### Modelling shallow gravity-driven solid-fluid mixtures over arbitrary topography

**Luca, I; Tai, YC; Kuo, CY ***(Academia Sinica)*

Wednesday 07 January 2009, 10:05-10:30

Seminar Room 1, Newton Institute

#### Abstract

The description of geophysical flows over arbitrary terrain is a topic comprising many challenging problems. One of them is the incorporation in the modelling equations of the influences due to the geometry of the basal surface. This issue was revigorated by the relatively recent introduction due to Bouchut and Westdickenberg of curvilinear coordinates, able to properly account for the arbitrariness of the bottom topography and for the shallowness of the avalanche mass. These coordinates are: two surface parameters on the bed surface, and the distance between the point in question and its orthogonal projection onto the topography; the depth of the avalanche is measured along the normal direction to the bottom surface.
We use these coordinates and develop depth-averaged models of gravity-driven saturated mixtures of solid grains and pore fluid on an arbitrary rigid basal surface. First, by only specifying the interaction force (as deduced by Schneider and Hutter within a thermodynamic analysis, and which is different from that used by Pitman and Le) and ordering approximations in terms of an aspect ratio between a typical length perpendicular to, and a typical length parallel to the topography, we derive governing equations for the shallow flowing mixture. In doing so, the non-uniformity through the avalanche depth of the constituent velocities and of the solid volume fraction is accounted for by coefficients of Boussinesq type.
Then, the rheology of both constituents is specified. One constituent is a Newtonian/non-Newtonian fluid with a viscosity so small, that in the governing equations only the basal shear stress survives, and this is parameterized by a viscous friction law. For the bulk stresses of the solid constituent we propose three models: one reduces to the inviscid fluid, and the other two are topography adapted versions of Iverson-Denlinger and Savage-Hutter models. Common to the proposed models is the assumption that two of the principal directions of the mean stress tensor are aligned with the principal directions of the mean surface stretching. The basal shear stress is parameterized by a Coulomb friction law.
With these mixture models at hand, we derived equations governing the motion of two shallow layers on arbitrary rigid basal surface: the layer near the bottom topography is essentially one of the three models of solid-fluid mixture, and the upper layer is the Newtonian/non-Newtonian fluid, present in the mixture layer. The interface between the two layers is a material surface for the solid constituent, and across it the jump conditions of mass and momentum balance equations corresponding to each constituent are envisaged.
Numerical results concerning the derived model equations are planned to be obtained in future.
Acknowledgements The authors are very grateful to Prof. K. Hutter for his neverending support and advice; the topography-adapted version of the Savage-Hutter model was developed following his

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