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Description
Reactive viscous fingering in porous media differs fundamentally from nonreactive miscible displacement because chemical reactions can sustain traveling fronts with fixed width and speed. As shown by A. De Wit (Phys. Rev. Lett., 2001), this property leads to time-independent dispersion relations under steady forcing, in contrast to purely diffusive fronts whose stability evolves in time. Building on this framework, Swernath and Pushpavanam (J. Chem. Phys., 2007) analyzed reactive viscous fingering with concentration and temperature-dependent viscosity under isothermal and adiabatic conditions, while retaining the assumption of constant injection and a steady traveling-wave base state.
In the present work, we extend these classical formulations by investigating reactive miscible displacement under time-dependent injection. A Darcy–reaction–diffusion–energy model is considered for a horizontal porous medium, with viscosity depending on concentration and temperature. While the steady-injection limit admits a traveling-wave base state consistent with earlier studies, temporal modulation of the injection velocity renders the base state intrinsically unsteady in the moving frame. Furthermore, the base state concentration is strongly governed by the ratio of the hydrodynamic to the chemical time scale (Damköhler number), transitioning from diffusion- to reaction-dominated morphologies, whereas the temperature is independent of this parameter.
Linear stability analysis performed about this time-dependent base state shows that injection unsteadiness fundamentally alters the classical dispersion-curve picture. Growth rates become time dependent, leading to shifts in the onset time of instability and in the most unstable wavelength relative to constant injection. Depending on the modulation parameters, injection unsteadiness can either delay the development of viscous fingering or enhance perturbation growth over specific time intervals. In adiabatic systems, these effects interact with thermal diffusion and temperature-dependent viscosity, modifying the stabilizing or destabilizing trends reported under steady injection. In the limit of zero modulation amplitude, the results recover the established steady-injection behavior, providing a consistent generalization of existing theory.
These results demonstrate that injection protocol design constitutes an additional control mechanism for reactive fingering and mixing in porous media.
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