Circulation around a closed curve is

$$\Gamma(t)=\oint_{C(t)}\mathbf v\cdot d\mathbf r.$$

If $C(t)$ is a material loop, every point of the curve moves with the fluid. Kelvin's circulation theorem identifies conditions under which the circulation of that moving loop cannot change.

Rate of change of circulation

Differentiate the line integral while following the material loop. The result can be written

$$\frac{d\Gamma}{dt} =\oint_{C(t)}\frac{D\mathbf v}{Dt}\cdot d\mathbf r +\oint_{C(t)}\mathbf v\cdot d\mathbf v.$$

The second integral is

$$\oint_{C(t)}d\left(\frac12|\mathbf v|^2\right)=0$$

because the curve is closed. Therefore

$$\boxed{ \frac{d\Gamma}{dt} =\oint_{C(t)}\frac{D\mathbf v}{Dt}\cdot d\mathbf r }.$$

The circulation changes only if the fluid acceleration has a nonzero closed-loop integral.

Insert the inviscid momentum equation

For inviscid flow with a conservative body force,

$$\frac{D\mathbf v}{Dt} =-\frac1\rho\nabla p-\nabla\Phi.$$

The body-force contribution integrates to zero around a closed loop:

$$\oint_C\nabla\Phi\cdot d\mathbf r=0.$$

The pressure term also becomes a pure gradient if the fluid is barotropic, meaning that pressure is a function of density alone:

$$\boxed{p=p(\rho)}.$$

Define

$$H(p)=\int^p\frac{dp'}{\rho(p')}.$$

Then

$$\nabla H =\frac1\rho\nabla p.$$

Hence

$$\frac{d\Gamma}{dt} =-\oint_C\nabla(H+\Phi)\cdot d\mathbf r=0.$$

Therefore

$$\boxed{\frac{d\Gamma}{dt}=0}$$

and

$$\boxed{\Gamma(t)=\text{constant}}.$$

This is Kelvin's circulation theorem.

A constant-density incompressible flow satisfies the same required gradient property directly:

$$\frac1\rho\nabla p =\nabla\left(\frac p\rho\right).$$

Thus Kelvin's result also holds for a smooth inviscid constant-density flow with conservative body forces, even though pressure need not be a thermodynamic function of density.

What the theorem actually says

Kelvin's theorem applies to a loop that moves and deforms with the fluid. It does not say that circulation around an arbitrary fixed curve in space is constant.

It also does not say that the velocity field itself is constant. A material loop may stretch, shrink, and change shape while its circulation remains unchanged.

Worked example: a shrinking circular material loop

Suppose an axisymmetric inviscid flow contains a circular material loop of radius $R$ around an axis. Assume the velocity on the loop has uniform tangential component $v_\theta$.

Its circulation is

$$\Gamma=\oint_C\mathbf v\cdot d\mathbf r =2\pi Rv_\theta.$$

Initially,

$$R_1=0.20,\mathrm m,$$

$$v_{\theta1}=1.5,\mathrm{m/s}.$$

Thus

$$\Gamma =2\pi(0.20)(1.5) =0.60\pi,\mathrm{m^2/s}.$$

If the material loop later contracts to

$$R_2=0.10,\mathrm m,$$

Kelvin's theorem gives

$$2\pi R_2v_{\theta2}=0.60\pi.$$

Therefore

$$v_{\theta2} =\frac{0.60\pi}{2\pi(0.10)} =\boxed{3.0,\mathrm{m/s}}.$$

The radius halves while the tangential speed doubles so that circulation remains constant.

When circulation can change

Kelvin's theorem depends on its assumptions. Circulation can be generated or destroyed when, for example,

• viscous stresses matter;
• the fluid is not barotropic, so pressure and density variations cannot be represented by one pressure potential;
• nonconservative body forces act;
• shocks, singular interfaces, or other nonsmooth regions invalidate the smooth-loop argument.

For a nonbarotropic inviscid fluid, misaligned pressure and density gradients can generate vorticity through baroclinic torque. This is one important mechanism by which the barotropic assumption can fail.

Kelvin's theorem is therefore not a universal statement that fluid circulation is always conserved. It is a precise dynamical result showing that, under ideal conditions, pressure and conservative body forces cannot change the circulation of a material loop.