We consider the linear stability of the cocurrent flow of two fluids of different viscosity in an infinite region (the viscous analogue of the classical Kelvin-Helmholtz problem). Attention is confined to the simplest case, Couette flow, and we solve the problem using both numerical and asymptotic techniques. We find that the flow is always unstable (in the absence of surface tension). The instability arises at the interface between the two fluids and occurs for short wavelengths, when viscosity rather than inertia is the dominant physical effect.
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