Linear stability of the dissipative, two-fluid, cylindrical Couette problem. Part 1. The stably-stratified hydrodynamic problem

Abstract

The stability of a two-fluid vortex is studied as a step towards understanding the separation and containment problems in a gaseous-core nuclear rocket. In particular, the linear hydrodynamic stability of two incompressible, immiscible, viscous fluids occupying separate annular regions of a cylindrical Couette apparatus is considered. Neglecting surface tension and gravity, a conservative assumption, the governing equations for arbitrary jumps in fluid properties are derived and numerical solutions to the resultant eigenvalue problems obtained. Results are presented for the effect on neutral stability of density and viscosity jumps, varying gap widths, and differing fluid-fluid interfacial positions. The solutions are limited, however, to the case of stably stratified fluids and a stationary outer cylinder.Two separate modes (multiple eigenvalues) have been discovered for all cases in which two fluids, differing in any property, are present. A rationale is presented for this phenomenon as well as for most of the other observed results.While most results are believed to be manifestations of the Taylor cylindrical Couette instability phenomenon, evidence is presented for the existence of additional ‘hidden’ eigenvalues attributable to the classical Kelvin–Helmholtz and/or the recently reported Yih viscosity-stratification instability phenomena.