In fluid mechanics

Fluid mechanics

Fluid mechanics is the study of fluids and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion...

, Kelvin's circulation theorem (named after the Irish scientist who published this theorem in 1869) states In an inviscid, barotropic

Barotropic

In meteorology, a barotropic atmosphere is one in which the pressure depends only on the density and vice versa, so that isobaric surfaces are also isopycnic surfaces . The isobaric surfaces will also be isothermal surfaces, hence the geostrophic wind is independent of height...

 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

William Thomson, 1st Baron Kelvin

William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...

. Stated mathematically:


where is the circulation around a material contour . 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 around a closed material contour is defined by:
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
where D/Dt is the convective derivative

Convective derivative

The material derivative is a derivative taken along a path moving with velocity v, and is often used in fluid mechanics and classical mechanics...

, ρ is the fluid density, p is the pressure and Φ is the potential for the body force. These are the Euler equations

Euler equations

In fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass , momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. Historically,...

 with a body force.

The condition of barotropicity implies that the density is a function only of the pressure, i.e. .

Taking the convective derivative of circulation gives

For the first term, we substitute from the governing equation, and then apply Stokes' theorem

Stokes' theorem

In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...

, thus:
The final equality arises since owing to barotropicity.

For the second term, we note that evolution of the material line element is given by
Hence
The last equality is obtained by applying Stokes theorem.

Since both terms are zero, we obtain the result

The theorem also applies to a rotating frame, with a rotation vector , if the
circulation is modified thus:


Here is the position of the area of fluid. From Stoke's theorem, this is: