Kelvin's circulation theorem

In an inviscid barotropic fluid subject to conservative body forces, circulation around a closed material curve remains constant as the fluid moves.

Circulation: The line integral of a velocity field around a closed curve, measuring net rotation or flow along the curve. The theorem states that this quantity is conserved for a loop moving with the fluid.
Vorticity: The curl of a velocity field, representing the local rotational tendency of a fluid. Stokes' theorem makes circulation the integrated vorticity through a surface.
Vortex line: A curve everywhere tangent to the vorticity vector in a fluid. Circulation conservation helps explain why vortex structures move with ideal fluid motion.
Viscosity: A fluid's resistance to deformation and relative motion between neighboring layers. Viscous stresses can change material-loop circulation, unlike the theorem's inviscid setting.
Material curve: A curve made of the same fluid particles as it moves and deforms with the flow. The conservation law applies to a loop that moves with the fluid, not a fixed spatial loop.
Baroclinicity: A condition in which pressure and density gradients are not parallel. Baroclinic fluids can generate vorticity and fall outside the theorem's barotropic assumption.
Vortex tube: A tube whose sides are formed by vortex lines in a fluid. Its circulation remains conserved under the theorem's assumptions.
Baroclinic vorticity generation: Vorticity production caused by misaligned pressure and density gradients. This source changes circulation when the fluid is not barotropic.
Euler equations: The equations of motion for an ideal fluid with negligible viscosity. Their momentum balance supplies the time derivative of circulation.
Lagrangian description: A description of motion that follows individual material particles through time. The moving curve is defined by following the same fluid particles.

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