Stability of heterogeneous flows to nonaxisymmetric disturbances

Abstract

The stability of heterogeneous fluids has been extensively investigated by TAYLOl~ (1931), SYI~OE (1933), MILES (196i), HOW~a~D and GuPTA (1962) and others. These results have also been extended to include the stability Of cylindrical masses of fluid but mostly for axisymmetric disturbances. When the fluid is homogeneous and incompressible and is having a 'solid body' rotation the stability for nonaxisymmetric disturbances has been investigated by HOWARD and GO-eTA (1962), LvI)WlEO (1961) and PEDLEY (1968). In the case of a homogeneous fluid, in addition to the solid body rotation, the presence of even ~ small axial shear makes the system unstable as has been shown by LIIDWlEG (1961) for a narrow gap and by PEDLEu (1968) without this restriction. In this paper we consider the stability for non-axisymmetrical disturbances of a cylindrical mass of heterogeneous fluid, with an exponential variation of density in the radial direction and having both axial and azimuthal velocities. Assuming the dependence of the radial perturbation velocity on r ~o be of the form u = ?l-'~H(r), we discuss two cases m = t/2 and 1. The general stability criteria, for both cases, have been derived. In the second case (m-----1), as an illustration the general stability criterion is applied to the Poiseuille type flow and a bound for instability is obtained. The growth rate of the most rapidly growing disturbances is also determined. 2. Formulation ol the Problem