Effect of cross-diffusion on the gravitational instability in a ternary mixture: Asymptotic and linear analyses

To consider the effect of cross diffusion more rigorously, the onset and the growth of the gravitational instabilities in a Hele-Shaw cell saturated with ternary solution are analyzed by considering all cross diffusion coefficients. Through the asymptotic analysis, we identify the double-diffusive (DD)-, diffusive-layer convection (DLC)- and extended double diffusive (EDD)-type instability regimes. To support the asymptotic stability analysis, new linear stability equations are derived in the global domain and then transformed into the similar domain. In the similar domain, we prove that initially the system is unconditionally stable. For transient stability analysis, the linear stability equations are solved by employing quasi-steady state approximations (QSSA's). To avoid the limit of the conventional QSSAz, we obtain the critical time for the onset of instability motion using the QSSA in the similar domain (QSSAζ).