Instability at the interface between two shearing fluids in a channel

We consider the linear stability Couette flow composed of two immiscible fluids in layers. The fluids have different viscosities but the same densities. It is known that the short wavelength asymptotics of the interfacial mode for the bounded and unbounded problems are identical. In this paper, we show that there is a critical Reynolds number above which the interfacial modes of the unbounded problem are approximated by those of the bounded problem for wavelengths outside the asymptotic short wavelength range. A full linear analysis reveals unstable situations missed out by the long and short wavelength asymptotic analyses but which have comparable orders of magnitudes for the growth rates. The inclusion of a density difference as well as a viscosity difference is discussed. In particular, the arrangement with the heavier fluid on top can be linearly stable in the presence of gravity if the viscosity stratification, volume ratio, surface tension, Reynolds number and Froude number are favorable.