Fluid mechanics in solidification problems was accurately described to estimate the time of the phase transformation, evolution of velocity, temperature and instantaneous location of the moving interface in cavities. The study includes two cases in which the non-Newtonian rheology of the molten fluid plays a leading role in heat convection during phase change, high temperature energy storage of ternary aluminium alloys, and low temperature fruit juices freezing. Fluid mechanics and energy equations are solved with the finite volume method by an efficient pressure-correction algorithm [1], for a generalized Herschel-Bulkley (HB) rheology model [2]. Fluid mechanics during phase change was characterized by an apparent viscosity changing dynamically with the liquid phase fraction, while the energy equation included an apparent specific heat calculated in terms of a temperature- dependent liquid phase fraction. Numerical results were obtained with a third order accuracy for calculating the fluid acceleration and the rate change of the internal energy, while velocity gradients in the shear rate were estimated with second order accuracy to determine the apparent viscosity of the HB fluids. Effects of power index, 0.1 ≤ n ≤ 1.9, and yield stress (0 ≤ τ0 ≤ 5 Pa) on the fluid dynamics, heat transfer, and solidification were carefully examined for Rayleigh numbers higher than 107. The main findings indicate that using a linear model for the variation of the liquid phase fraction with temperature instead of a highly non-linear one originated a 75% difference in the solidification time when τ0 ≥ 0.5 Pa for Al-Cu-Si alloys. A 50 times increased value of the yield stress, from 0.01 Pa, provoked a high reduction in fluid velocity, deformation and cooling rates, with significant increments in solidification time. Experimental results of temperature in air and liquid foods validated fluid mechanics results of food freezing in turbulent airflow.