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 sigma(2)rho = 1, where sigma is the ratio of the fluid angular velocities at z = +/-infinity and rho is the density ratio of the two fluids. Under this restriction the problem reduces to one with two independent parameters sigma and mu, the viscosity ratio of the fluids. Numerical results of the resulting system of ordinary differential equations are found for selected values of mu and sigma, and it is shown that similarity solutions exist for sigma(c)(mu) <= sigma <= 1, where sigma(c)(mu) < 0 (i.e., counterrotating flows). For sigma < 0 the lower fluid can become divided into two distinct recirculation regions between which fluid cannot transfer.