This paper proposes a thin layer depth-averaged two-phase model provided by a dissipative energy balance
to describe avalanches of solid-fluid mixtures. This model is derived from a 3D two-phase model based
on the equations proposed by Jackson [R. Jackson, The Dynamics of Fluidized Particles, 2000]
which takes into account the force of buoyancy and the forces of interaction between the solid and fluid phases.
Jackson's model is based on mass and momentum conservation within the two phases,
i.e. two vector and two scalar equations. This system has five unknowns: the solid volume fraction,
the solid and fluid pressures and the solid and fluid velocities, i.e. three scalars and two vectors.
As a result, an additional equation is necessary to close the system. Surprisingly,
this issue is inadequately accounted for in the models that have been developed on the basis of Jackson's work.
In particular, Pitman and Le [E.B. Pitman, L. Le, Phil. Trans. R. Soc. A, 2005] replaced
this closure simply by imposing an extra boundary condition. If the pressure is assumed to be hydrostatic,
this condition can be considered as a closure condition. However, the corresponding model cannot account
for a dissipative energy balance. We propose here a closure equation to complete Jackson's model,
imposing incompressibility of the solid phase. We prove that the resulting whole 3D model is compatible
with a dissipative energy balance. From this model, we deduce a 2D depth-averaged model and we also
prove that the energy balance associated with this model is dissipative. Finally, we propose a
numerical scheme to approximate the depth-averaged model. We present several numerical tests
for the 1D case that are compared to the results of the model proposed by Pitman and Le.
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