Large Cross-diffusion Research Articles

A Parabolic Cross-Diffusion System for Granular Materials

We analyze a cross-diffusion system of parabolic equations for the relative concentration and the dynamic repose angle of a mixture of two different granular materials in a long rotating drum. The main feature of the system is the ability to describe the axial segregation of the two granular components. The existence of global-in-time weak solutions is shown for arbitrary large cross-diffusion by using entropy-type inequalities and approximation arguments. The uniqueness of solutions is proved if cross-diffusion is not too large. Furthermore, we derive a sufficient condition on the parameters to have nonsegregation. Finally, numerical simulations illustrate the long-time coarsening of the segregation bands in the drum.

Double-diffusive convection caused by coupled molecular diffusion

Double-diffusive convection is studied for the case where a large coupled diffusion (or cross-diffusion) effect is present. The Soret effect is a familiar example of this cross-diffusion where the flux of the solute depends not only on its own spatial gradient but also on the in situ temperature gradient. The linear stability analysis of double-diffusive convection has been extended to include the two cross-diffusion flux terms and it has been shown that, with a sufficiently large coupled diffusion effect, fingers can form even when the concentrations of both components make the fluid's density gradient statically stable. The conditions under which the diffusive instability can occur are compared with those for the formation of fingers and it is shown that these two types of double-diffusive convection cannot occur together in any particular set of linear property gradients. We then consider finite-amplitude, steady, infinitely long fingers and show that a sufficiently large cross-diffusion effect can again allow fingers to exist when the concentrations of both solutes increase with depth. It is also shown that the diffusion of properties from an initially sharp interface may set up vertical gradients that are favourable for the formation of fingers.