Vector Field Flow through and around a Circle

The vector flow across a circle depends on the divergence of the given field: it is always zero when there are no sinks, sources, or singularities. Similarly, the vector flow around the circle depends on rotation (or curl). Here the circle is taken as parametrized in the counterclockwise sense.

The flow of the vector field F = ((Subscript[f, 1]) (x, y), (Subscript[f, 2 ])(x, y) ) along the curve C = {(x(t),y(t) ), a≤t≤b} is given by the line integral (∫_C)F·d(OverVector[r]), which can also be written as (∫_a^b)(Subscript[f, 1])(x(t),y(t)) x'(t)+(Subscript[f, 2])(x(t),y(t))y'(t)dt.

Using Green's theorem, the flow along a closed curve C is (∫_C)F. d(OverVector[r]) = ∫(∫_( D))rot FdA, where D is the region enclosed by the curve C and the rotation of the vector field is rot F =(∂/( ∂x))(Subscript[f, 2])(x,y)-(∂/(∂y))(Subscript[f, 1])(x,y).

The flow of the vector field F across the given closed curve is

∫(∫_( D))div FdA = (∫_a^b)-(Subscript[f, 2])(x(t),y(t)) x'(t)+(Subscript[f, 1])(x(t),y(t))y'(t)dt,

where the divergence of the vector field F is div F= (∂/(∂x))(Subscript[f, 1])(x,y)+(∂/(∂y))(Subscript[f, 2])(x,y).

If the divergence or rotation is constant, the flow across or around any closed curve is constant, too. For example, if the divergence depends only on y, like  F=( 1,(x^2)+(y^2) ), dragging the circle to the left or right does not change the flow.

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Contributed by: Gosia Konwerska

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