On the rigid-lid approximation of shallow water Bingham

Bilal Al Taki; Khawla Msheik; Jacques Sainte-Marie

doi:10.3934/dcdsb.2020146

Article Contents

2021, Volume 26, Issue 2: 875-905. Doi: 10.3934/dcdsb.2020146

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On the rigid-lid approximation of shallow water Bingham

1.
Sorbonne University, Laboratory of Jacques-Louis Lions, 4 Place Jussieu 75005, Paris, France, INRIA Paris, ANGE Team, 2 Rue Simone IFF, 75012 Paris, France
2.
Savoie Mont Blanc University, LAMA UMR5127 CNRS, 73376 Le Bourget du Lac, France, Lebanese University, Faculty of sciences 1, Laboratory of Mathematics-EDST, Hadath, Lebanon

^* Corresponding author: Bilal Al Taki
^* Corresponding author: Bilal Al Taki

Received: April 2019

Revised: January 2020

Early access: May 2020

Published: February 2021

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Abstract

This paper discusses the well posedness of an initial value problem describing the motion of a Bingham fluid in a basin with a degenerate bottom topography. A physical interpretation of such motion is discussed. The system governing such motion is obtained from the Shallow Water-Bingham models in the regime where the Froude number degenerates, i.e taking the limit of such equations as the Froude number tends to zero. Since we are considering equations with degenerate coefficients, then we shall work with weighted Sobolev spaces in order to establish the existence of a weak solution. In order to overcome the difficulty of the discontinuity in Bingham's constitutive law, we follow a similar approach to that introduced in [G. DUVAUT and J.-L. LIONS, Springer-Verlag, 1976]. We study also the behavior of this solution when the yield limit vanishes. Finally, a numerical scheme for the system in 1D is furnished.

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Mathematics Subject Classification: Primary: 35Q30, 76N10, 35B65; Secondary: 35D35.

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Figure 1. Chosen profile for $ b(x) $

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Figure 2. (a) variations of the velocity $ u $ and (b) variations of the pressure $ p $ in the fluid domain at time $ t = T/2 $

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Figure 3. (a) variations of the quantity $ \tilde\sigma $ and (b) variations of $ \,\partial_x u $ at time $ t = T/2 $

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