Miscible displacements in Hele-Shaw cells: two-dimensional base states and their linear stability

Miscible fingering in a Hele-Shaw cell is studied by means of Stokes simulations and linear stability analysis. The two-dimensional simulations of miscible displacements in a gap indicate the existence of a quasi-steady state near the tip of the displacement front for sufficiently large Peclet numbers and viscosity ratios, in agreement with earlier work by other authors. The front thickness of this quasi-steady state is seen to scale with Pe(-1/2), while it depends only weakly on the viscosity ratio. The nature of the viscosity-concentration relationship is found to have a significant influence on the quasi-steady state. For the exponential relationship employed throughout most of the investigation, we find that the tip velocity increases with Pe for small viscosity ratios, while it decreases with Pe for large ratios. In contrast, for a linear viscosity-concentration relationship the tip velocity is seen to increase with Pe for all viscosity ratios. The simulation results suggest that in the limit of high Pe and large viscosity contrast, the width and tip velocity of the displacement front asymptote to the same values as their immiscible counterparts in the limit of large capillary numbers. In a subsequent step, the stability of the quasi-steady front to spanwise perturbations is examined, based on the three-dimensional Stokes equations. For all values of Pe, the maximum growth rate is found to increase monotonically with the viscosity ratio. The influence of Pe on the growth of the instability is non-uniform. For mild viscosity contrasts, a larger Pe is found to be destabilizing, while for large viscosity contrasts an increase in Pe has a slightly stabilizing influence. A close inspection of the instability eigenfunction reveals the presence of two sets of counter-rotating roll-like structures, with axes aligned in the cross-gap and streamwise directions, respectively. The former lead to the periodic acceleration and deceleration of the front, while the latter result in the widening and narrowing of the front. These roll-like structures are aligned in such a way that the front widens where it speeds up, and narrows where it slows down. The findings from the present stability analysis are discussed and compared with their Darcy counterparts, as well as with experimental data by other authors for miscible and immiscible flows.