Steady flow of one uniformly rotating fluid layer above another immiscible uniformly rotating fluid layer

The steady laminar flow of two immiscible, uniformly rotating fluid layers is studied and exact similarity solutions of the axisymmetric Navier-Stokes equations in cylindrical polar coordinates are found. The similarity solutions occur with a flat interface at z = 0 under the parameter restriction that σ²ρ = 1 where σ is the ratio of the fluid angular velocities at z = ±∞ and ρ is the density ratio of the two fluids. Under this restriction the problem reduces to one with two independent parameters, σ and μ, which is the viscosity ratio of the fluids. Numerical results of the resulting system of ODEs are found for selected values of μ and σ, and it is shown that similarity solutions exist for σc(μ)≤ σ ≤ 1 where σc(μ) ˂ 0 (i.e. counter-rotating flows). For σ ˂ 0 the lower fluid can become divided into two distinct recirculation regions between which fluid cannot transfer.