Existence of Invariant Region to Damköhler number in a Reactive Viscous Fingering*

Chemical reactions can have a profound impact on hydrodynamic instability by modifying the viscosity and density at the interface between two fluids [1]. A prominent example of this phenomenon is viscous fingering (VF), which occurs within a porous medium when a less viscous fluid displaces a more viscous fluid [2]. It is prevalent in various transport phenomena, such as secondary oil recovery, chromatography separation, and CO₂ sequestration [2]. We study a reactive displacement that involves the chemical reaction A + B → C with miscible fluids undergoing radial flow. The flow dynamics are modeled using Darcy’s law coupled with convection-reaction-diffusion equations. The viscosity depends on the concentrations of

the reactants and product and is characterized by two dimensionless parameters, R_b and R_c [3]. We solve this system using a hybrid scheme comprising the pseudospectral method and the compact finite difference method [4]. Numerical and experimental results have shown that chemical reactions can induce instability and radial flow requires a minimum viscosity contrast to trigger the instability [4, 5]. Further, through linear stability analysis and nonlinear simulations, we illustrate the impact of the Damköhler number (Da) on viscous fingering for various values of (R_b, R_c). Interestingly, we found an invariant region in the (R_b, R_c) phase plane that remains unaffected by changes in Da [6].