Starting Flow in a Channel With Two Immiscible Fluids

The starting flow due to a sudden pressure gradient in a channel containing two layers of different fluids is studied for the first time. The necessary eigenvalues and eigenfunctions, including orthogonality, for the composite regions are developed. Infinite series analytic solution is obtained for the starting transient. The properties of the instantaneous velocity profiles depend on the thickness ratio of the layers, the viscosity ratio, and the density ratio. Starting times are determined for the important cases of air over water and oil over water. The bulk flow is greatly increased when there exists a low-viscosity layer buffeting the channel wall. An important conclusion is that, in general, Navier's partial slip condition cannot be applied to unsteady starting flows.