A Simplified Analytical Modeling of One‐Dimensional Solidification in Plane Couette Flow With Temperature‐Dependent Viscosity

Abstract

ABSTRACT This work is conducted to study the solidification of a liquid undergoing a plane Couette flow between two flat and parallel plates with variable dynamic viscosity over temperature. The flow and thermal characteristics of the circulating liquid during the solidification phase change were predicted by solving analytically the non‐dimensional governing conservation equations of continuity, momentum and energy. The obtained analytical solution expresses the transient dimensionless temperatures of the different phases (solid and liquid), the instantaneous dimensionless solid layer thickness, the dimensionless thickness of the solid layer at the steady state, and the standardized instantaneous power required to maintain the uniform motion of the moving wall. The main parameters affecting the thermal behavior of the problem are then investigated through a parametric study, which reveals that the dimensionless solid layer thickness increases with time and reaches its maximum at the steady‐state regime. For fixed Brinkman number, Br, and dimensionless temperature, , the higher the Biot number, Bi, the higher the dimensionless solid layer thickness. However, increasing the Brinkman, Br, number and dimensionless temperature, , leads to a decrease in the dimensionless solid layer thickness. For 0 < Br ≤ 10, the relative decrease in the solid layer thickness at the steady state for is estimated at 75.53%, 74.92%, and 74.4%, respectively, for Bi = 5, 65.49%, 64.16%, and 62.93% for Bi = 20, and 62%, 60.67%, and 59.26% for the limit case, that is, . When the Brinkman number, Br, and Biot number, Bi, are held constant, increasing the moving wall dimensionless temperature, , leads to a reduction in the required power and therefore in the energy consumed, by as much as about 60%.