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)

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

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

B. Listing, 3: MATLAB code for Figure 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)

, Cdrag=linspace(1000,10000,5)'; (300,length(Cdrag))

, length(Cdrag), vol.100

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

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

*. 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(j)

, omega(1:100,j)=omega111

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

, omega, p.333

, theta(1:100,j)=theta222

, Lambda(1:100,j)=lambda333

, 94 plot(r,omega(201:300,1),'r',r,omega, vol.300, 0201.

, 95 grid on

. C-_-{21}=1000,

. C-_-{21}=7750, C _ {21}=10000

, 104 grid on

. C-_-{21}=1000,

. C-_-{21}=7750, C _ {21}=10000, vol.106

, plot(r,Lambda(1:100,1),'r',r,Lambda(1:100,2),'b',r,Lambda(1:100,3),'g',r, Lambda(1:100,4),'k',r,Lambda(1:100,5)

. C-_-{21}=1000,

. C-_-{21}=7750, , vol.116

, /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)

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

B. Listing, , vol.4

, % 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)

, 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,

_. {1}=10, 108 'c _ {1}=100 and c _ {2}=200, vol.107

_. {1}=1000,

_. {1}=10000,

L. , :100,1),'r',r,Lambda(1:100,2),'b',r,Lambda(1:100,3),'m',r, Lambda(1:100,4)

_. {1}=10, 120 'c _ {1}=100 and c _ {2}=200, vol.119

_. {1}=1000,

_. {1}=10000,

, Relative pressure of the mixture for different drag values.') 124 xlabel('r') 125 ylabel

, /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)

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

B. Listing, 5: MATLAB code of the auxiliary function integra

, 6 f1(j)=ci+trapz(s,nu1)

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, 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