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 is the circulation of the vortex core. Navier-Stokes equations lead to
which, subject to the conditions that it is regular at and becomes unity as , leads to
where is the kinematic viscosity of the fluid. At , we have a potential vortex with concentrated vorticity at the axis; and this vorticity diffuses away as time passes.
The only non-zero vorticity component is in the 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 obtained by looking for solutions of the form
that leads to the equation
Self-similar solution exists for the coordinate , provided , where is a constant, in which case . The solution for may be written according to Rott (1958) as
where is an arbitrary constant. For , the classical Lamb–Oseen vortex is recovered. The case corresponds to the axisymmetric stagnation point flow, where is a constant. When , , a Burgers vortex is a obtained. For arbitrary , the solution becomes , where is an arbitrary constant. As , Burgers vortex is recovered.
See also
The Rankine vortex and Kaufmann (Scully) vortex are common simplified approximations for a viscous vortex.