrodynamics, it can be the electric or the magnetic field.
In aerodynamics, it finds applications in the calculation of lift, for which circulation was first used independently by Frederick Lanchester,<ref>Template:Cite book</ref> Ludwig Prandtl,<ref>Template:Cite book</ref> Martin Kutta and Nikolay Zhukovsky.<ref>Anderson, John D. (1984), Fundamentals of Aerodynamics, Section 2.13, McGraw Hill</ref> It is usually denoted Template:Math (uppercase gamma).
Definition and propertiesEdit
If Template:Math is a vector field and Template:Math is a vector representing the differential length of a small element of a defined curve, the contribution of that differential length to circulation is Template:Math: <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, Template:Math is the angle between the vectors Template:Math and Template:Math.
The circulation Template:Math of a vector field Template:Math around a closed curve Template:Math is the line integral:<ref>Template:Cite book</ref><ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</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 potential.<ref name=":0" />
Relation to vorticity and curlEdit
Circulation can be related to curl of a vector field Template:Math 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 Template:Math is the boundary or perimeter of an open surface Template:Math, whose infinitesimal element normal Template:Math is oriented according to the 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. Template:ISBN</ref>
UsesEdit
Kutta–Joukowski theorem in fluid dynamicsEdit
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In fluid dynamics, the lift 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 <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">Template:Cite book</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 electromagnetismEdit
In electrodynamics, the Maxwell-Faraday law of induction can be stated in two equivalent forms:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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.