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 $z=\pm \infty$ 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 ${\sigma }_c\left(\mu \right)\le \sigma \le 1$, where ${\sigma }_c\left(\mu \right)<0$ (i.e., counterrotating flows). For $\sigma <0$ the lower fluid can become divided into two distinct recirculation regions between which fluid cannot transfer.

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• Shear flows
• Navier-Stokes equation