The Hydrodynamic Stability of Two Viscous Incompressible Fluids in Parallel Uniform Shearing Motion

The stability problem of a thin film of a viscous incompressible fluid bounded on one side by another more viscous and less dense incompressible fluid of semi-infinite extent and on the other side by a fixed wall, where both fluids are in steady motion parallel to their interface and each fluid has a linear velocity profile, is solved for large values of the Reynolds number and small values of the viscosity ratio. Neutral stability curves of the Reynolds number versus the wave number are presented, parametrized with either the density ratio or the viscosity ratio as the family parameters.