Theoretical and numerical analyses of the effect of cross-diffusion on the gravitational instability in ternary mixtures

Abstract

To understand the effect of cross diffusion on the onset and the growth of the gravitational instabilities in a ternary solution more rigorously, theoretical and numerical analyses are conducted by considering all cross diffusion coefficients. We clearly showed that the stable concentration field is possible even for the complex-eigenvalues systems, such as acetone(1)-benzene(2)-CCl4(common solvent) ternary mixture. By employing Faddeeva functions, we extended the previous asymptotic analysis into the complex eigenvalue systems. In addition, by considering all possible cross diffusion coefficients, i.e., complex conjugate, and real and distinct eigenvalues system, we derive the linear stability equations in the uncoupled form, solve them, and conduct nonlinear numerical simulations employing the linear stability result as an initial condition. Through the present asymptotic, linear and nonlinear analyses, the instability motions in the double-diffusive (DD), diffusive-layer convection (DLC) and extended double diffusive (EDD) regimes are clearly identified. The present asymptotic, linear and nonlinear analyses support each other and are in good agreement with the previous theoretical, numerical and experimental work.