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A vector proportional to the local angular velocity of a fluid flow. The vorticity, , is a derived quantity in fluid mechanics, defined, for a flow field with velocity , by Eq. (1).
Closely related to vorticity is the fluid circulation, &Ggr;, defined, for any closed contour, C, in a fluid, by Eq. (2).
Vortex line and vortex tube
A vortex line is defined as a line that is everywhere tangent to the local vorticity vector (analogous to a streamline). A series of adjacent vortex lines is referred to as a vortex tube. The first Helmholtz vortex law states that at any instant in time the circulation about all loops taken around the exterior of a vortex tube is the same. Thus, vortex tubes must either form loops entirely within a fluid or terminate at some fluid boundary. See Vortex
Kelvin's theorem considers how the circulation &Ggr; around a material loop in a fluid (a loop that moves with the fluid) varies in time. Starting with the Navier-Stokes equations, Lord Kelvin showed that if (1) the fluid is inviscid along the loop, (2) the fluid is subject only to potential body forces, and (3) the fluid pressure is a function of density alone, then the rate of change of &Ggr; is 0. In other words, the circulation around a material loop is time-independent. Kelvin's theorem may also be stated slightly differently: subject to the above three constraints, vortex lines are material lines, convected with the local fluid velocity. See Kelvin's circulation theorem
Kelvin's theorem can tell what happens when vorticity is already present in a flow, but it sheds no light on how vorticity is generated. To answer this question, it is useful to consider situations for which Kelvin's theorem is inapplicable: flow with viscosity, with nonpotential body forces, and for which the pressure is not solely a function of the density.
The action of viscosity has two effects on vorticity. One effect of viscosity is to cause the diffusion of vorticity in a fluid. A second effect of viscosity is the generation of vorticity at a wall where there is a pressure gradient at the wall. See Viscosity
A common example of a nonpotential body force is the Coriolis force, which is present in a rotating frame of reference. This force generates vorticity in a fluid, and is a major cause of the large-scale circulation in the atmosphere and oceans.
There are many flows for which the pressure may not be solely a function of the density (so-called baroclinic flows), such as the flow of gas with heat addition and the flow of water with salinity variations. Pressure gradients in such flows generate vorticity. This source of vorticity is called baroclinic torque, and is important in atmospheric flow, buoyancy-driven flow, and oceanographic flow.