Solidification of circular Couette flow with viscous dissipation

Abstract

A semi-analytic solution is presented for the solidification of laminar circular Couette flow within a one-dimensional annular region with a rotating outer cylinder and stationary inner cylinder. Viscous dissipation in the liquid is taken into account. Closed-form expressions for the dimensionless temperature distribution in the solid and liquid regions, Nusselt number at the solid–liquid interface, dimensionless power and torque per unit length, dimensionless steady-state freeze front location, and dimensionless pressure distribution in the liquid are derived as a function of liquid-to-solid thermal conductivity ratio, Brinkman number, annulus radius ratio, and Stefan number, which is assumed to be small (<0.1) but non-vanishing. The instantaneous dimensionless solid–liquid interface location is determined using numerical integration. The results show that the power and torque requirements can increase by a factor of greater than seven for a thin-gap annulus (radius ratio >0.9). It is also shown that the size of the liquid region within the annular gap has a more controlling influence on the solidification rate at latter times (when the Brinkman number is small) while a Brinkman number of order unity has a dominant influence at earlier times.