Vorticity

In continuum mechanics, the vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. In continuum mechanics, the vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the flow. Conceptually, vorticity could be determined by marking parts of a continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow. The vorticity vector would be twice the mean angular velocity vector of those particles relative to their center of mass, oriented according to the right-hand rule. This quantity must not be confused with the angular velocity of the particles relative to some other point. More precisely, the vorticity is a pseudovector field ω→, defined as the curl (rotational) of the flow velocity u→ vector. The definition can be expressed by the vector analysis formula: where ∇ is the del operator. The vorticity of a two-dimensional flow is always perpendicular to the plane of the flow, and therefore can be considered a scalar field. The vorticity is related to the flow's circulation (line integral of the velocity) along a closed path by the (classical) Stokes' theorem. Namely, for any infinitesimal surface element C with normal direction n→ and area dA, the circulation dΓ along the perimeter of C is the dot product ω→ ∙ (dA n→) where ω→ is the vorticity at the center of C. Many phenomena, such as the blowing out of a candle by a puff of air, are more readily explained in terms of vorticity, rather than the basic concepts of pressure and velocity. This applies, in particular, to the formation and motion of vortex rings. The name vorticity was created by Horace Lamb in 1916. In a mass of continuum that is rotating like a rigid body, the vorticity is twice the angular velocity vector of that rotation. This is the case, for example, in the central core of a Rankine vortex. The vorticity may be nonzero even when all particles are flowing along straight and parallel pathlines, if there is shear (that is, if the flow speed varies across streamlines). For example, in the laminar flow within a pipe with constant cross section, all particles travel parallel to the axis of the pipe; but faster near that axis, and practically stationary next to the walls. The vorticity will be zero on the axis, and maximum near the walls, where the shear is largest.