J. C. Adams, mudpack: Multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations, Applied Mathematics and Computation, vol.34, issue.2, pp.113-146, 1989.

A. Robert, J. J. Adams, and . Fournier, , 2003.

E. C. Bingham, Fluidity and Plasticity, 1922.

F. Boyer and P. Fabrie, Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles, 2005.
DOI : 10.1007/3-540-29819-3

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00777731

B. Button, L. Cai, C. Ehre, M. Kesimer, D. B. Hill et al., Periciliary brush promotes the lung health by separating the mucus layer from airway epithelia, Science, vol.337, issue.6097, pp.937-941, 2012.

G. Carbou and P. Fabrie, Boundary layer for a penalization method for viscous incompressible flow, Differential Equations, vol.8, issue.12, pp.1453-1480, 2003.
URL : https://hal.archives-ouvertes.fr/hal-00295077

P. Carreau, Rheological equations from molecular network theories, Trans. Soc. Rheol, vol.16, p.99, 1972.
DOI : 10.1122/1.549276

R. Chatelin, D. Anne-archard, M. Murris-espin, D. Sanchez, M. Thiriet et al., Chapter 5-Modeling Cystic Fibrosis and Mucociliary Clearance, Modeling of Microscale Transport in Biological Processes, pp.113-154, 2017.
DOI : 10.1016/b978-0-12-804595-4.00005-5
URL : https://hal.inria.fr/hal-01476216/document

R. Chatelin, D. Anne-archard, M. Murris-espin, M. Thiriet, and P. Poncet, Numerical and experimental investigation of mucociliary clearance breakdown in cystic fibrosis, Journal of Biomechanics, vol.53, pp.56-63, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01476172

R. Chatelin and P. Poncet, A hybrid grid-particle method for moving bodies in 3D stokes flow with variable viscosity, SIAM Journal on Scientific Computing, vol.35, issue.4, pp.925-949, 2013.
URL : https://hal.archives-ouvertes.fr/hal-02010645

R. Chatelin and P. Poncet, Hybrid grid-particle methods and penalization: A Sherman-MorrisonWoodbury approach to compute 3D viscous flows using FFT, Journal of Computational Physics, vol.269, pp.314-328, 2014.
DOI : 10.1016/j.jcp.2014.03.023

R. Chatelin and P. Poncet, A parametric study of mucociliary transport by numerical simulations of 3d non-homogeneous mucus, Journal of Biomechanics, vol.49, issue.9, pp.1772-1780, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01996103

R. Chatelin, D. Sanchez, and P. Poncet, Analysis of the penalized 3d variable viscosity stokes equations coupled to diffusion and transport, ESAIM: Mathematical Modelling and Numerical Analysis, vol.50, issue.2, pp.565-591, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01872062

G. Cottet, R. Hildebrand, P. Koumoutsakos, and C. Mimeau, Iraj Mortazavi, and Philippe Poncet. Passive and active flow control using vortex methods, 6th International Conference on Vortex Flows and Vortex Models, 2014.

G. H. Cottet, Vortex methods: theory and practice, 2001.

M. Veronica, D. Angelo, H. Auradou, C. Allain, and J. Hulin, Pore scale mixing and macroscopic solute dispersion regimes in polymer flows inside two-dimensional model networks, Physics of Fluids, vol.19, issue.3, p.33103, 2007.

A. Decoene, S. Martin, and B. Maury, Microscopic modelling of active bacterial suspensions, Mathematical Modelling of Natural Phenomena, vol.6, issue.5, pp.98-129, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00601842

L. Diening, Theoretical and Numerical Results for Electrorheological Fluids, 2002.

L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol.2017, 2011.

L. Diening, P. Hästö, and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, FSDONA04 Proceedings, vol.3858, 2004.

G. Duvaut and J. L. Lions, Les inéquations en mécanique et en physique, 1972.

V. John, B. Fahy, and . Dickey, Airway mucus function and dysfunction. The New England journal of medicine, vol.363, p.21121836, 2010.

C. Foias and R. Temam, Remarques sur les équations de navier-stokes stationnaires et les phénomènes successifs de bifurcation, vol.5, pp.29-63, 1978.

L. M. Joana, M. Ganter, M. Milas, and . Rinaudo, On the viscosity of sodium poly(styrene sulphonate), a flexible polyelectrolyte, Polymer, vol.33, issue.1, pp.113-116, 1992.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 1983.

H. Giesekus, A simple constitutive equation for polymer fluids based on the concept of deformation dependent tensorial mobility, J. Non-Newtonian Fluid Mech, vol.11, p.69, 1982.

J. L. Guermond, P. Minev, and J. Shen, An overview of projection methods for incompressible flows, Computer Methods in Applied Mechanics and Engineering, vol.195, pp.6011-6045, 2006.

W. H. Herschel and R. Bulkley, Konsistenzmessungen von Gummi-Benzollösungen, Kolloid Zeitschrift, vol.39, pp.291-300, 1926.

C. P. Kelco, Keltrol/Kelzan, Xanthan gum book 8th edition, 2007.

M. Krotkiewski, I. S. Ligaarden, K. Lie, and D. W. Schmid, On the importance of the stokes-brinkman equations for computing effective permeability in karst reservoirs, Communications in Computational Physics, 2011.

K. Samuel, Y. Lai, D. Wang, J. Wirtz, and . Hanes, Micro-and macrorheology of mucus, Advanced Drug Delivery Reviews, vol.61, issue.2, pp.86-100, 2009.

P. Lindqvist, Notes on the p-Laplace equation, 2006.

J. Lions, Quelques méthodes de résolution des problemes aux limites non linéaires, vol.31, 1969.

J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, vol.1, 1968.

J. J. Monaghan, Extrapolating B Splines for Interpolation, Journal of Computational Physics, vol.60, issue.2, pp.253-262, 1985.

, Jind?ich Ne?as. Les méthodes directes en théorie des équations elliptiques. Academia, 1967.

B. Noetinger, L. Hume, R. Chatelin, and P. Poncet, The effective viscosity of a random mixture of fluids, Physical Review Fluids, vol.3, p.14103, 2018.

J. G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. London, p.523, 1950.

E. Puchelle, D. Zahm, and . Quemada, Rheological properties controlling mucociliary frequency and respiratory mucus transport, Biorheology, vol.24, issue.6, p.3502756, 1987.

M. Le-ravalec, B. Noetinger, and L. H. Hu, The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning gaussian simulations, Mathematical Geology, vol.32, pp.701-723, 2000.

M. J. Sanderson and M. A. Sleigh, Ciliary activity of cultured rabbit tracheal epithelium: beat pattern and metachrony, Journal of Cell Science, vol.47, issue.1, pp.331-347, 1981.

D. J. Smith, E. A. Gaffney, and J. R. Blake, Modelling mucociliary clearance, Respiratory Physiology & Neurobiology, vol.163, pp.178-188, 2008.

G. A. Stahl and D. N. Schulz, Water-Soluble Polymers for Petroleum Recovery, 2012.

P. Swarztrauber and R. Sweet, Efficient FORTRAN subprograms for the solution of elliptic partial differential equations (abstract), SIGNUM Newsl, vol.10, issue.4, 1975.

M. D. Torres, B. Hallmark, D. I. Wilson, and L. Hilliou, Natural Giesekus fluids: Shear and extensional behavior of food gum solutions in the semidilute regime, AIChE Journal, vol.60, issue.11, pp.3902-3915, 2014.

E. Zeidler, Nonlinear Functional Analysis and its Applications II/B, 1990.

L. Zhong, M. Oostrom, M. J. Truex, V. R. Vermeul, and J. E. Szecsody, Rheological behavior of xanthan gum solution related to shear thinning fluid delivery for subsurface remediation, Journal of Hazardous Materials, pp.160-170, 2013.