Mathematical Modelling and Numerical Analysis Will be set by the publisher Modélisation Mathématique et Analyse Numérique A TWO-PHASE SHALLOW DEBRIS FLOW MODEL WITH ENERGY BALANCE

Abstract. 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 i