Inward solidification of Couette flow in an annulus with translating inner cylinder

An analytical solution is presented for one-dimensional solidification of fully developed, laminar Couette flow within an annular region with viscous dissipation. The solution is valid for small but non-vanishing Stefan numbers. A translating inner cylinder induces the shear-driven motion in the liquid region. A closed-form expression for the instantaneous location of the solid-liquid interface is derived. In addition, expressions for the temperature distribution in the solidified region, temperature distribution in the liquid region, Nusselt number at the solid-liquid interface, and power per unit length are all derived as a function of pertinent dimensionless parameters. The analytical solution reduces to a classical result in the appropriate asymptotic limits.