Time-dependent Solidification in a Square Cavity with a Temperature-modulated Liquid Layer Cooled from Above

In this work, we investigate the dynamic response of a solid phase formed during unidirectional solidification below a cooling top wall of a square cavity filled with distilled water and subjected to timevarying heating temperatures at its bottom wall. Assuming a quasisteady state condition, we have formulated a one-dimensional model that predicts the average thickness of the forming solid phase. While non-dimensionalizing the model equations, three important non-dimensional parameters are identified, namely the Biot number based on the solid phase thickness at steady state, the Stefan number based on the temperature difference between the cooling upper wall and the liquid temperatures, and the Stefan number based on the heating bottom wall and the liquid temperatures. A perturbation solution of the quasi-steady state formulation has been developed for small amplitude temperature variations on the heating bottom wall. The perturbation solution has been extensively tested against a full two-dimensional numerical solution that uses boundary-tracking techniques for tracing the solid-liquid interface, and good general agreements have been confirmed. The solid phase thickness variation with the time and its phase delays have been expressed as a function of the non-dimensional angular frequency of the heating bottom temperature and the above-mentioned three non-dimensional parameters. The implications of this study and its potential for employment in education and practical engineering have been also addressed.