Circulation (fluid dynamics)

The circulation is the contour integral of the vector field through a closed path. When currents it is a measure of the vorticity in the area enclosed by the path.

The term is used in the vector analysis in fluid mechanics and electrodynamics. The circulation occurs in particular in the Stokes' theorem, which plays a central role in the electrodynamics.

Mathematical formulation

Is a piecewise smooth, closed and oriented in the way ( of particular importance here is the ) and along this path integrable vector field, ie as

Circulation along.

If a vector field on an oriented and piecewise smooth bounded surface differentiable, then the circulation of along the edge is oriented to related by the theorem of Stokes equal to the surface integral of the rotation of more than:

Example

Circulation of the magnetic field of a current thread

A lying on the z -axis current thread that is traversed in the positive z - direction with the current from the magnetic field

Surrounded. The circulation of this vector field along a circle and an arbitrary positive radius is equal to the current:

This example demonstrates that for the applicability of Stokes' integral theorem the vector field corresponding to a bounded by the closed curve surface must be differentiable. The vector field of this example is not defined on the z- axis. However, the circulation is formed along a circle which encloses the z-axis. The Stokes' integral theorem is therefore not applicable in this case. This is confirmed by the fact that the circulation of along the circle has the nonzero value even though the vector field over its entire domain of definition is irrotational ( for ).