In the first part of our talk, we present experimental observations concerning the flow of a densely packed grain collection down a two-dimensional inclined channel. These results oppose the the predictions of the kinetic theory. We evidence that that continuous paths of transient contacts are effective for transporting momentum and energy through the bulk, so that the binary collision hypothesis (which is at the basis of the kinetic theories) is inadequate to describe dense flows. We propose an alternative model, which succeeds in accounting for the observed velocity profiles, and for the and the paradoxical nonzero shear rate in the vicinity of the free surface. In the second part, we emphasize some remarkable features exhibited by dry grain avalanches in laboratory experiments. According to the slope angle, the rear front propagates either upwards or downwards, with velocity approximately equal to the depth averaged velocity of the avalanche. As a counterpart, in both regimes, the velocity magnitude of the head front remains of the order of twice the depth averaged avalanche velocity. We suggest simple elementary mechanisms capable of accounting for these observations. We propose then an analytical modelling aimed at describing the combined processes governing the avalanche expansion. The two solutions that we obtain for the growth regimes and for the avalanche shapes resemble very closely the observations made in the laboratory and in the field.

• Event: "Numerical models and methods for compressible and two-phase flow" (2010)

The evolution of large amplitude internal solitary waves propagating towards the shore is studied. The mathematical model describing solitary waves interaction and decaying has been derived. It is a variant of the Choi-Camassa equations for two-layer and three-layer flows. The exact solution representing the waves of permanent form for sharp interfaces is found. It is shown by the comparison between experimental data and numerical results that the rate of wave decay before and after interaction of solitary waves moving in opposite directions can be predicted by the model to a high accuracy.

• Event: "Numerical models and methods for compressible and two-phase flow" (2010)

Thin fluid films can have surprising behavior depending on the boundary conditions enforced, the energy input and the specific Reynolds number of the fluid motion. Here we study the equations of motion for a thin fluid film with a free boundary and its other interface in contact with a solid wall. Although shear dissipation increases for thinner layers and the motion can generally be described in the limit as viscous inertial modes can always be excited for a sufficiently high input of energy. We derive the minimal set of equations containing inertial effects in this strongly dissipative regime.

• Event: "Numerical models and methods for compressible and two-phase flow" (2010)

We study numerically an instability mechanism for the formation of shear bands at high strain-rate deformations of metals. We use a reformulation of the problem that exploits scaling properties of the model, in conjunction with adaptive nite element methods of any order in the spatial discretization and implicit Runge-Kutta methods with variable step in time. The numerical schemes are of implicit-explicit type and provide adequate resolution of shear bands up to full development. We nd that already from the initial stages, shear band formation is associated with collapse of stress diusion across the band and that process intensi es as the band fully forms. For fully developed bands, heat conduction plays an important role in the subsequent evolution by causing a delay or even stopping the development of the band.

• Event: "Numerical models and methods for compressible and two-phase flow" (2010)

We study how the Reynolds approximation is perturbed by a non-regular roughnes of the boundary. We explicit the critical values of abrupt changes in the profile. The lubrification approximation is mathematically justified through a variant of the noton of twoscale convergence.

• Event: "Numerical models and methods for compressible and two-phase flow" (2010)

In this talk we present preliminary results on the Faraday instability simulation using a two-fluid approach.

• Event: "Numerical models and methods for compressible and two-phase flow" (2010)