This work treats the vertically stratified system of two homogeneous fluid layers confined between horizontal infinite rotating disks which rotate steadily about a common vertical axis. Allowance is made for uniform injection of fluid at either disk. With appropriate restrictions on disk rotational speeds and injection rates a flat interface is possible, and the problem admits similarity solutions to the Navier-Stokes equations of the Kármán-Bödewadt-Batchelor variety. This type of flow allows for a uniformly accessible surface of interphase mass and heat transfer at the two-fluid interface, and, with that as the primary motivation, the present work provides exploratory numerical solutions of the above equations for both corotation and counter-rotation combined with injection.
A linearized theory is given for the case of nearly rigid rotation, with explicit analytical results for the large-Reynolds-number boundary-layer limit. Also, we offer a theoretical discussion of the inviscid limit for arbitrary rotation and injection rates. Based on the type of Euler-cell solutions identified in previous work, we derive the remarkably simple formula
\[
\rho_1\omega_1^2\cot^2\frac{\omega_1d_1}{V_1} = \rho_2\omega^2_2\cot^2\frac{\omega_2d_2}{V_2}
\]
connecting densities ρ, depths d, rotation speeds ω and injection velocities V.
Sample calculations and comparisons are given for property ratios typical of water-kerosene layers. In this case, the linearized theory works exceedingly well for corotation with small injectional Rossby numbers V/ωd. The simple inviscid theory cited above shows excellent agreement with the numerical computations for Reynolds numbers greater than 500 and for Rossby numbers > 1/π, corresponding to strong blowing in the inviscid regime. The larger-wavelength inviscid cell structure appears to provide the kind of stagnation-flow pattern essential to the application envisaged.