Dam break with Coulomb friction: A model for granular slumping?

Axisymmetric and two-dimensional dam-break problems are solved under the shallow layer approximation and with basal Coulomb friction in a first attempt to model recent granular slumping experiments [G. Lube, H. E. Huppert, R. S. J. Sparks, and M. A. Hallworth, J. Fluid Mech. 508, 175 (2004); E. Lajeunesse, A. Mangeney-Castelnau, and J. P. Vilotte, Phys. Fluids 16, 2371 (2004); N. J. Balmforth and R. R. Kerswell, J. Fluid Mech. (in press)]. The axisymmetric model predicts a scaling law h0∕h∞∝a for the ratio of initial to final maximum heights which compares well with the slumping data over a wide range of a≔h0∕r0, the initial aspect ratio (h0 and r0 are the initial height and radius). The predicted runout scaling (r∞−r0)∕r0∝a, however, significantly overestimates the data runout which behaves more like a (r∞ being the maximum final radius). This situation may be improved by redefining the runout to take account of the finite granule size but a noticeable discrepancy still remains. The runout scaling prediction in the two-dimensional case mirrors the axisymmetric result, whereas in two dimensions h0∕h∞∝aα with α→23 as a→∞ and a≔h0∕l0 where l0 is the initial base length of the column. The limitations of the model and possible improvements are discussed.