Effect of viscosity in the dynamics of two point vortices: Exact results

An exact, unstationary, two-dimensional solution of the Navier-Stokes equations for the flow generated by two point Vortices is obtained. The viscosity v is introduced as a Brownian motion in the Hamiltonian dynamics of point vortices. The point vortices execute a stochastic motion whose probability density can be computed from a Fokker-Planck equation, equivalent to the original Navier-Stokes equation. The derived solution describes, in particular, the merging process of two Lamb vortices, and the development of the characteristic spiral structure in the topology of the vorticity. The viscous effects are thoroughly investigated by an asymptotic analysis of the solution. In particular, the selection mechanism of a specific pattern among the infinity satisfying the v=0 (Euler) equation is discussed.