A class of exact solutions of the Navier-Stokes equation

The unsteady problem of a viscous incompressible flow in free space is investigated, and a class of exact solutions of the Navier-Stokes equation is given for a general initial condition. This flow field represents several shear layers superimposed on an irrotational, three-dimensional straining flow. This solution incorporates the three main features of vortex motion, i.e., stretching, convection and viscous diffusion of vorticity. The solution is exemplified for several kinds of initial condition. One of them represents a flow approaching a steady state in which the above three effects are brought to an equilibrium. Another solution shows collision of two shear layers in various arrangements. Two parallel shear layers merge into a single layer, while two antiparallel shear layers in which the vortices in each layer are in the opposite directions disappear as pair cancellation, and two layers merge like vectors by oblique collision. It is also shown that N shear layers, in general, merge like vectors.