Circulation (physics)

If {{math|V}} is a vector field and {{math|dl}} is a vector representing the differential length of a small element of a defined curve, the contribution of that differential length to circulation is {{math|dΓ}}:

$$\backslash mathrm\{d\}\backslash Gamma\; =\; \backslash mathbf\{V\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}\; =\; \backslash left|\backslash mathbf\{V\}\backslash right|\; \backslash left|\backslash mathrm\{d\}\backslash mathbf\{l\}\backslash right|\; \backslash cos\; \backslash theta.$$

Here, {{math|θ}} is the angle between the vectors {{math|V}} and {{math|dl}}.

The circulation {{math|Γ}} of a vector field {{math|V}} around a closed curve {{math|C}} is the line integral:{{cite book

| title = Introduction to Fluid Mechanics

| author1 = Robert W. Fox

| author2 = Alan T. McDonald

| author3 = Philip J. Pritchard

| edition = 6

| publisher = Wiley

| year = 2003

| isbn = 978-0-471-20231-8

}}{{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}}

$$\backslash Gamma\; =\; \backslash oint\_\{C\}\backslash mathbf\{V\}\; \backslash cdot\; \backslash mathrm\; d\; \backslash mathbf\{l\}.$$

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.

Circulation can be related to curl of a vector field {{math|V}} and, more specifically, to vorticity if the field is a fluid velocity field,

$$\backslash boldsymbol\{\backslash omega\}\; =\; \backslash nabla\backslash times\backslash mathbf\{V\}.$$

By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter,

$$\backslash Gamma\; =\; \backslash oint\_\{\backslash partial\; S\}\; \backslash mathbf\{V\}\backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}\; =\; \backslash iint\_S\; \backslash nabla\; \backslash times\; \backslash mathbf\{V\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{S\}=\backslash iint\_S\; \backslash boldsymbol\{\backslash omega\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{S\}$$

Here, the closed integration path {{math|∂S}} is the boundary or perimeter of an open surface {{math|S}}, whose infinitesimal element normal {{math|1=dS = ndS}} 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.Anderson, John D. (1984), Fundamentals of Aerodynamics, section 3.16. McGraw-Hill. {{ISBN|0-07-001656-9}}

{{main|Kutta–Joukowski theorem}}

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 $\backslash rho$, and the speed of the body relative to the free-stream $v\_\{\backslash infty\}$:

$$L\text{'}\; =\; \backslash rho\; v\_\{\backslash infty\}\; \backslash Gamma$$

This is known as the Kutta–Joukowski theorem.{{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}}

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.

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.

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

In electrodynamics, the Maxwell-Faraday law of induction can be stated in two equivalent forms:{{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}} that the curl of the electric field is equal to the negative rate of change of the magnetic field,

$$\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$$

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

$$\backslash oint\_\{\backslash partial\; S\}\; \backslash mathbf\{E\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}\; =\; \backslash iint\_S\; \backslash nabla\backslash times\backslash mathbf\{E\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{S\}\; =$$

- \frac{\mathrm{d}}{\mathrm{d}t} \int_{S} \mathbf{B} \cdot \mathrm{d}\mathbf{S}.

Circulation of a static magnetic field is, by Ampère's law, proportional to the total current enclosed by the loop

$$\backslash oint\_\{\backslash partial\; S\}\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}\; =\; \backslash mu\_0\; \backslash iint\_S\; \backslash mathbf\{J\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{S\}\; =\; \backslash mu\_0\; I\_\backslash text\{enc\}.$$

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

{{Continuum mechanics| cTopic=Fluid mechanics}}

• Maxwell's equations
• Biot–Savart law in aerodynamics
• Kelvin's circulation theorem

{{Clear}}

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Category:Fluid dynamics

Category:Physical quantities

Category:Electromagnetism