Miscible viscous fingering of finite width samples with adsorption

Influence of adsorption on miscible viscous fingering of a finite width sample of higher viscous fluid displaced by a lower viscous fluid is studied in porous media. We assume that the viscosity of the sample depends on the mobile phase concentration $c_m$ of a given solute. This concentration $c_m$ is related to the stationary phase concentration $c_s$ of the solute adsorbed onto the porous matrix by a linear adsorption isotherm model. The model equations governed by Darcy's law coupled with the evolution equation for $c_m$ incorporating a retention factor $K$ are numerically integrated using a Fourier spectral method. In the presence of adsorption, fingering similar to that of the non-adsorptive case is observed but appears on a slower time scale and with a different wavelength. The fingering pattern with adsorption adopts a traveling wave solution with a speed which depends on the adsorption parameter $K$. Fingering contributes transiently to the broadening of the peak in time by increasing its variance in both adsorptive and non-adsorptive cases but the variance with adsorption is always smaller than without adsorption. Relevance of the results is discussed in relation with transport of viscous samples in chromatographic columns with retention.