Viscous fingering in reaction-diffusion systems

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Abstract
The problem of viscous fingering is studied in the presence of simultaneous chemical reactions. The flow is governed by the usual Darcy equations, with a concentration-dependent viscosity. The concentration field in turn obeys a reaction–convection–diffusion equation in which the rate of chemical reaction is taken to be a function of the concentration of a single chemical species and admits two stable equilibria separated by an unstable one. The solution depends on four dimensionless parameters: R, the log mobility ratio, Pe, the Peclet number, α, the Damköhler number or dimensionless rate constant, and d, the dimensionless concentration of the unstable equilibrium. The resulting nonlinear partial differential equations are solved by direct numerical simulation over a reasonably wide range of Pe, α, and d. We find new mechanisms of finger propagation that involve the formation of isolated regions of either less or more viscous fluid in connected domains of the other. Both the mechanism of formation of these regions and their effects on finger propagation are studied in some detail.