Numerical analysis of a SUSHI scheme for an elliptic-parabolic system modeling miscible fluid flows in porous media

In this paper, we prove the convergence of a schema using stabilisation and hybrid interfaces of partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations:an anisotropic diffusion equation on the pressure and a convection-diffusion dispersion equation on the concentration of invading fluid. The anisotropicdiffusion operators in both equations require special care while discretizing bya finite volume method SUSHI. Later, we present some numerical experiments.

Interfacial phenomena in immiscible liquids subjected to vibrations in microgravity

We consider the response to periodic forcing between 5 Hz and 50 Hz of an interface separating immiscible fluids under the microgravity conditions of a parabolic flight. Two pairs of liquids with viscosity ratios differing by one order of magnitude are investigated. By combining experimental data with numerical simulations, we describe a variety of dynamics including harmonic and subharmonic (Faraday) waves, frozen waves and drop ejection, determining their thresholds and scaling properties when possible. Interaction between these various modes is facilitated in microgravity by the relative ease with which the interface can move, altering its orientation with respect to the forcing axis. The effects of key factors controlling pattern selection are analysed, including vibrational forcing, viscosity ratio, finite-size effects and residual gravity. Complex behaviour often arises with features on several spatial scales, such as Faraday waves excited on the interface of a larger columnar structure that develops due to the frozen wave instability – this type of state was previously seen in miscible fluid experiments but is described for the first time here in the immiscible case.

Thermal convection instability of two miscible viscous fluids in a Hele-Shaw cell

The onset of convection in two superimposed miscible fluid layers is investigated in the configuration of a geometric Hele-Shaw cell using linear stability analysis. The two fluids have different densities. We neglect the surface tension and chemical diffusion at the interface which is assumed of small amplitude. We consider only the asymptotic case, where the Prandtl number’s order is of the order of unity or larger than unity. We show, in the Hele-Shaw configuration, which can simulate convection in porous media, that the onset of convection can be either stationary or oscillatory depending on the Buoyancy number, B (the ratio of the stabilizing chemical density anomaly to the destabilizing thermal density anomaly), which depends on the viscosity and layer height ratios. When the buoyancy number is lower than a critical value, Bc, oscillating convection occurs in the whole cell height, however beyond Bc, the stratified regime develops without deformation of the interface and convection occurs separately in each layer. In this paper, the transition from oscillatory regime to stratified regime is visualised by using the streamlines at the onset of convection