MISCIBLE DISPLACEMENT IN A HELE-SHAW CELL

We formulated a theory of simple mixtures of incompressible miscible liquids in terms of the mass averaged velocity a and the solenoidal volume averaged velocity W We derived simplified equations for miscible displacement in a Hele-Shaw cell. We obtained a steady solution of these equations corresponding to displacement under gravity with prescribed values of concentration and mean normal stress at the inlet and exit of the cell. We studied the stability of this steady flow. This differs from previous works which treat the stability of unsteady miscible displacement using a quasi-static assumption and classical equations based on div u = 0. In our problem, replacing u with W gives rise to a difference in the mean normal stress, which alters the pressure drop across the cell and changes the velocity of free fall. We found that the stability equations are the same in the two formulations, but the boundary conditions are slightly different; however the difference will be small if diffusion is slow or the thickness of the cell is small. The results show that steady miscible displacement in a Hele-Shaw cell is stable to long and short waves. Within certain ranges of parameters, the displacement of glycerin into water can be unstable. This instability is basically of a Rayleigh-Taylor type, regularized by diffusion. As the diffusion parameter S becomes smaller, the waves of disturbances become finer and are confined to an increasingly thin diffusion layer. Water displacing glycerin is always stable. This is due to the fact that the steady equilibrium profile is not steep enough to create a fingering instability.