B. Listing, 1: MATLAB code for figures 3

, % This functions solves the DAE system using the function ode15s

, %\Lambda _ {rel}'=4rC _ {21}(\gamma _ 1\theta _ 2\gamma _ 2\theta _ 1)\omega _ {rel}^2

, %\theta _ 1\theta _ 2\Lambda _ {rel}=\theta _ 1\theta _ 2C _ {21}\left(\rho _ 1+\rho _ 2\ right)r^2\omega _ {rel}^2

, %+c _ 2\theta _ 1\left(\theta _ 2^2\theta _ c^2\right)

, %c _ 1\theta _ 2\left(\theta _ 1^2\theta _ {1,c}^2\right)

, Presion=sqrt(2) * A2./mus./Rf.^2, vol.28

M. , &. , and E. ,

, /2 * (Presiontheta2. * lambda). * (theta2. * gamma2. * theta1 * Cdrag. * r.^2) .^1); ,theta1,'r',r,theta2, p.75

B. Appendix, MATLAB codes for Section 3.3 76 legend('\theta _ 1','\theta _ 2') 77 title('Concentration of the fluid and the solid phases') ,omega1,'r * ',r,omega2,'b') 82 grid on

, Lambda1=(lambda+gamma2 * Lambda2')/gamma1; 96 plot(r,Lambda1,'r',r,Lambda2,'b') 97 legend('\Lambda _ 1','\Lambda _ 2') 98 title('Lagrange multipliers of the fluid and the solid phases

, /2 * (Presiontheta2 * lambda) * (theta2 * gamma2 * theta1 * Cdrag * r^2)^1)

, )=lambda * theta2 * theta1+(theta2 * gamma2+gamma1 * theta1) * Cdrag * r^2 * velr^2 * theta2 * theta1

. Fval=zeros, , vol.10

, Fval(k)] = fsolve(@(x) FindBoundaryConditions(x,Presion,thetac2, thetac1,c2,c1,gamma1,gamma2,Cdrag,cbvel,Rf),ci(k))

, sys=(Presion+2/3 * gamma2 * x * (1x) * Cdrag * Rf^2 * cbvel^2) * (1x)

, +Rf^2 * cbvel^2 * Cdrag * (gamma1+(gamma2gamma1) * x) * x * (1x)

, 137 +c2 * (x^2thetac2^2) * (1x)c1 * x * ((1x)^2thetac1^2)

B. Listing, MATLAB code for Figure, vol.2

, % This functions solves the DAE system using the function ode15s

, %\Lambda _ {rel}'=4rC _ {21}(\gamma _ 1\theta _ 2\gamma _ 2\theta _ 1)\omega _ {rel}^2

, %\theta _ 1\theta _ 2\Lambda _ {rel}=\theta _ 1\theta _ 2C _ {21}\left(\rho _ 1+\rho _ 2\ right)r^2\omega _ {rel}^2

, %+c _ 2\theta _ 1\left(\theta _ 2^2\theta _ c^2\right)

, %c _ 1\theta _ 2\left(\theta _ 1^2\theta _ {1,c}^2\right)

. A=linspace,

B. Appendix, MATLAB codes for Section 3.3 j=1:1:length(A) 26

*. A1=cbtheta1,

, Presion=sqrt(2) * A2./mus./Rf.^2, vol.34

M. , &. , and E. , 48 options = odeset('Mass

*. A1=cbtheta1,

, A2=A(j)A1

, omega(j,3)=velr(end)

A. and O. ,

, 79 grid on; 80 legend('\omega _ 1(R _ 1)','\omega _ 2(R _ 1)') 81 title

, /2 * (Presiontheta2 * lambda) * (theta2 * gamma2 * theta1 * Cdrag * r^2)^1)

, )=lambda * theta2 * theta1+(theta2 * gamma2+gamma1 * theta1) * Cdrag * r^2 * velr^2 * theta2 * theta1

, Rf), vol.109, p.2

. Fval=zeros, , vol.10

, = fsolve(@(x) FindBoundaryConditions(x,Presion,thetac2, thetac1,c2,c1,gamma1,gamma2,Cdrag,cbvel,Rf),ci(k)); sys=FindBoundaryConditions(x,Presion

, sys=(Presion+2/3 * gamma2 * x * (1x) * Cdrag * Rf^2 * cbvel^2) * (1x)

, %\Lambda _ {rel}'=4rC _ {21}(\gamma _ 1\theta _ 2\gamma _ 2\theta _ 1)\omega _ {rel}^2

, %\theta _ 1\theta _ 2\Lambda _ {rel}=\theta _ 1\theta _ 2C _ {21}\left(\rho _ 1+\rho _ 2\ right)r^2\omega _ {rel}^2

, %+c _ 2\theta _ 1\left(\theta _ 2^2\theta _ c^2\right)

, %c _ 1\theta _ 2\left(\theta _ 1^2\theta _ {1,c}^2\right)

, length(Cdrag), vol.100

. Lambda=zeros, length(Cdrag), vol.100

, 26 for j=1:1:length(c11)

*. A1=cbtheta1,

, Presion=sqrt(2) * A2./mus./Rf.^2, vol.35

M. , &. , and E. , 49 options = odeset('Mass

, /2 * (Presiontheta2. * lambda). * (theta2. * gamma2. * theta1 * Cdrag. * r.^2) .^1)

, omega(1:100,j)=omega111

, 87 omega(101:200,j)=omega222

, omega, p.333

, theta(1:100,j)=theta222

, Lambda(1:100,j)=lambda333

O. , 300,1),'r',r,omega(201:300,2),'b',r,omega(201:300,3),'m',r, omega(201:300,4)

_. {1}=10, 97 'c _ {1}=100 and c _ {2}=200

_. {1}=1000,

_. {1}=10000,

L. B?lilescu, J. San-martín, and T. Takahashi, On the Navier-Stokes system with the Coulomb friction law boundary condition. submitted, 2016.

L. B?lilescu, J. San-martín, and T. Takahashi, Fluid-Rigid Structure Interaction System with Coulomb's Law, SIAM J. Math. Anal, vol.49, issue.6, pp.4625-4657, 2017.

T. Barker, D. G. Schaeffer, M. Shearer, and J. M. Gray, Well-posed continuum equations for granular flow with compressibility and µ(i)-rheology, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol.473, 2017.

H. Beirão and . Veiga, On the existence of strong solutions to a coupled fluid-structure evolution problem, J. Math. Fluid Mech, vol.6, issue.1, pp.21-52, 2004.

E. C. Bingham, An investigation of the laws of plastic flow, Bulletin of the Bureau of Standards, vol.13, pp.309-353, 1916.

C. Bost, G. 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, vol.48, issue.4, pp.1313-1337, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00352808

M. Boulakia, Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid, J. Math. Pures Appl, vol.84, issue.9, pp.1515-1554, 2005.

M. Boulakia, Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid, J. Math. Fluid Mech, vol.9, issue.2, pp.262-294, 2007.

M. Boulakia and S. Guerrero, A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.26, issue.3, pp.777-813, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00538038

M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluidstructure interaction problem, ESAIM Control Optim. Calc. Var, vol.14, issue.1, pp.1-42, 2008.
URL : https://hal.archives-ouvertes.fr/inria-00542535

M. Boulakia, E. Schwindt, and T. Takahashi, Existence of strong solutions for the motion of an elastic structure in an incompressible viscous fluid, Interfaces Free Bound, vol.14, issue.3, pp.273-306, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00765176

M. Bravin, On the weak uniqueness of viscous incompressible fluid + rigid body system with Navier slip-with-friction conditions in a 2D bounded domain, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01740859

H. Brenner and R. G. Cox, The resistance to a particle of arbitrary shape in translational motion at small reynolds numbers, Journal of Fluid Mechanics, vol.17, issue.4, pp.561-595, 1963.

L. B?lilescu, J. San-martín, and T. Takahashi, On the Navier-Stokes system with the Coulomb friction law boundary condition, Z. Angew. Math. Phys, vol.68, issue.1, 2017.

M. Bulí-?ek and J. Málek, On unsteady internal flows of Bingham fluids subject to threshold slip on the impermeable boundary. In Recent developments of mathematical fluid mechanics, Adv. Math. Fluid Mech, pp.135-156, 2016.

M. Bulí?ek, E. Feireisl, and J. Málek, A navier-stokes-fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Analysis: Real World Applications, vol.10, issue.2, pp.992-1015, 2009.

M. Bulí?ek, R. Lewandowski, and J. Málek, On evolutionary navier-stokesfourier type systems in three spatial dimensions, Commentationes Mathematicae Universitatis Carolinae, vol.52, pp.89-114, 2011.

M. Bulí?ek, J. Málek, and K. Rajagopal, Navier's slip and evolutionary navier-stokes-like systems with pressure and shear-rate dependent viscosity, Indiana University Mathematics Journal, vol.56, issue.1, pp.51-85, 2007.

N. V. Chemetov and ?. Ne?asová, The motion of the rigid body in the viscous fluid including collisions. Global solvability result, Nonlinear Anal. Real World Appl, vol.34, pp.416-445, 2017.

N. V. Chemetov, ?. Ne?asová, and B. Muha, Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition, 2017.

C. Conca, On the application of the homogenization theory to a class of problems arising in fluid mechanics, J. Math. Pures Appl, vol.64, issue.9, pp.31-75, 1985.

C. Conca, J. San-martín, H. , and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, vol.25, issue.5-6, pp.1019-1042, 2000.

L. Consiglieri, Weak solutions for a class of non-newtonian fluids with energy transfer, Journal of Mathematical Fluid Mechanics, vol.2, issue.3, pp.267-293, 2000.

L. Consiglieri, J. F. Rodrigues, and T. Shilkin, On the navier-stokes equations with energy-dependent nonlocal viscosities, Journal of Mathematical Sciences, vol.130, issue.4, pp.4814-4826, 2005.

M. D. Cooley and M. E. O'neill, On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere, Mathematika, vol.16, issue.1, pp.37-49, 1969.

D. Coutand and S. Shkoller, Motion of an elastic solid inside an incompressible viscous fluid, Arch. Ration. Mech. Anal, vol.176, issue.1, pp.25-102, 2005.

D. Coutand and S. Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Ration. Mech. Anal, vol.179, issue.3, pp.303-352, 2006.

P. Cumsille and T. Takahashi, Wellposedness for the system modelling the motion of a rigid body of arbitrary form in an incompressible viscous fluid, Czechoslovak Math. J, vol.58, issue.133, pp.961-992, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00169087

P. A. Cundall and O. D. Strack, A discrete numerical model for granular assemblies, Géotechnique, vol.29, issue.1, pp.47-65, 1979.

S. Frédéric-da-cruz, M. Emam, J. Prochnow, F. Roux, and . Chevoir, Rheophysics of dense granular materials: Discrete simulation of plane shear flows, Phys. Rev. E, vol.72, p.21309, 2005.

E. J. Dean, R. Glowinski, and G. Guidoboni, On the numerical simulation of Bingham visco-plastic flow: Old and new results, Journal of Non-Newtonian Fluid Mechanics, vol.142, issue.1, pp.36-62, 2007.

B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal, vol.146, issue.1, pp.59-71, 1999.

B. Desjardins and M. J. Esteban, On weak solutions for fluid-rigid structure interaction: compressible and incompressible models, Comm. Partial Differential Equations, vol.25, issue.7-8, pp.1399-1413, 2000.

R. J. Diperna and P. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math, vol.98, issue.3, pp.511-547, 1989.

D. C. Drucker and W. Prager, Soil mechanics and plastic analysis or limit design, Quarterly of Applied Mathematics, vol.10, issue.2, pp.157-165, 1952.

G. Duvaut, Mathématiques appliquées pour la maîtrise. Dunod, 1998.

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

G. Duvaut and J. Lions, Transfert de chaleur dans un fluide de bingham dont la viscosity depend de la temperature, Journal of Functional Analysis, vol.11, issue.1, pp.93-110, 1972.

G. Duvaut and J. Lions, Grundlehren der Mathematischen Wissenschaften, p.219, 1976.

J. I. Díaz and G. Galiano, Existence and uniqueness of solutions of the boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal, vol.11, issue.1, pp.59-82, 1998.

I. Liu, A solid-fluid mixture theory of porous media, International Journal of Engineering Science, vol.84, pp.133-146, 2014.

I. Liu, A solid-fluid mixture theory of porous media, Internat. J. Engrg. Sci, vol.84, pp.133-146, 2014.

J. Málek and K. R. , A thermodynamic framework for a mixture of two liquids, Nonlinear Anal. Real World Appl, vol.9, issue.4, pp.1649-1660, 2008.

J. Málek, J. Ne?as, and K. R. , Global analysis of the flows of fluids with pressure-dependent viscosities, Archive for Rational Mechanics and Analysis, vol.165, issue.3, pp.243-269, 2002.

E. Maringova and J. Zabensky, On a navier-stokes-fourier-like system capturing transitions between viscous and inviscid fluid regimes and between no-slip and perfectslip boundary conditions, Nonlinear Analysis: Real World Applications, vol.41, pp.152-178, 2018.

B. Maury, Direct simulations of 2D fluid-particle flows in biperiodic domains, J. Comput. Phys, vol.156, issue.2, pp.325-351, 1999.

C. Meruane, A. Tamburrino, and O. Roche, Dynamics of dense granular flows of smalland-large-grain mixtures in an ambient fluid, Phys. Rev. E, vol.86, p.26311, 2012.

P. P. Mosolov and V. P. Miasnikov, Variational methods in the theory of the fluidity of a viscous-plastic medium, Journal of Applied Mathematics and Mechanics, vol.29, issue.3, pp.545-577, 1965.

P. P. Mosolov and V. P. Miasnikov, On stagnant flow regions of a viscous-plastic medium in pipes, Journal of Applied Mathematics and Mechanics, vol.30, issue.4, pp.841-854, 1966.

J. Necas, C. G. Simader, ?. Necasová, G. Tronel, and A. Kufner, Direct Methods in the Theory of Elliptic Equations, 2011.

?. Ne?asová and ;. Ferrara, On the motion of several rigid bodies in an incompressible nonnewtonian and heat-conducting fluid, vol.55, pp.325-352, 2009.

W. Jace, E. K. Nunziato, and . Walsh, On ideal multiphase mixtures with chemical reactions and diffusion. Archive for Rational Mechanics and Analysis, vol.73, pp.285-311, 1980.

J. G. Oldroyd, A rational formulation of the equations of plastic flow for a Bingham solid, Proc. Cambridge Philos. Soc, vol.43, pp.100-105, 1947.

J. Ortega, L. Rosier, and T. Takahashi, On the motion of a rigid body immersed in a bidimensional incompressible perfect fluid, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.24, issue.1, pp.139-165, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00142401

J. H. Ortega, L. Rosier, and T. Takahashi, Classical solutions for the equations modelling the motion of a ball in a bidimensional incompressible perfect fluid, M2AN Math. Model. Numer. Anal, vol.39, issue.1, pp.79-108, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00091625

P. D. Panagiotopoulos, Inequality problems in mechanics and applications, Convex and nonconvex energy functions, 1985.

S. L. Passman, J. W. Nunziato, and E. K. Walsh, A Theory of Multiphase Mixtures, pp.286-325, 1984.

S. Passman, Mixtures of granular materials, International Journal of Engineering Science, vol.15, issue.2, pp.117-129, 1977.

W. Prager, On slow visco-plastic flow, Studies in mathematics and mechanics presented to Richard von Mises, pp.208-216, 1954.

A. Putz, A. Ian, and . Frigaard, Creeping flow around particles in a bingham fluid, Journal of Non-Newtonian Fluid Mechanics, vol.165, issue.5, pp.263-280, 2010.

L. Tartar, The general theory of homogenization, Lecture Notes of the Unione Matematica Italiana, vol.7, 2009.

R. Temam, Navier-Stokes equations, Studies in Mathematics and its Applications, vol.2, 1979.

R. Temam, Mathematical Problems in Plasticity, Modern Applied Mathematics Series. Gauthier-Villars, 1985.

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Society for Industrial and Applied Mathematics, 1995.

,. Ho-touloukian and K. Powell, TPRC Data Series, vol.1, issue.3, 1970.

C. , Rational thermodynamics, 1984.

J. L. Vázquez and E. Zuazua, Large time behavior for a simplified 1D model of fluid-solid interaction, Comm. Partial Differential Equations, vol.28, issue.9, pp.1705-1738, 2003.

J. L. Vázquez and E. Zuazua, Lack of collision in a simplified 1D model for fluid-solid interaction, Math. Models Methods Appl. Sci, vol.16, issue.5, pp.637-678, 2006.

Y. Wang and K. Hutter, A constitutive model of multiphase mixtures and its application in shearing flows of saturated solid-fluid mixtures, Granular Matter, vol.1, issue.4, pp.163-181, 1999.

H. F. Weinberger, On the steady fall of a body in a Navier-Stokes fluid, Partial differential equations, vol.XXIII, pp.421-439, 1971.

J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mech, vol.9, issue.1, pp.104-138, 2007.

W. P. Ziemer, Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, 2012.

, Nous supposons dans un premier temps que la dynamique du matériau granulaire est modélisée à l'aide d'une approche fondée sur la théorie du mélange. D'autre part pour le deux modèles restant, nous considérons que le fluide granulaire est modélisé à l'aide d'une approche multiphase associant des structures et des fluides rigides. Plus exactement : -Dans le premier modèle, nous avons obtenu un ensemble d'équations basées sur la théorie du mélange en utilisant des outils d'homogénéisation et une procédure thermodynamique. Ces équations reflètent deux propriétés essentielles des fluides granulaires : la nature visqueuse du fluide interstitiel et un comportement de type Coulomb de la composante granulaire. Avec nos équations, Cette thèse vise à obtenir et à développer des modèles mathématiques pour comprendre certains aspects de la dynamique des fluides granulaires hétérogènes. Plus précisément, le résultat attendu consiste à développer trois modèles

. Dans-le-deuxième-modèle, Nous obtenons un problème à frontière libre dû au mouvement du corps rigide. Le résultat est obtenu en utilisant un problème approximé et en le passant à la limite. Les problèmes approchés considèrent la régularisation des termes convexes dans le fluide de Bingham. Nous utilisons une méthode de pénalisation pour prendre en compte la présence du corps rigide. -Dans le troisième modèle, nous considérons le mouvement d'un corps rigide conducteur thermique parfait dans un fluide newtonien conducteur de la chaleur. Les équations 3D de Fourier-Navier-Stokes modélisent le fluide, tandis que les lois de Newton et l'équilibre de l'énergie interne modélisent le déplacement du corps rigide. Notre principal objectif dans cette partie est de prouver l'existence d'une solution faible pour le système correspondant. La formulation faible est composée de l'équilibre entre la quantité du mouvement et l'équation de l'énergie totale, qui inclut la pression du fluide, et implique une limite libre due au mouvement du corps rigide. Pour obtenir une pression intégrable, nous considérons une condition au limite de glissement de Navier pour la limite extérieure et l'interface mutuelle, nous considérons le mouvement d'un corps rigide dans un matériau viscoplastique. Les équations 3D de Bingham modélisent ce matériau et les lois de Newton régissent le déplacement du corps rigide

. Mots-clés, Solution faible, fluides de Bingham, interaction fluide-structure, théorie des mélanges