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.