Comparison of Korteweg stresses effect on the fingering instability of higher or less viscous miscible slices: Linear stability analysis

Abstract

The dispersion of the localized higher or less viscous sample/slices than the displacing fluid viscosity can be affected by the viscous fingering (VF) that arises at the interfaces of the underlying fluids. VF dynamics are greatly influenced by the viscosity contrast of the two fluids, which in turn again depends upon the concentration gradient between them. However, if the fluid diffuses at a slower rate, the gradient remains very sharp during a longer time period. Such a steep concentration gradient gives rise to a stress, called Korteweg stress, which acts as a stabilizing factor at the miscible diffusive interface of the two fluids. The fingering dynamics have been analyzed through a linear stability analysis (LSA) based on the self-similar-quasi-steady-state-approximation (SS-QSSA) method. In this paper a comparison for the influences of the Korteweg stresses on the onset of instability between high and less viscous miscible slices has been presented. Such miscible slices of high or less viscosity give rise to the opposite sign of the log-mobility ratio R and lead to the existence of VF at opposite interfaces. It is observed that due to the presence of Korteweg stresses the onset of instability delays more when the more viscous fluid displaces a sample of lesser viscous one (R<0R<0) than the case of less viscous fluid displacing a higher viscous sample (R>0R>0). Under both the cases of R>0R>0 and R<0R<0 the unstable regions in the wave number-diffusive time phase plane are found to be asymmetric in the absence of Korteweg stresses and with the increasing influences of the Korteweg stresses the unstable region observed to be more symmetrical. Qualitative analysis has been conducted for various other flow parameters such as Korteweg constant δδ, extent of the finite sample, l. These results are obtained by performing LSA study on the model obtained by coupling the convection-diffusion equation for the concentration with the Darcy-Korteweg equation. The findings of the present study are in good qualitative agreements with the experimental evidence and the nonlinear simulation results available in the literature.