Hyperbolicity and Change of Type in the Flow Between Rotating Cylinders When the Inner Cylinder Is Corrugated

Abstract

We may think of this problem as the flow in a gear box. We get this flow by perturbing Couette flow between two concentric rotating cylinders, putting a wave on the inner cylinder. As in the previous examples, the vorticity equation changes type from elliptic to hyperbolic whenever the velocity at some point in the domain exceeds the speed of shear waves into rest. There are two cases: the inner cylinder rotates and the outer cylinder rotates. The isovorticity lines even in weakly subcritical flow lie in directions swept out by characteristics. This direction is uniquely determined in the regions where the two families lie in the same direction. In the strongly supercritical case, high Mach numbers, one finds that hyperbolicity dominates the flow field. The isovorticity lines are oriented along forward facing characteristics and the vorticity of the perturbed motion is then swept out along these characteristics.

This chapter is based on the thesis of Riccius [1989b].