Stability Regions of Cellular Fluid Flow

Abstract

It is well known that the Navier-Stokes equation of motion for a homogeneous incompressible fluid has a unique stationary solution, if the body forces acting on the fluid and the prescribed values of the velocity at the boundary are sufficiently small. We shall call this solution the primary solution. While the primary solution is a possible solution for all values of the prescribed parameters, it is not unique anymore when the parameters exceed certain critical values. The simplest kind of nonuniqueness is caused by stationary secondary solutions for which the principle of exchange of stability holds, i.e. the secondary solution branching off the primary solution becomes stable in place of the primary solution for values of the relevant parameter beyond the critical branching point. Physically this process corresponds to the appearance of a secondary motion superimposed onto the primary state of the fluid system. The secondary motion is usually char-acterized by a typical scale which is not shown by the primary state and which is exhibited in the form of a cell-like subdivision of the flow field. It is for this reason that the secondary motion for which the principle of exchange of stability holds has been called cellular flow.