Abstract
We propose a new model for the motion of a viscous incompressible fluid. More precisely, we consider the Navier–Stokes system with a boundary condition governed by the Coulomb friction law. With this boundary condition, the fluid can slip on the boundary if the tangential component of the stress tensor is too large. We prove the existence and uniqueness of weak solution in the two-dimensional problem and the existence of at least one solution in the three-dimensional case, together with regularity properties and an energy estimate. We also propose a fully discrete scheme of our problem using the characteristic method, and we present numerical simulations in two physical examples.
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References
Brezis, H.: Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983. Théorie et applications. [Theory and applications]
Conca, C., San Martín, J., Tucsnak, M.: Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Commun. Partial Differ. Equ. 25, 1019–1042 (2000)
Duvaut, G., Lions, J.L.: Les inéquations en méchanique et en physique. Dunod, Paris (1972). (in French)
Gérard-Varet, D., Hillairet, M.: Existence of weak solutions up to collision for viscous fluid-solid systems with slip. Commun. Pure Appl. Math. 67, 2022–2075 (2014)
Gérard-Varet, D., Hillairet, M., Wang, C.: The influence of boundary conditions on the contact problem in a 3D Navier–Stokes flow. J. Math. Pures Appl. 9(103), 1–38 (2015)
Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier–Stokes Equations. Springer, Berlin (1979)
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer Series in Computational Physics, Springer, New York (1984)
Hecht, F.: New development in FreeFem++. J. Numer. Math. 20, 251–265 (2012)
Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incomprehensible Flow. Gordon and Breach Science Publishers, New York (1969)
Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris (1969)
Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications, vol. I. Springer, New York: Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181 (1972)
Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier–Stokes equations. Numer. Math. 38, 309–332 (1982)
San Martín, J., Scheid, J.-F., Smaranda, L.: A modified Lagrange–Galerkin method for a fluid-rigid system with discontinuous density. Numer. Math. 122, 341–382 (2012)
Martin, J.S., Scheid, J.-F., Takahashi, T., Tucsnak, M.: Convergence of the Lagrange–Galerkin method for the equations modelling the motion of a fluid-rigid system. SIAM J. Numer. Anal. 43, 1536–1571 (2005). (electronic)
San Martín, J.A., Starovoitov, V., Tucsnak, M.: Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Ration. Mech. Anal. 161, 113–147 (2002)
Süli, E.: Convergence and nonlinear stability of the Lagrange–Galerkin method for the Navier–Stokes equations. Numer. Math. 53, 459–483 (1988)
Temam, R.: Navier–Stokes equations, North-Holland Publishing Co., Amsterdam, third, (eds.): Theory and numerical analysis. With an appendix by F. Thomasset (1984)
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Bălilescu, L., San Martín, J. & Takahashi, T. On the Navier–Stokes system with the Coulomb friction law boundary condition. Z. Angew. Math. Phys. 68, 3 (2017). https://doi.org/10.1007/s00033-016-0744-x
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DOI: https://doi.org/10.1007/s00033-016-0744-x