no. 6
A nonlinear Stokes–Biot model for the interaction of a non-Newtonian fluid with poroelastic media
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 1915-1955

We develop and analyze a model for the interaction of a quasi-Newtonian free fluid with a poroelastic medium. The flow in the fluid region is described by the nonlinear Stokes equations and in the poroelastic medium by the nonlinear quasi-static Biot model. Equilibrium and kinematic conditions are imposed on the interface. We establish existence and uniqueness of a solution to the weak formulation and its semidiscrete continuous-in-time finite element approximation. We present error analysis, complemented by numerical experiments.

DOI : 10.1051/m2an/2019061
Classification : 76S05, 76D07, 74F10, 35M33, 65M60, 65M12
Keywords: Fluid-poroelastic structure interaction, Stokes–Biot model, fractured poroelastic media, non-Newtonian fluid

Ambartsumyan, Ilona 1 ; Ervin, Vincent J. 1 ; Nguyen, Truong 1 ; Yotov, Ivan 1

1 
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     title = {A nonlinear {Stokes{\textendash}Biot} model for the interaction of a {non-Newtonian} fluid with poroelastic media},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1915--1955},
     year = {2019},
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Ambartsumyan, Ilona; Ervin, Vincent J.; Nguyen, Truong; Yotov, Ivan. A nonlinear Stokes–Biot model for the interaction of a non-Newtonian fluid with poroelastic media. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 1915-1955. doi: 10.1051/m2an/2019061

[1] G. Acosta, T. Apel, R. Durán and A. Lombardi, Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra. Math. Comput. 80 (2011) 141–163. | MR | Zbl | DOI

[2] I. Ambartsumyan, E. Khattatov, I. Yotov and P. Zunino, A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model. Numer. Math. 140 (2018) 513–553. | MR | Zbl | DOI

[3] S. Badia, A. Quaini and A. Quarteroni, Coupling Biot and Navier-Stokes equations for modelling fluid-poroelastic media interaction. J. Comput. Phys. 228 (2009) 7986–8014. | MR | Zbl | DOI

[4] G.S. Beavers and D.D. Joseph, Boundary conditions at a naturally impermeable wall. J. Fluid. Mech. 30 (1967) 197–207. | DOI

[5] M. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12 (1941) 155–164. | JFM | DOI

[6] R.B. Bird, R.C. Armstrong, O. Hassager and C.F. Curtiss, In: Vol. 1 of Dynamics of Polymeric Liquids. Wiley New York (1977).

[7] D. Boffi and L. Gastaldi, Analysis of finite element approximation of evolution problems in mixed form. SIAM J. Numer. Anal. 42 (2004) 1502–1526. | MR | Zbl | DOI

[8] D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. In: Vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013). | MR | Zbl

[9] F. Brezzi, D. Boffi, L. Demkowicz, R.G. Durán, R.S. Falk and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications. Springer (2008). | MR

[10] M. Bukač, I. Yotov, R. Zakerzadeh and P. Zunino, Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche’s coupling approach. Comput. Methods Appl. Mech. Eng. 292 (2015) 138–170. | MR | Zbl | DOI

[11] M. Bukač, I. Yotov and P. Zunino, An operator splitting approach for the interaction between a fluid and a multilayered poroelastic structure. Numer. Methods Partial Differ. Equ. 31 (2015) 1054–1100. | MR | Zbl | DOI

[12] M. Bukač, I. Yotov and P. Zunino, Dimensional model reduction for flow through fractures in poroelastic media. ESAIM: M2AN 51 (2017) 1429–1471. | MR | Zbl | Numdam

[13] S. Caucao, G.N. Gatica, R. Oyarzúa and I. Šebestová, A fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity. J. Numer. Math. 25 (2017) 55–88. | MR | Zbl | DOI

[14] A. Cesmelioglu, Analysis of the coupled Navier–Stokes/Biot problem. J. Math. Anal. Appl. 456 (2017) 970–991. | MR | Zbl | DOI

[15] A. Cesmelioglu, H. Lee, A. Quaini, K. Wang and S.-Y. Yi, Optimization-based decoupling algorithms for a fluid-poroelastic system. In: Topics in Numerical Partial Differential Equations and Scientific Computing. Vol. 160 of IMA Vol. Math. Appl. Springer, New York (2016) 79–98. | MR | Zbl | DOI

[16] S.-S. Chow and G.F. Carey, Numerical approximation of generalized Newtonian fluids using Powell–Sabin–Heindl elements: I. Theoretical estimates. Int. J. Numer. Methods Fluids 41 (2003) 1085–1118. | MR | Zbl | DOI

[17] M. Dauge, Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. In: Vol. 1341 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1988). | MR | Zbl

[18] D. Di Pietro and J. Droniou, A hybrid high-order method for Leray-Lions elliptic equations on general meshes. Math. Comput. 86 (2017) 2159–2191. | MR | Zbl | DOI

[19] M. Discacciati, E. Miglio and A. Quarteroni, Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math. 43 (2002) 57–74. | MR | Zbl | DOI

[20] S. Dominguez, G.N. Gatica, A. Márquez and S. Meddahi, A primal-mixed formulation for the strong coupling of quasi-Newtonian fluids with porous media. Adv. Comput. Math. 42 (2016) 675–720. | MR | Zbl | DOI

[21] R. Durán, Error analysis in Lp, 1 ≤ p ≤ ∞, for mixed finite element methods for linear and quasi-linear elliptic problems. ESAIM: M2AN 22 (1988) 371–387. | MR | Zbl | Numdam | DOI

[22] V.J. Ervin, E.W. Jenkins and S. Sun, Coupled generalized nonlinear Stokes flow with flow through a porous medium. SIAM J. Numer. Anal. 47 (2009) 929–952. | MR | Zbl | DOI

[23] V.J. Ervin, E.W. Jenkins and S. Sun, Coupling nonlinear Stokes and Darcy flow using mortar finite elements. Appl. Numer. Math. 61 (2011) 1198–1222. | MR | Zbl | DOI

[24] L. Formaggia, A. Quarteroni and A. Veneziani. In: Vol. 1 of Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System. Springer Science & Business Media (2010). | MR | Zbl

[25] S. Frei, B. Holm, T. Richter, T. Wick and H. Yang, Fluid-structure interaction: modeling, adaptive discretisations and solvers, In: Vol. 20 of Radon Series on Computational and Applied Mathematics. De Gruyter(2017) . | DOI

[26] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-state Problems. Springer Science & Business Media (2011). | MR | Zbl

[27] G.P. Galdi and R. Rannacher, Fundamental trends in fluid-structure interaction, Vol. 1 of Contemporary Challenges in Mathematical Fluid Dynamics and Its Applications. World Scientific Publishing Co., Pte. Ltd., Hackensack, NJ, 2010. | MR | DOI

[28] V. Girault, M.F. Wheeler, B. Ganis and M.E. Mear, A lubrication fracture model in a poro-elastic medium. Math. Models Methods Appl. Sci. 25 (2015) 587–645. | MR | Zbl | DOI

[29] P. Grisvard, Elliptic Problems in Nonsmooth Domains. SIAM (2011). | MR | Zbl | DOI

[30] B. Guerciotti and C. Vergara, Computational comparison between Newtonian and non-Newtonian blood rheologies in stenotic vessels. In: Biomedical Technology. Vol 84 of Lecture Notes in Applied and Computational Mechanics. Springer (2018) 169–183. | DOI

[31] F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | MR | Zbl | DOI

[32] J. Janela, A. Moura and A. Sequeira, A 3D non-Newtonian fluid–structure interaction model for blood flow in arteries. J. Comput. Appl. Math. 234 (2010) 2783–2791. | MR | Zbl | DOI

[33] W.J. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2003) 2195–2218. | MR | Zbl | DOI

[34] S. Lee, A. Mikelić, M.F. Wheeler and T. Wick, Phase-field modeling of proppant-filled fractures in a poroelastic medium. Comput. Methods Appl. Mech. Eng. 312 (2016) 509–541. | MR | Zbl | DOI

[35] X. Lopez, P.H. Valvatne and M.J. Blunt, Predictive network modeling of single-phase non-Newtonian flow in porous media. J. Colloid Interface Sci. 264 (2003) 256–265. | DOI

[36] J. Necas, J. Málek, M. Rokyta and M. Ruzicka, Vol. 13 of Weak and Measure-valued Solutions to Evolutionary PDEs. CRC Press (1996). | MR | Zbl

[37] R.G. Owens and T.N. Phillips, Vol. 14 of Computational Rheology. World Scientific (2002). | MR | Zbl | DOI

[38] J.R.A. Pearson and P.M.J. Tardy, Models for flow of non-Newtonian and complex fluids through porous media. J. Non-Newton. Fluid Mech. 102 (2002) 447–473. | Zbl | DOI

[39] M. Renardy and R.C. Rogers, Vol. 13 of An Introduction to Partial Differential Equations. Springer Science & Business Media (2006). | Zbl | MR

[40] B. Rivière and I. Yotov, Locally conservative coupling of Stokes and Darcy flows. SIAM J. Numer. Anal. 42 (2005) 1959–1977. | MR | Zbl | DOI

[41] P.G. Saffman, On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50 (1971) 93–101. | Zbl | DOI

[42] D. Sandri, Sur l’approximation numérique des écoulements quasi-newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau. ESAIM: M2AN 27 (1993) 131–155. | MR | Zbl | Numdam | DOI

[43] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | MR | Zbl | DOI

[44] R.E. Showalter, Poroelastic filtration coupled to Stokes flow. Control Theory of Partial Differential Equations. Vol. 242 of Lect. Notes Pure Appl. Math. Chapman & Hall/CRC, Boca Raton, FL (2005) 229–241. | MR | Zbl | DOI

[45] R.E. Showalter, Nonlinear degenerate evolution equations in mixed formulation. SIAM J. Math. Anal. 42 (2010) 2114–2131. | MR | Zbl | DOI

[46] R.E. Showalter, Vol. 49 of Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Soc. (2013). | MR | Zbl | DOI

[47] D. Vassilev, C. Wang and I. Yotov, Domain decomposition for coupled Stokes and Darcy flows. Comput. Methods Appl. Mech. Eng. 268 (2014) 264–283. | MR | Zbl | DOI

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