Linear stability analysis and direct numerical simulation of two layer   channel flow
  We study the stability of two-fluid flow through a plane channel at Reynolds numbers of a hundred to a thousand in the linear and nonlinear regimes. The two fluids have the same density but different viscosities. The fluids, when miscible, are separated from each other by a mixed layer of small but finite thickness, across which viscosity changes from that of one fluid to that of the other. When immiscible, the interface is sharp. Our study spans a range of Schmidt numbers, viscosity ratios and location and thickness of the mixed layer. A region of instability distinct from that of the Tollmien-Schlichting mode is obtained at moderate Reynolds numbers. We show that the overlap of the layer of viscosity-stratification with the critical layer of the dominant disturbance provides a mechanism for this instability. At very low values of diffusivity, the miscible flow behaves exactly like the immiscible in terms of stability characteristics. High levels of miscibility make the flow more stable. At intermediate levels of diffusivity however, in both linear and non-linear regimes, miscible flow can be more unstable than the corresponding immiscible flow without surface tension. 
Kirti Chandra Sahu and Rama Govindarajan
physics.flu-dyn