Boundary layers at the interface of two different shear flows

We present solutions for the boundary layer between two uniform shear flows flowing in the same direction. In the upper layer, the flow has shear strength a, fluid density ρ1, and kinematic viscosity ν1, while the lower layer has shear strength b, fluid density ρ2, and kinematic viscosity ν2. Similarity transformations reduce the boundary-layer equations to a pair of ordinary differential equations governed by three dimensionless parameters: the shear strength ratio γ = b/a, the density ratio ρ = ρ2/ρ1, and the viscosity ratio ν = ν2/ν1. Further analysis shows that an affine transformation reduces this multi-parameter problem to a single ordinary differential equation which may be efficiently integrated as an initial-value problem. Solutions of the original boundary-value problem are shown to agree with the initial-value integrations, but additional dual and quadruple solutions are found using this method. We argue on physical grounds and through bifurcation analysis that these additional solutions are not tenable. The present problem is applicable to the trailing edge flow over a thin airfoil with camber.