Book review

Abstract

Exact similarity solutions for the impingement of two viscous, immiscible oblique stagnation flows forming a flat interface are given. The problem is governed by three parameters: the ratios of density ρ = ρ1ρ2 and of viscosity μ = μ1μ2 of the two fluids and R = tanθ1tanθ2 where θ1 and θ2 are the asymptotic angles of the incident streamlines in each fluid layer. For given values of ρ, μ, and θ2, the compatible flows in the lower fluid, as measured by the strain rate ratio β = β1β2 of the two fluids and the asymptotic angle of incidence θ1, are found such that the interface remains horizontal in a uniform gravitational field. For ρ = 1, explicit solutions show that a family of co-current and counter-current shears supporting a flat interface exist for all finite, nonzero values of R. For ρ ≠ 1, the normal stress interfacial boundary conditions restricts the flow to a unique combination of asymptotic far-field shear and Hiemenz stagnation-point flow in each fluid layer. The displacement thicknesses in each layer are always positive when the fluid densities are not equal, but vanish simultaneously as ρ → 1. At each value of ρ the interfacial velocities increase with increasing viscosity ratio μ. As a generalization of the present oblique two-fluid stagnation-point flow problem, we discuss how the flat interface may be inclined with respect to the horizontal in a uniform gravitational field.