Unstable miscible displacements in radial flow with chemical reactions

Abstract

The effects of the A + B → C chemical reaction on miscible viscous fingering in a radial source flow are analysed using linear stability theory and numerical simulations. This flow and transport problem is described by a system of nonlinear partial differential equations consisting of Darcy’s law for an incompressible fluid coupled with nonlinear advection–diffusion–reaction equations. For an infinitely large Péclet number (Pe), the linear stability equations are solved using spectral analysis. Further, the numerical shooting method is used to solve the linearized equations for various values of Pe including the limit Pe → ∞. In the linear analysis, we aim to capture various critical parameters for the instability using the concept of asymptotic instability, i.e. in the limit τ → ∞, where τ represents the dimensionless time. We restrict our analysis to the asymptotic limit Da∗ (= Daτ ) → ∞ and compare the results with the non-reactive case (Da = 0) for which Da∗ = 0, where Da is the Damköhler number. In the latter case, the dynamics is controlled by the dimensionless parameter RPhys = −(RA − βRB). In the former case, for a fixed value of RPhys, the dynamics is determined by the dimensionless parameter RChem = −(RC − RB − RA). Here, β is the ratio of reactants’ initial concentration and RA, RB and RC are the log-viscosity ratios. We perform numerical simulations of the coupled nonlinear partial differential equations for large values of Da. The critical values RPhys,c and RChem,c for instability decrease with Pe and they exhibit power laws in Pe. In the asymptotic limit of infinitely large Pe they exhibit a power-law dependence on Pe (RChem,c ∼ Pe−1/2 as Pe → ∞) in both the linear and nonlinear regimes