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Asymptotic expansion of the velocity field within the front of viscoplastic surges: comparison with experiments

Published online by Cambridge University Press:  17 December 2019

G. Chambon Open the ORCID record for G. Chambon [Opens in a new window] ,
P. Freydier ,
M. Naaim  and
J.-P. Vila
G. Chambon*
Affiliation:
Univ. Grenoble Alpes, INRAE, ETNA, F-38402Saint-Martin-d’Hères, France
P. Freydier
Affiliation:
Univ. Grenoble Alpes, INRAE, ETNA, F-38402Saint-Martin-d’Hères, France Laboratoire FAST, Univ. Paris Sud, CNRS, Univ. Paris-Saclay, F-91405Orsay, France
M. Naaim
Affiliation:
Univ. Grenoble Alpes, INRAE, ETNA, F-38402Saint-Martin-d’Hères, France
J.-P. Vila
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Univ. Toulouse, CNRS, INSA, F-31077Toulouse, France
*
†Email address for correspondence: [email protected]
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Abstract

In this article, we investigate the internal dynamics (velocity profiles and shear rate) of free-surface surges made of viscoplastic fluids. Compared with fluids without a yield stress, additional complexity arises from the possible coexistence of sheared and unsheared (or pseudo-plug) zones in the flow. Expanding on the thin-layer approach of Fernandez-Nieto et al. (J. Non-Newtonian Fluid Mech., vol. 165, 2010, pp. 712–732), we derive formal asymptotic expansions of the velocity field and discharge up to $O(\unicode[STIX]{x1D716})$ with respect to flow aspect ratio $\unicode[STIX]{x1D716}$. Detailed comparisons between these theoretical predictions and experimental data reveal that, although the leading-order approximation (equivalent to a lubrication model) satisfactorily accounts for the global dynamics of the flow, considering $O(\unicode[STIX]{x1D716})$ correction terms is required to capture the evolution of velocity and shear rate close to the tip. Notably, these correction terms are responsible for the vanishing of the unsheared layer in the tip region, a feature clearly observed in the experiments. Differences between the leading-order and $O(\unicode[STIX]{x1D716})$ models appear to be enhanced by the viscoplastic character of the fluid. In particular, $O(\unicode[STIX]{x1D716})$ correction terms related to the existence of $O(1)$ plastic normal stresses in the pseudo-plug layer play a critical role. This study provides important insights for future development of consistent shallow-water models adapted to viscoplastic materials.

JFM classification

Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 884 , 10 February 2020 , A43
Copyright
© 2019 Cambridge University Press

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