THE STABILITY OF A VISCOUS HETEROGENEOUS SHEAR FLOW,

A numerical method is used to compute the stability of a shear layer embedded between two homogeneous fluid regions of different densities that are moving at uniform but different velocities. The mean flow is specified by a hyperbolic tangent velocity profile and an exponential of a hyperbolic tangent density profile. Numerical results are presented that show the changes in the stability of the layer with the Reynolds number, Froude number, wavenumber of the disturbance, and the gradients of shear and density. Two modes of instability are found one propagates upstream relative to the moving fluid, the other propagates downstream at a velocity that is always less than the average of the two outer regions. Viscosity tends to stabilize waves that are already stable and to destabilize waves that are unstable. The neutral stability curves appear to have only one branch therefore, no critical Reynolds numbers were found. Gravity tends to stabilize one mode and to destabilize the other but when gravity forces are very large, both modes are stabilized. One mode was unstable when the minimum Richardson number across the shear layer exceeded 14 therefore, the critical Richardson number for a viscous fluid must be larger than that for an inviscid fluid. Author