Solidification of Two-Dimensional Viscous, Incompressible Stagnation Flow

Abstract

The history of the study of fluid solidification in stagnation flow is limited to a few cases. Among these few studies, only some articles have considered the fluid viscosity and yet pressure variations along the thickness of the viscous layer have not been taken into account and the energy equation has been assumed to be one-dimensional. In this study the solidification of stagnation flows is modeled as an accelerated flat plate moving toward an impinging fluid. The unsteady momentum equations, taking the pressure variations along viscous layer thickness into account, are reduced to ordinary differential equations by the use of proper similarity variables and are solved by using a fourth-order Runge-Kutta integrating method at each prescribed interval of time. In addition, the energy equation is numerically solved at any step for the known velocity and the problem is presented in a two-dimensional Cartesian coordinate. Comparisons of these solutions are made with existing special ranges of past solutions. The fluid temperature distribution, transient velocity component distribution, and, most important of all the rate of solidification or the solidification front are presented for different values of nondimensional Prandtl and Stefan numbers. The results show that an increase of the Prandtl numbers (up to ten times) or an increase of the heat diffusivity ratios (up to two times) causes a decrease of the ultimate frozen thickness by almost half, while the Stefan number has no effect on this thickness and its effect is only on the freezing time.