(125b) Effect of Concentration-Dependent Diffusion Coefficient of Nanoparticles on Viscous Fingering Instability

In fluid flow through porous media, one often encounters a situation when one fluid is displaced by another fluid, either miscible or immiscible with the fluid being displaced. If the displacing fluid is less viscous than the displaced fluid, the interface becomes unstable leading to viscous fingering instability which manifests in the form of finger-like patterns of less viscous fluid penetrating into the high viscosity fluid. The viscous fingering instability is prevalent in porous media flow encountered in enhanced oil recovery, liquid chromatography column, CO2 sequestration, and filtration.

The linear stability theory can be employed to examine the onset of instability and predict the growth rate and wavelength of finger formation in the stratified fluid flow through porous media. We adopt the widely used rectilinear Hele-Shaw cell as a prototype for the homogenous porous media flow described using the Darcy’s law. Further, the penetration of the displacing fluid in the miscible displaced fluid within the porous medium is governed by the velocity-dependent dispersion in addition to the molecular diffusion. In the present study, we investigate the viscous fingering stability in the scenario in which the displacing fluid is laden with nanoparticles. The linear stability theory is employed to construct the spectrum of disturbance growth rates using the Chebyshev collocation technique. The largest growth rate represents the rate of finger formation instability. The role of nanoparticles concentration and concentration dependent diffusion coefficient on the onset of viscous fingering instability is analysed. The diffusion coefficient is taken as a function of solute concentration for dilute suspension. In the case of concentrated suspension, particle-particle interaction is significant therefore diffusion coefficient is assumed to be a function of local particle concentration in addition to the solute concentration. In comparison to the constant diffusion coefficient, the incorporation of concentration dependent diffusivity is found to significantly alter not only the growth rate but also the critical wavelength of fingering instability. For dilute suspension, the instability is always found to be weaker as growth rate decreases in the case of non-uniform diffusion coefficient while the stability is observed to be time-dependent for concentrated suspension. The growth rate curve shows a minimum at intermediate time. The mechanisms responsible for stabilization of the flow are described in the terms of modified viscosity profiles changing from monotonic to non-monotonic. The impact of velocity-induced dispersion is also investigated in addition to the concentration-dependent Brownian diffusion coefficient. It is observed that dispersion strengthens the instability, leading to higher growth rate and wavenumber.