Analysis of the penalized 3D variable viscosity stokes equations coupled to diffusion and transport
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 565-591.

The analysis of the penalized Stokes problem, in its variable viscosity formulation, coupled to convection-diffusion equations is presented in this article. It models the interaction between a highly viscous fluid with variable viscosity and immersed moving and deformable obstacles. Indeed, while it is quite common to couple Poisson equations to diffusion-transport equations in plasma physics or fluid dynamics in vorticity formulations, the study of complex fluids requires to consider together the Stokes problem in complex moving geometry and convection-diffusion equations. The main result of this paper shows the existence and the uniqueness of the solution to this equations system with regularity estimates. Then we show that the solution to the penalized problem weakly converges toward the solution to the physical problem. Numerical simulations of fluid mechanics computations in this context are also presented in order to illustrate the practical aspects of such models: lung cells and their surrounding heterogeneous fluid, and porous media flows. Among the main original aspects in the present study, one can highlight the non linear dynamics induced by the coupling, and the tracking of the time-dependence of the domain.

Reçu le :
DOI : 10.1051/m2an/2015056
Classification : 35Q30, 76D03, 76D07, 65M25, 68U20, 76Z05, 92B05
Mots clés : Stokes equations, moving geometry, variable viscosity flows, porous media flows, biomathematics
Chatelin, Robin 1 ; Sanchez, David 2 ; Poncet, Philippe 3

1 Universitéde Lyon, ENISE, LTDS UMR CNRS 5513, 58 rue Jean Parot, 42023 Saint-Étienne cedex 02, France.
2 Toulouse Mathematics Institute, UMR CNRS 5219, Team MIP, INSA, GMM 135 avenue de Rangueil, 31077 Toulouse, France.
3 LMAP, UMR CNRS 5142, IPRA, UPPA, avenue de l’Université, BP 1155, 64013 Pau, France.
@article{M2AN_2016__50_2_565_0,
     author = {Chatelin, Robin and Sanchez, David and Poncet, Philippe},
     title = {Analysis of the penalized {3D} variable viscosity stokes equations coupled to diffusion and transport},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {565--591},
     publisher = {EDP-Sciences},
     volume = {50},
     number = {2},
     year = {2016},
     doi = {10.1051/m2an/2015056},
     mrnumber = {3482555},
     zbl = {1343.35185},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015056/}
}
TY  - JOUR
AU  - Chatelin, Robin
AU  - Sanchez, David
AU  - Poncet, Philippe
TI  - Analysis of the penalized 3D variable viscosity stokes equations coupled to diffusion and transport
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2016
SP  - 565
EP  - 591
VL  - 50
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2015056/
DO  - 10.1051/m2an/2015056
LA  - en
ID  - M2AN_2016__50_2_565_0
ER  - 
%0 Journal Article
%A Chatelin, Robin
%A Sanchez, David
%A Poncet, Philippe
%T Analysis of the penalized 3D variable viscosity stokes equations coupled to diffusion and transport
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2016
%P 565-591
%V 50
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2015056/
%R 10.1051/m2an/2015056
%G en
%F M2AN_2016__50_2_565_0
Chatelin, Robin; Sanchez, David; Poncet, Philippe. Analysis of the penalized 3D variable viscosity stokes equations coupled to diffusion and transport. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 565-591. doi : 10.1051/m2an/2015056. http://www.numdam.org/articles/10.1051/m2an/2015056/

J.C. Adams, mudpack: Multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations. Appl. Math. Comput. 34 (1989) 113–146. | Zbl

R.A. Adams and J.J.F. Fournier, Sobolev Spaces. Academic Press (2003). | MR | Zbl

S. Agmon, Lectures on Elliptic Boundary Value Problems. Prepared for Publication by B. Frank Jones, with the Assistance of George W. Batten (1965). | MR | Zbl

Ph. Angot, Analysis of singular perturbations on the brinkman problem for fictitious domain models of viscous flowsh. Math. Methods Appl. Sci. 22 (1999) 1395–1412. | DOI | MR | Zbl

Ph. Angot, Ch.-H. Bruneau and P. Fabrie, A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81 (1999) 497–520. | DOI | MR | Zbl

G.J. Besseris and D.B. Yeates, Rotating magnetic particle microrheometry in biopolymer fluid dynamics: Mucus microrheology. J. Chem. Phys. 127 (2007) 105106–105106. | DOI

C. Bost, G.-H. Cottet and E. Maitre, Convergence analysis of a penalization method for the three-dimensional motion of a rigid body in an incompressible viscous fluid. SIAM J. Numer. Anal. 48 (2010) 1313–1337. | DOI | MR | Zbl

F. Boyer and P. Fabrie, Eléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles. Springer (2005). | MR | Zbl

G. Carbou and P. Fabrie, Boundary layer for a penalization method for viscous incompressible flow. Adv. Differ. Eq. 8 (2003) 1453–1480. | MR | Zbl

R. Chatelin, Méthodes numériques pour l’écoulement de Stokes 3D: fluides à viscosité variable en géométrie complexe mobile; application aux fluides biologiques. Ph.D. thesis, Université Toulouse 3 Paul Sabatier (2013).

R. Chatelin and Ph. Poncet, A hybrid grid-particle method for moving bodies in 3D stokes flow with variable viscosity. SIAM J. Sci. Comput. 35 (2013) B925–B949. | DOI | MR | Zbl

R. Chatelin and Ph. Poncet, Hybrid grid–particle methods and penalization: A Sherman–Morrison–Woodbury approach to compute 3D viscous flows using FFT. J. Comput. Phys. 269 (2014) 314–328. | DOI | MR | Zbl

G.-H. Cottet and P. Poncet, Advances in direct numerical simulations of 3D wall-bounded flows by vortex-in-cell methods. J. Comput. Phys. 193 (2004) 136–158. | DOI | MR | Zbl

M. Coquerelle and G.-H. Cottet, A vortex level set method for the two-way coupling of an incompressible fluid with colliding rigid bodies. J. Comput. Phys. 227 (2008) 9121–9137. | DOI | MR | Zbl

G.-H. Cottet and P.D. Koumoutsakos, Vortex Methods: Theory and Practice. Cambridge University Press (2000). | MR | Zbl

R.V. Craster and O.K. Matar, Surfactant transport on mucus films. J. Fluid Mech. 425 (2000) 235–258. | DOI | Zbl

R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1−6. Springer (2000). | Zbl

G. Duvaut and J.L. Lions, Les inéquations en mécanique et en physique (1972). | MR | Zbl

P.A. Edwards and D.B. Yeates, Magnetic Rheometry of Bronchial Mucus. In vol. 489 of Viscoelasticity of Biomaterials, ACS Symposium Series. American Chemical Society (1992) 249–267.

M. El Ossmani and P. Poncet, Efficiency of multiscale hybrid grid-particle vortex methods. Multiscale Model. Simul. 8 (2010) 1671–1690. | DOI | MR | Zbl

C. Foias and R. Temam, Remarques sur les équations de navier-stokes stationnaires et les phénomènes successifs de bifurcation. Ann. Sc. Norm. Super. Pisa - Cl. Sci. 5 (1978) 29–63. | Numdam | MR | Zbl

R. Glowinski, T.-W. Pan, T.I. Hesla and D.D. Joseph, A distributed lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 25 (1999) 755–794. | DOI | MR | Zbl

G. Hou, J. Wang and A. Layton, Numerical methods for fluid-structure interaction – a review. Commun. Comput. Phys. 12 (2012) 337–377. | DOI | MR | Zbl

H.H. Hu, Direct simulation of flows of solid-liquid mixtures. Int. J. Multiphase Flow 22 (1996) 335–352. | DOI | Zbl

J. Hussong, N. Schorr, J. Belardi, O. Prucker, J. Rhe and J. Westerweel, Experimental investigation of the flow induced by artificial cilia. Lab on a Chip 11 (2011) 2017–2022. | DOI

J. Hussong, W.-P. Breugem and J. Westerweel, A continuum model for flow induced by metachronal coordination between beating cilia. J. Fluid Mech. 684 (2011) 137–162. | DOI | MR | Zbl

J. Janela, A. Lefebvre and B. Maury. A penalty method for the simulation of fluid-rigid body interaction. ESAIM: Procs. 14 (2005) 115–123. | DOI | MR | Zbl

M. Krotkiewski, I.S. Ligaarden, K.-A. Lie and D.W. Schmid, On the importance of the stokes-brinkman equations for computing effective permeability in karst reservoirs. Commun. Comput. Phys. 10 (2011) 1315–1332. | DOI | Zbl

A. Lefebvre, Fluid-particle simulations with FreeFem++. ESAIM: Procs. 18 (2007) 120–132. | DOI | MR | Zbl

J.L. Lions-Magenes, Problèmes aux limites non homogènes et applications. In vol. 1 (1968). | MR | Zbl

O.K. Matar, R.V. Craster and M.R.E. Warner, Surfactant transport on highly viscous surface films. J. Fluid Mech. 466 (2002) 85–111. | DOI | Zbl

B. Maury, Direct simulations of 2D fluid-particle flows in biperiodic domains. J. Comput. Phys. 156 (1999) 325–351. | DOI | MR | Zbl

B. Maury, Numerical analysis of a finite element/volume penalty method. SIAM J. Numer. Anal. 47 (2009) 1126–1148. | DOI | MR | Zbl

S.M. Mitran, Metachronal wave formation in a model of pulmonary cilia. Comput. Struct. 85 (2007) 763–774. | DOI

J. Nečas, Les méthodes directes en théorie des équations elliptiques. Academia (1967). | MR | Zbl

E. Puchelle, J.M. Zahm and C. Duvivier, Spinability of bronchial mucus. relationship with viscoelasticity and mucous transport properties. Biorheology 20 (1983) 239–249. PMID: 6871438. | DOI

E. Puchelle, J.M. Zahm and D. Quemada, Rheological properties controlling mucociliary frequency and respiratory mucus transport. Biorheology 24 (1987) 557–563. PMID: 3502756. | DOI

M.J. Sanderson and M.A. Sleigh, Ciliary activity of cultured rabbit tracheal epithelium: beat pattern and metachrony. J. Cell Sci. 47 (1981) 331–347. | DOI

D.J. Smith, E.A. Gaffney and J.R. Blake, Modelling mucociliary clearance. Respiratory Phys. Neurobiology 163 (2008) 178–188. | DOI

D.J. Smith, E.A. Gaffney and J.R. Blake. A viscoelastic traction layer model of muco-ciliary transport. Bull. Math. Biology 69 (2007) 289–327. | DOI | MR | Zbl

P. Swarztrauber and R. Sweet, Efficient FORTRAN subprograms for the solution of elliptic partial differential equations (abstract). SIGNUM Newsl. 10 (1975). | DOI

R.A. Sweet. A parallel and vector variant of the cyclic reduction algorithm. SIAM J. Sci. Statist. Comput. 9 (1988) 761–765. | DOI | MR | Zbl

M. Thiriet, Tissue Functioning and Remodeling in the Circulatory and Ventilatory Systems. In vol. 5 in Biomathematical and Biomechanical Modeling of the Circulatory and Ventilatory Systems. Springer, Dordrecht (2012).

Cité par Sources :