Numerical simulations using a pseudo-spectral method based on the Hartley transform are conducted to examine the viscous fingering instability of miscible flow displacements involving heat transfer between the fluids. The flow geometry consists of a rectilinear homogeneous porous media where the fluids and the media are assumed in thermal equilibrium. The physical model is represented using a continuity equation, Darcy's law, and volume-averaged forms of the convection-diffusion equation for mass and energy balance. The dependence of the fluids viscosities on both temperature and concentration is represented by two dimensionless parameters; the solutal viscosity ratio; βC, and the thermal viscosity ratio; βT. In particular, the study analyzed the effects of varying important parameters such as the Lewis number, the thermal lag coefficient, and the thermal mobility ratio on the dynamics of the flow. The development of new finger structures is analyzed by examining contours of the concentration and characterizing them qualitatively through a spectral analysis of the average concentration and an analysis of the variations of the mixing length and the relative contact area. It is found that diffusion of heat and its redistribution between the fluid and solid phases have a strong effect on the growth of the fingers. Furthermore, close and intricate interactions between the fluid and thermal fronts affect the subsequent development of instability, even when the two fronts are separated, with the thermal front lagging behind the fluid one.
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Islam et al. (2010) studied this question.