The Kelvin circulation theorem
(published by Lord Kelvin
in 1867) is an important result in fluid dynamics
. Under the assumption the fluid is inviscid it tells us that circulation is in some sense conserved.
Kelvin circulation theorem:
Let C(t) be a closed curve that is fixed in a fluid with velocity field u. Then the circulation
K = ∫C(t)u.dl
An integral is a limit of approximating sums, so in order to differentiate it you essentially do it term-by-term, which works out as integrating the partial derivative of the integrand round the curve. Since in this case the curve C(t) that we are integrating round moves with the fluid we have to use slightly different methods in order to get the same term-by-term effect. In each term we have to take into account that the integrand changes due to the change in position as well as the change in time, and also that the material line element dl changes. Therefore we use substantial derivatives rather than partial derivatives and include dl in the differentiation.
Using the Euler momentum equation and the expression for the substantial derivative of a material line element gives
dK/dt = ∫C(t)D(u.dl)/Dt =
∫C(t)(Du/Dt).dl + u.(Ddl/Dt) =
∫C(t)(∇(p+G)).dl + ∫C(t)u.((dl.∇)u) =
∫C(t)(∇(p + G + u2/2)).dl = 0
since the line integral of a gradient of a single-valued function round a closed curve vanishes. QED.
One of the important points to note about the proof is that we do not make any appeal to the curve C(t) being spannable by a surface, as we would have to do e.g. in order to apply Stokes' theorem. This means that we use the condition of zero viscosity only on the curve C(t) itself. The theorem will therefore hold true in fluid flows with non-zero viscosity too, as long as the viscous effects are negligable near C(t).
An example of how the circulation theorem can be used is to understand how circulation is generated round the wing of an airplane. We analyse the flow as being two-dimensional, which is reasonable since the wing is wide.
The wing is designed so as to leave behind a vortex when it is set in motion. By Stokes' theorem there is some circulation K round a curve enclosing the vortex. According to the circulation theorem there is no circulation round a curve enclosing both the wing and the vortex, since the circulation was initially zero. If we consider a curve that encloses the wing but not the starting vortex it must therefore have a circulation -K round it.
By the Kutta-Joukowski lift theorem this causes a lift proportional to K on the wing.