Cross-diffusion-driven gravitational instability in a Hele-Shaw cell saturated with a ternary solution

The effect of cross diffusion on the onset of the gravitational instabilities in a Hele-Shaw cell saturated with a ternary solution is analyzed. Based on the linear stability theory, new stability equations are derived in the global domain and then transformed into the similar domain. These stability equations are solved by employing various methods such as an initial value problem approach and quasi-steady state approximations (QSSA's). Through the initial growth rate analysis without the QSSA, we prove that initially the system is unconditionally stable. However, the QSSA in the global domain showed that the system can be initially unstable for a certain condition. Based on the QSSA in the similar domain (QSSAζ), we obtain the critical time for the onset of instability motion. As expected, the higher |δ₂₁β| makes the system more unstable, i.e., accelerates the onset of instability motion; here δ₂₁ and β represent the normalized cross diffusion coefficient and the ratio of densification coefficients, respectively. Based on the linear analysis, fully nonlinear analyses are also conducted by using the Fourier spectral method. The present nonlinear analyses show that the double-diffusive and diffusive-layer convection-type of instabilities are possible for the positive and negative δ₂₁β-values, respectively. From the present nonlinear analysis, the system having δ₂₂ > 1 prefers the instabilities with a larger wavelength than the system having δ₂₂ < 1. Here δ₂₂ is the normalized normal diffusion coefficient of component B.