Two-dimensional similarity solutions for finite-mass granular avalanches with Coulomb- and viscous-type frictional resistance
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This paper is concerned with the motion of an unconfined finite mass of granular material down an inclined plane when released from a rest position in the shape of a circular or elliptical paraboloid. The granular mass is treated as a frictional Coulomb-like continuum with a constant angle of internal friction. The basal friction force is assumed to be composed of a Coulomb-type component with a bed-friction angle that is position-dependent and a viscous Voellmy-type resistive stress that is proportional to the velocity squared. The model equations are those of Hutter and others (in press b) and form a spatially two-dimensional set for the evolution of the avalanche height and the depth averaged in-plane velocity components; they hold for a motion of a granular mass along a plane surface. Similarity solutions, i.e. solutions which preserve the shape and the structure of the velocity field, are constructed by decomposing the motion into that of the centre of mass and the deformation relative to it. This decomposition is possible provided the effect of the Voellmy drag on the deformation is ignored. With it, the depth and velocities relative to those of the centre of mass of the moving pile can be determined analytically. It is shown that the pile has a parabolic cap shape and contour lines are elliptical. The semi-axes and the position and velocity of the centre of mass are calculated numerically. We explicitly show that (i) For two-dimensional spreading, a rigid-body motion does not exist, no matter what be the values of the bed-friction angle and the coefficient of viscous drag. (ii) A steady final velocity of the centre of the mass cannot be assumed, but the motion of the centre of mass depends strongly on the value of the Voellmy coefficient. (iii) The geometry of the moving pile depends on the variation of the bed-friction angle with position, as well as on the value of the coefficient of viscous drag.
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