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February  2021, 26(2): 875-905. doi: 10.3934/dcdsb.2020146

On the rigid-lid approximation of shallow water Bingham

Bilal Al Taki 1,, , Khawla Msheik 2, and  Jacques Sainte-Marie 1,

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

Received  April 2019 Revised  January 2020 Published  May 2020

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.

Keywords: Bingham fluid, Muckenhoupt weights, Variational inequality, Low Froude number.
Mathematics Subject Classification: Primary: 35Q30, 76N10, 35B65; Secondary: 35D35.
Citation: Bilal Al Taki, Khawla Msheik, Jacques Sainte-Marie. On the rigid-lid approximation of shallow water Bingham. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 875-905. doi: 10.3934/dcdsb.2020146
References:
[1]

N. Aïssiouene, M.-O. Bristeau, E. Godlewski, A. Mangeney, C. Parés and J. Sainte-Marie, A Two-dimensional Method for a Dispersive Shallow Water Model, https://hal.archives-ouvertes.fr/hal-01632522, Working paper or preprint, 2017. Google Scholar

[2]

B. Al Taki, Viscosity effect on the degenerate lake equations, Nonlinear Anal., 148 (2017), 30-60.  doi: 10.1016/j.na.2016.09.017.  Google Scholar

[3]

E. Bingham, Fluidity and Plasticity, McGraw-Hill, 1922. Google Scholar

[4]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/b93802.  Google Scholar

[5]

C. Bourdarias and S. Gerbi, A finite volume scheme for a model coupling free surface and pressurised flows in pipes, J. Comput. Appl. Math., 209 (2007), 109-131.  doi: 10.1016/j.cam.2006.10.086.  Google Scholar

[6]

C. Bourdarias, S. Gerbi and M. Gisclon, A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes, J. Comput. Appl. Math., 218 (2008), 522-531.  doi: 10.1016/j.cam.2007.09.009.  Google Scholar

[7]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183 of Applied Mathematical Sciences, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[8]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.  Google Scholar

[9]

D. Bresch, E. D. Fernández-Nieto, I. R. Ionescu and P. Vigneaux, Augmented Lagrangian method and compressible visco-plastic flows: Applications to shallow dense avalanches, in New Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser Verlag, Basel, (2010), 57–89.  Google Scholar

[10]

M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska Gwiazda, On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., 44 (2012), 2756-2801.  doi: 10.1137/110830289.  Google Scholar

[11]

A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762.  doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[12]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[13]

D. E. Edmunds and R. Hurri-Syrjänen, Weighted Hardy inequalities, J. Math. Anal. Appl., 310 (2005), 424-435.  doi: 10.1016/j.jmaa.2005.01.066.  Google Scholar

[14]

R. Farwig and H. Sohr, Weighted $L^q$-theory for the Stokes resolvent in exterior domains, J. Math. Soc. Japan, 49 (1997), 251-288.  doi: 10.2969/jmsj/04920251.  Google Scholar

[15]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, Cham, 2017, Second edition of [MR2499296].  Google Scholar

[16]

E. Fernández-Nieto, P. Noble and J. Vila, Shallow water equations for power law and Bingham fluids, Sci. China Math., 55 (2012), 277-283.  doi: 10.1007/s11425-011-4358-7.  Google Scholar

[17]

A. Fröhlich, The Stokes operator in weighted $L^q$-spaces. Ⅱ. Weighted resolvent estimates and maximal $L^p$-regularity, Math. Ann., 339 (2007), 287-316.  doi: 10.1007/s00208-007-0114-2.  Google Scholar

[18]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, NY, 2006, Unabridged republication of the 1993 original.  Google Scholar

[19]

A. Kał amajska, Coercive inequalities on weighted Sobolev spaces, Colloq. Math., 66 (1994), 309-318.   Google Scholar

[20]

C. D. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography, Nonlinearity, 14 (2001), 1493-1515.  doi: 10.1088/0951-7715/14/6/305.  Google Scholar

[21]

J.-L. Lions, Remarks on some nonlinear evolution problems arising in Bingham flows, Israel J. Math., 13 (1972), 155–172 (1973).  Google Scholar

[22] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2, vol. 10 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar
[23]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207–226, URL https://doi-org.camphrier-1.grenet.fr/10.2307/1995882. doi: 10.1090/S0002-9947-1972-0293384-6.  Google Scholar

[24]

M. Naaim and A. Bouchet, Etude Expérimentale des Écoulements D'avalanches de Neige Dense, Mesures et interprétations des profils de vitesse en écoulements quasi permanents et pleinement développés. Rapport scientifique. UR ETNA, Grenoble. Google Scholar

[25]

A. Nekvinda, Characterization of traces of the weighted Sobolev space $W^{1, p}(\Omega, d^\epsilon_M)$ on $M$, Czechoslovak Math. J., 43 (1993), 695-711.   Google Scholar

[26]

K. Nishimura and N. Maeno, Contribution of viscous forces to avalanche dynamics, Annals of Glaciology, 13 (1989), 202-206.   Google Scholar

[27]

R. Perla, T. Cheng and D. McClung, A two–parameter model of snow–avalanche motion, Ann. Glaciol., 26 (1980), 197-207.   Google Scholar

[28]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[29]

B. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, vol. 1736 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. URL https://doi-org.camphrier-1.grenet.fr/10.1007/BFb0103908. Google Scholar

[30]

A. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.  doi: 10.1007/s00222-016-0666-4.  Google Scholar

show all references

References:
[1]

N. Aïssiouene, M.-O. Bristeau, E. Godlewski, A. Mangeney, C. Parés and J. Sainte-Marie, A Two-dimensional Method for a Dispersive Shallow Water Model, https://hal.archives-ouvertes.fr/hal-01632522, Working paper or preprint, 2017. Google Scholar

[2]

B. Al Taki, Viscosity effect on the degenerate lake equations, Nonlinear Anal., 148 (2017), 30-60.  doi: 10.1016/j.na.2016.09.017.  Google Scholar

[3]

E. Bingham, Fluidity and Plasticity, McGraw-Hill, 1922. Google Scholar

[4]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/b93802.  Google Scholar

[5]

C. Bourdarias and S. Gerbi, A finite volume scheme for a model coupling free surface and pressurised flows in pipes, J. Comput. Appl. Math., 209 (2007), 109-131.  doi: 10.1016/j.cam.2006.10.086.  Google Scholar

[6]

C. Bourdarias, S. Gerbi and M. Gisclon, A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes, J. Comput. Appl. Math., 218 (2008), 522-531.  doi: 10.1016/j.cam.2007.09.009.  Google Scholar

[7]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183 of Applied Mathematical Sciences, Springer, New York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[8]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.  doi: 10.1007/s00220-003-0859-8.  Google Scholar

[9]

D. Bresch, E. D. Fernández-Nieto, I. R. Ionescu and P. Vigneaux, Augmented Lagrangian method and compressible visco-plastic flows: Applications to shallow dense avalanches, in New Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Birkhäuser Verlag, Basel, (2010), 57–89.  Google Scholar

[10]

M. Bulíček, P. Gwiazda, J. Málek and A. Świerczewska Gwiazda, On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., 44 (2012), 2756-2801.  doi: 10.1137/110830289.  Google Scholar

[11]

A. J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp., 22 (1968), 745-762.  doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[12]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[13]

D. E. Edmunds and R. Hurri-Syrjänen, Weighted Hardy inequalities, J. Math. Anal. Appl., 310 (2005), 424-435.  doi: 10.1016/j.jmaa.2005.01.066.  Google Scholar

[14]

R. Farwig and H. Sohr, Weighted $L^q$-theory for the Stokes resolvent in exterior domains, J. Math. Soc. Japan, 49 (1997), 251-288.  doi: 10.2969/jmsj/04920251.  Google Scholar

[15]

E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, Cham, 2017, Second edition of [MR2499296].  Google Scholar

[16]

E. Fernández-Nieto, P. Noble and J. Vila, Shallow water equations for power law and Bingham fluids, Sci. China Math., 55 (2012), 277-283.  doi: 10.1007/s11425-011-4358-7.  Google Scholar

[17]

A. Fröhlich, The Stokes operator in weighted $L^q$-spaces. Ⅱ. Weighted resolvent estimates and maximal $L^p$-regularity, Math. Ann., 339 (2007), 287-316.  doi: 10.1007/s00208-007-0114-2.  Google Scholar

[18]

J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Dover Publications, Inc., Mineola, NY, 2006, Unabridged republication of the 1993 original.  Google Scholar

[19]

A. Kał amajska, Coercive inequalities on weighted Sobolev spaces, Colloq. Math., 66 (1994), 309-318.   Google Scholar

[20]

C. D. Levermore and M. Sammartino, A shallow water model with eddy viscosity for basins with varying bottom topography, Nonlinearity, 14 (2001), 1493-1515.  doi: 10.1088/0951-7715/14/6/305.  Google Scholar

[21]

J.-L. Lions, Remarks on some nonlinear evolution problems arising in Bingham flows, Israel J. Math., 13 (1972), 155–172 (1973).  Google Scholar

[22] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2, vol. 10 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar
[23]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207–226, URL https://doi-org.camphrier-1.grenet.fr/10.2307/1995882. doi: 10.1090/S0002-9947-1972-0293384-6.  Google Scholar

[24]

M. Naaim and A. Bouchet, Etude Expérimentale des Écoulements D'avalanches de Neige Dense, Mesures et interprétations des profils de vitesse en écoulements quasi permanents et pleinement développés. Rapport scientifique. UR ETNA, Grenoble. Google Scholar

[25]

A. Nekvinda, Characterization of traces of the weighted Sobolev space $W^{1, p}(\Omega, d^\epsilon_M)$ on $M$, Czechoslovak Math. J., 43 (1993), 695-711.   Google Scholar

[26]

K. Nishimura and N. Maeno, Contribution of viscous forces to avalanche dynamics, Annals of Glaciology, 13 (1989), 202-206.   Google Scholar

[27]

R. Perla, T. Cheng and D. McClung, A two–parameter model of snow–avalanche motion, Ann. Glaciol., 26 (1980), 197-207.   Google Scholar

[28]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2. North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[29]

B. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, vol. 1736 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. URL https://doi-org.camphrier-1.grenet.fr/10.1007/BFb0103908. Google Scholar

[30]

A. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.  doi: 10.1007/s00222-016-0666-4.  Google Scholar

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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