300px|thumb|Field lines of a vector field , around the boundary of an open curved surface with infinitesimal line element along boundary, and through its interior with the infinitesimal surface element and the [[unit vector|unit normal to the surface. Top: Circulation is the line integral of around a closed loop . Project along , then sum. Here is split into components perpendicular (⊥) parallel ( ‖ ) to , the parallel components are tangential to the closed loop and contribute to circulation, the perpendicular components do not. Bottom: Circulation is also the flux of vorticity through the surface, and the curl of is heuristically depicted as a helical arrow (not a literal representation). Note the projection of along and curl of 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 electrodynamics, it can be the electric or the magnetic field.
The term circulation was introduced by William Thomson (later Lord Kelvin) in 1869 to denote the line integral of velocity around a closed curve as a kinematic measure of rotational motion in a fluid, independent of any particular application.
In aerodynamics, circulation appears in a more specialised context in relation to the calculation of lift, where it is evaluated on contours enclosing a body under additional flow assumptions. In this context, circulation was first used independently by Frederick Lanchester, Ludwig Prandtl, Martin Kutta and Nikolay Zhukovsky. It is usually denoted by (uppercase gamma).
Definition and properties
If is a vector field and 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 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, is the angle between the vectors and .
The circulation of a vector field around a closed curve is the line integral:
<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.
Uses
Kutta–Joukowski theorem in fluid dynamics
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.
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. 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
- Maxwell's equations
- Biot–Savart law in aerodynamics
- Kelvin's circulation theorem