In fluid dynamics, the Lamb–Oseen vortex models a line vortex that decays due to viscosity. This vortex is named after Horace Lamb and Carl Wilhelm Oseen.

Mathematical description

Oseen looked for a solution for the Navier–Stokes equations in cylindrical coordinates with velocity components of the form where \Gamma is the circulation of the vortex core. Navier-Stokes equations lead to which, subject to the conditions that it is regular at r=0 and becomes unity as, leads to where \nu is the kinematic viscosity of the fluid. At t=0, we have a potential vortex with concentrated vorticity at the z axis; and this vorticity diffuses away as time passes. The only non-zero vorticity component is in the z direction, given by The pressure field simply ensures the vortex rotates in the circumferential direction, providing the centripetal force where ρ is the constant density

Generalized Oseen vortex

The generalized Oseen vortex may be obtained by looking for solutions of the form that leads to the equation Self-similar solution exists for the coordinate, provided , where a is a constant, in which case. The solution for \varphi(t) may be written according to Rott (1958) as where c is an arbitrary constant. For \gamma=0, the classical Lamb–Oseen vortex is recovered. The case \gamma=k corresponds to the axisymmetric stagnation point flow, where k is a constant. When c=-\infty,, a Burgers vortex is a obtained. For arbitrary c, the solution becomes, where \beta is an arbitrary constant. As, Burgers vortex is recovered.