Oblique two-fluid stagnation-point flow

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 = ρ 1 ρ 2 and of viscosity μ = μ 1 μ 2 of the two fluids and R = tanθ 1 tanθ 2 where θ 1 and θ 2 are the asymptotic angles of the incident streamlines in each fluid layer. For given values of ρ, μ, and θ 2, the compatible flows in the lower fluid, as measured by the strain rate ratio β = β 1 β 2 of the two fluids and the asymptotic angle of incidence θ 1, 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.