Kelvin's circulation theorem

In fluid mechanics, Kelvin's circulation theorem (named after the Irish scientist who published this theorem in 1869) states In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve (which encloses the same fluid elements) moving with the fluid remains constant with time. The theorem was developed by William Thomson, 1st Baron Kelvin. Stated mathematically:

\[\frac{\mathrm{D}\Gamma}{\mathrm{D}t} = 0\]

where \(\Gamma\) is the circulation around a material contour \(C(t)\). Stated more simply this theorem says that if one observes a closed contour at one instant, and follows the contour over time (by following the motion of all of its fluid elements), the circulation over the two locations of this contour are equal.

This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example a coriolis force) or non-barotropic pressure-density relations.

Mathematical proof

The circulation \(\Gamma\) around a close material contour \(C(t)\) is defined by: \[\Gamma(t) = \oint_C \boldsymbol{u} \cdot \boldsymbol{ds}\] where u is the velocity vector, and ds is an element along the closed contour.

The governing equation for an inviscid fluid with a conservative body force is \[\frac{\mathrm{D} \boldsymbol{u}}{\mathrm{D} t} = - \frac{1}{\rho}\boldsymbol{\nabla}p + \boldsymbol{\nabla} \Phi\] where D/Dt is the convective derivative, ρ is the fluid density, p is the pressure and Φ is the potential for the body force. These are the Euler equations with a body force.

The condition of barotropicity implies that the density is a function only of the pressure, i.e. \(\rho=\rho(p)\).

Taking the convective derivative of circulation gives \[ \frac{\mathrm{D}\Gamma}{\mathrm{D} t} = \oint_C \frac{\mathrm{D} \boldsymbol{u}}{\mathrm{D}t} \cdot \boldsymbol{\mathrm{d}s} + \oint_C \boldsymbol{u} \cdot \frac{\mathrm{D} \boldsymbol{\mathrm{d}s}}{\mathrm{D}t}. \]

For the first term, we substitute from the governing equation, and then apply Stokes' theorem, thus: \[ \oint_C \frac{\mathrm{D} \boldsymbol{u}}{\mathrm{D}t} \cdot \boldsymbol{ds} = \int_A \boldsymbol{\nabla} \times \left( -\frac{1}{\rho} \boldsymbol{\nabla} p + \boldsymbol{\nabla} \Phi \right) \cdot \boldsymbol{n} \, \mathrm{d}S = \int_A \frac{1}{\rho^2} \left( \boldsymbol{\nabla} \rho \times \boldsymbol{\nabla} p \right) \cdot \boldsymbol{n} \, \mathrm{d}S = 0. \] The final equality arises since \(\boldsymbol{\nabla} \rho \times \boldsymbol{\nabla} p=0\) owing to barotropicity.

For the second term, we note that evolution of the material line element is given by \[\frac{\mathrm{D} \boldsymbol{\mathrm{d}s}}{\mathrm{D}t} = \left( \boldsymbol{\mathrm{d}s} \cdot \boldsymbol{\nabla} \right) \boldsymbol{u}.\] Hence \[\oint_C \boldsymbol{u} \cdot \frac{\mathrm{D} \boldsymbol{\mathrm{d}s}}{\mathrm{D}t} = \oint_C \boldsymbol{u} \cdot \left[ \left( \boldsymbol{\mathrm{d}s} \cdot \boldsymbol{\nabla} \right) \boldsymbol{u} \right] = \frac{1}{2} \oint_C \boldsymbol{\nabla} \left( |\boldsymbol{u}|^2 \right) \cdot \boldsymbol{\mathrm{d}s} = 0.\] The last equality is obtained by applying Stokes theorem.

Since both terms are zero, we obtain the result \[\frac{\mathrm{D}\Gamma}{\mathrm{D}t} = 0.\]

The theorem also applies to a rotating frame, with a rotation vector \( \boldsymbol{\Omega} \), if the circulation is modified thus:

\[\Gamma(t) = \oint_C (\boldsymbol{u} + \boldsymbol{\Omega} \times \boldsymbol{r}) \cdot \boldsymbol{ds}\]

Here \( \boldsymbol{r} \) is the position of the area of fluid. From Stoke's theorem, this is:

\[\Gamma(t) = \int_A \boldsymbol{\nabla} \times (\boldsymbol{u} + \boldsymbol{\Omega} \times \boldsymbol{r}) \cdot \boldsymbol{n} \, \mathrm{d}S = \int_A (\boldsymbol{\nabla} \times \boldsymbol{u} + 2 \boldsymbol{\Omega}) \cdot \boldsymbol{n} \, \mathrm{d}S\]

See also

• Helmholtz's theorems

Notes

ja:ケルビンの渦定理