Instabilities of particle-laden layers in the stably stratified environment

Abstract

The stability of the interface formed by fine suspended particles is studied through linear stability analysis. Our derivation using the regular perturbation expansion with respect to the particle’s settling velocity shows that the unstable modes are independent of the gravitational settling of individual particles. These modes can be obtained from the six-order ordinary differential equation obtained from the analysis of zero-order quantities. In addition to the four boundary conditions applied at the interface in the traditional Rayleigh-Taylor problem in the semi-infinite domain, two conditions based on the continuity of the concentration of the background stratification agent and its gradient are introduced. Our stability results show transition of modes from a small value in a regime of Rayleigh-Taylor instability to the large values of double-diffusive convection when the background density stratification becomes increasingly significant. In the latter case, our analysis shows growth of small perturbations with dominant wavelengths scaled by the double-diffusion length scale. The transition of unstable modes depends on the density ratio, the Prandtl number of the stratification agent, and the viscosity ratio between the two fluid layers. The analysis is further confirmed by the results from the direct numerical simulation.

Formation of drops and rings in double-diffusive sedimentation

“…These are obtained from the theoretical results in Chandrasekhar (1961) and then rescaled with the double-diffusion length scale L. Figure 24(c,d) show that the dominant RTI modes are less than half of the present dominant modes in both cases, which indicates the dominance of the double-diffusive sedimentation in the present cases. A more detailed derivation for the transition between RTI and double-diffusive convection of the instability modes can be found in a recently submitted paper by Chou et al (2019).…”