Linear stability analysis of Korteweg stresses effect on miscible viscous fingering in porous media

Abstract

Viscous fingering (VF) is an interfacial hydrodynamic instability phenomenon observed when a fluid of lower viscosity displaces a higher viscous one in a porous media. In miscible viscous fingering, the concentration gradient of the undergoing fluids is an important factor, as the viscosity of the fluids are driven by concentration. Diffusion takes place when two miscible fluids are brought in contact with each other. However, if the diffusion rate is slow enough, the concentration gradient of the two fluids remains very large during some time. Such steep concentration gradient, which mimics a surface tension type force, called the effective interfacial tension, appears in various cases such as aqua-organic, polymer-monomer miscible systems, etc. Such interfacial tension effects on miscible VF is modeled using a stress term called Korteweg stress in the Darcy's equation by coupling with the convection-diffusion equation of the concentration. The effect of the Korteweg stresses at the onset of the instability has been analyzed through a linear stability analysis using a self-similar Quasi-steady-state-approximation (SS-QSSA) in which a self-similar diffusive base state profile is considered. The quasi-steady-state analyses available in literature are compared with the present SS-QSSA method and found that the latter captures appropriately the unconditional stability criterion at an earlier diffusive time as well as in long wave approximation. The effects of various governing parameters such as log-mobility ratio, Korteweg parameters, disturbances' wave number, etc., on the onset of the instability are discussed for, (i) the two semi-infinite miscible fluid zones and (ii) VF of the miscible slice cases. The stabilizing property of the Korteweg stresses effect is observed for both of the above mentioned cases. Critical miscible slice lengths are computed to have the onset of the instability for different governing parameters with or without Korteweg stresses. These stabilizing properties of the Korteweg stresses captured in this present study are in agreement with the numerical simulations of fully nonlinear problem and the experimental observations reported in the literature.