In this thesis, the theory of poroelasticity, namely the Mixture Theory version -- a homogenized, macroscopic scale approach used to describe fluid flow through a porous medium -- is employed in three separate cases pertaining to a biological phenomenon. The first investigation explores the behavior of interstitial fluid pressure (IFP) in solid tumors. Thus, in Chapter 2, a Mixture Theory based approach is developed to describe the evolution of the IFP from that in a healthy interstitium to the elevated levels in cancerous tumors. Attention is focused on angiogenesis, a tightly regulated process in healthy tissue that provides all necessary nutrients through the creation of new blood vessels. Once this process becomes unruly within a tumor, angiogenesis gives rise to an abnormal vasculature by forming convoluted and leaky blood vessels. Thus, the primary focus of the model is on the capillary filtration coefficient and vascular density as they increase in time, which in turn elevates the tumor IFP. Later, the Mixture Theory model is extended to simulate the effects of vascular normalization, where the cancer therapy not only prunes blood vessels, but reverts the chaotic vasculature to a somewhat normal state, thereby temporarily lowering the tumor IFP. In Chapter 3, the validity of an assumption that was made in order to facilitate the mathematical calculations is investigated. In addition to all of the Mixture Theory assumptions, it is assumed that the pore pressure p is proportional to the tissue dilatation e. This assumption is examined to determine how appropriate and accurate it is, by using a heat type equation without the presence of sources and sinks under the assumption of a spherical geometry. The results obtained under the proportionality of p and e, are compared with the results obtained without this assumption. A substantial difference is found, which suggests that great care must be exercised in assuming the proportionality of p and e. The last application is reported in Chapter 4 and it investigates the pathogenesis of normal pressure hydrocephalus. In a normal brain, cerebrospinal fluid (CSF) is created by the choroid plexus, circulates around the brain and the spinal cord without any impediment, and then is absorbed at various sites. However, normal pressure hydrocephalus occurs when there is an imbalance between the production and absorption of CSF in the brain that causes the impaired clearance of CSF and the enlargement of ventricles; however, the ventricular pressure in this case is frequently measured to be normal. Thus, a mathematical model using Mixture Theory is formulated to analyze a possible explanation of this brain condition. Levine (1999) proposed the hypothesis that CSF seeps from the ventricular space into the brain parenchyma and is efficiently absorbed in the bloodstream. To test this hypothesis, Levine used the consolidation theory version of poroelasticity theory, with the addition of Starling's law to account for the absorption of CSF in the brain parenchyma at steady state. However, the Mixture Theory model does not agree with the results obtained by Levine (1999) which leads one to conclude that the pathogenesis of normal pressure hydrocephalus remains unknown. To conclude the thesis, all three applications of Mixture Theory are discussed and the importance and contribution of this work is highlighted. In addition, possible future directions are indicated based on the findings of this thesis.

We consider the system of equations arising from mantle dynamics introduced by McKenzie (J. Petrology, 1985). In this multi-phase model, the fluid melt velocity obeys Darcy's law while the deformable "solid" matrix is governed by a highly viscous Stokes equation. The system is then coupled through mass conservation and compaction relations. Together these equations form a coupled Darcy-Stokes system on a continuous single-domain mixture of fluid and matrix. The porosity φ, representing the relative volume of fluid melt to the bulk volume, is assumed to be much smaller than one. When coupled with solute transport and thermal evolution in a time-dependent problem, the model transitions dynamically from a non-porous single phase solid to a two-phase porous medium. Such mixture models have an advantage for numerical approximation since the free boundary between the one and two-phase regions need not be determined explicitly. The equations of mantle dynamics apply to a wide range of applications in deep earth physics such as mid-ocean ridges, subduction zones, and hot-spot volcanism, as well as to glacier dynamics and other two-phase flows in porous media. Mid-ocean ridges form when viscous corner flow of the solid mantle focuses fluid toward a central ridge. Melt is believed to migrate upward until it reaches the lithospheric "tent" where it then moves toward the ridge in a high porosity band. Simulation of this physical phenomenon required confidence in numerical methods to handle highly heterogeneous porosity as well as the single-phase to two-phase transition. In this work we present a standard mixed finite element method for the equations of mantle dynamics and investigate its limitations for vanishing porosity. While stable and optimally convergent for porosity bounded away from zero, the stability estimates we obtain suggest, and numerical results show, the method becomes unstable as porosity approaches zero. Moreover, the fluid pressure is no longer a physical variable when the fluid phase disappears and thus is not a good variable for numerical methods. Inspired by the stability estimates of the standard method, we develop a novel stable mixed method with uniqueness and existence of solutions by studying a linear degenerate elliptic sub-problem akin to the Darcy part of the full model: [mathematical equation], where a and b satisfy a(0)=b(0)=0 and are otherwise positive, and the porosity φ ≥ 0 may be zero on a set of positive measure. Using scaled variables and mild assumptions on the regularity of φ, we develop a practical mass-conservative method based on lowest order Raviart-Thomas finite elements. Finally, we adapt the numerical method for the sub-problem to the full system of equations. We show optimal convergence for sufficiently smooth solutions for a compacting column and mid-ocean ridge-like corner flow examples, and investigate accuracy and stability for less regular problems / text

New unsaturated coupled models have been developed for fluid transport in deformable rock by using modified mixture theory rather than a fully mechanics-based approach. These models include the following: an unsaturated hydro-mechanical coupled model for both non-swelling and swelling materials, in which a new coupled formulation for hydration swelling rock has been included; and an unsaturated hydro-mechanical-chemo coupled model, incorporating a new coupled formulation including osmosis flow and an unsaturated version of Darcy's law which has been extended by including osmosis effects.Modified mixture theory is mainly based on non-equilibrium thermodynamics. Helmholtz free energy is used to give the energy relationship between the fluids and solid and, by using the Gibbs-Duhem equation, the interactions between different fluids such as gas, water and chemical can be obtained. In this research, general coupled formulations for both large small and deformations have been obtained. For swelling rocks, the water between the clay platelets can be modeled by including the difference between the free energy of whole domain and that of the pore water plus the solid skeleton. By assuming small deformations, the final equations can be compared with those derived using the mechanics approach.The new coupled models have been tested by carrying out simple benchmark numerical simulations using finite elements. Problems analyzed include: (1) the consolidation of saturated swelling rocks in which the hydration swelling effects on consolidation have been analysed in detail; (2) the desaturation and resaturation of seasonally affected rocks around tunnels; (3) the desaturation stage for swelling rocks used in the containment of nuclear waste disposal; (4) chemical transport in very low permeability rock used for nuclear waste disposal, in which particular attention has been focused on osmosis flow and chemical consolidation. In summary, this thesis extends modified mixture theory and develops new coupled formulations which can be applied to deep nuclear waste disposal, including tunnelling, drilling and chemical transport in low permeability host rock.

Following trauma, such as an impact injury or stroke, brain tissue can swell. Swelling is the result of water accumulation in the tissue that is driven by pathological changes, such as increased permeability of the capillary walls and osmotic pressure changes within the tissue. Swelling causes increased intracranial pressure and mechanical deformation of the brain tissue, exacerbating the original injury. Furthermore, prolonged local swelling can lead to the spread of damage to the (initially undamaged) surrounding tissue, since compression and increased intracranial pressure may restrict blood flow in this tissue. In this thesis, we develop mathematical models to examine the consequences of pathophysiological damage mechanisms on the swelling, and associated stress and strain, experienced by brain tissue. Mixture theory is used to represent brain tissue as a mixture of elastic solid, fluid and solutes. This modelling approach allows elastic deformations to be coupled with hydrodynamic pressure and osmotic gradients; the consequences of different mechanisms of damage may then be quantified. We consider three particular problems motivated by experimental observations of swelling brain tissue. Firstly, we investigate the swelling of isolated, damaged, brain tissue slices; we show that mechanisms leading to an osmotic pressure difference between the tissue slice and its surroundings can explain experimental observations for swollen tissue slices. Secondly, we use our modelling approach to demonstrate that local changes in capillary permeability can cause significant stresses and strains in the surrounding tissue. Thirdly, we investigate the conditions under which a locally swollen, damaged, region can cause compression of the vasculature within the surrounding tissue, and potentially result in damage propagation. To do this, we propose a coupled model for the oxygen concentration within, and mechanical deformation of, brain tissue. We use our model to assess the impact of treatment strategies on damage propagation through the tissue, and show that performing a craniectomy reduces the extent of propagation.

Low back pain represents a significant concern in the United States, with 70% of individuals experiencing symptoms at some point in their lifetime. Although the specific cause of low back pain remains unclear, symptoms have been strongly associated with degeneration of the intervertebral disc. Insufficient nutritional supply to the disc is believed to be a major mechanism for tissue degeneration. Understanding nutrients' transport in intervertebral disc is crucial to elucidate the mechanisms of disc degeneration, and to develop strategies for tissue repair (in vivo), and tissue engineering (in vitro). Transport in intervertebral disc is complex and involves a series of electromechanical, chemical and biological coupled events. Despite of the large amount of studies performed in the past, transport phenomena in the disc are still poorly understood. This is partly due to the limited number of available experimental techniques for investigating transport properties, and the paucity of theoretical or numerical methods for systematically predicting the mechanisms of solute transport in intervertebral disc. In this dissertation, a theoretical and experimental approach was taken in order to investigate the mechanisms of solute transport and binding interactions in intervertebral disc. New imaging techniques were developed for the experimental determination of diffusive and binding parameters in biological tissues. The techniques are based on the principle of fluorescence recovery after photobleaching, and allow the determination of the anisotropic diffusion tensor, and the rates of binding and unbinding of a solute to the extracellular matrix of a biological tissue. When applied to the characterization of transport properties of intervertebral disc, these methods allowed the establishment of a relationship between solute anisotropic and inhomogeneous diffusivity and the unique morphology of human lumbar annulus fibrosus. A mixture theory for charged hydrated soft tissues was presented as a framework for theoretical investigations on solute transport and binding interactions in cartilaginous tissues. Based on this theoretical framework and on experimental observations, a finite element model was developed to predict solute diffusive-convective-reactive transport in cartilaginous tissues. The numerical model was applied to simulate the effect of mechanical loading on solute transport and binding interactions in cartilage explants and intervertebral disc.

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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. 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 les 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, nous étudions le problème de Couette entre deux cylindres infinis d’un écoulement hétérogène granulaire dense, composé d’un fluide newtonien et d’une composante solide. • Dans le deuxième modèle, 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. Notre résultat principal est d’établir l’existence d’une solution faible pour le système correspondant. • 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 / This Ph.D. thesis aims to obtain and to develop some mathematical models to understand some aspects of the dynamics of heterogeneous granular fluids. More precisely, the expected result is to develop three models, one where the dynamics of the granular material is modeled using a mixture theory approach, and the other two, where we consider the granular fluid is modeled using a multiphase approach involving rigid structures and fluids. More precisely : • In the first model, we obtained a set of equations based on the mixture theory using homogenization tools and a thermodynamic procedure. These equations reflect two essential properties of granular fluids : the viscous nature of the interstitial fluid and a Coulomb-type of behavior of the granular component. With our equations, we study the problem of a dense granular heterogeneous flow, composed by a Newtonian fluid and a solid component in the setting of the Couette flow between two infinite cylinders. • In the second model, we consider the motion of a rigid body in a viscoplastic material. The 3D Bingham equations model this material, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. • In the third model, we consider the motion of a perfect heat conductor rigid body in a heat conducting Newtonian fluid. The 3D Fourier-Navier-Stokes equations model the fluid, and the Newton laws and the balance of internal energy model the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is composed by the balance of momentum and the balance of total energy equation which includes the pressure of the fluid, and it involves a free boundary (due to the motion of the rigid body). To obtain an integrable pressure, we consider a Navier slip boundary condition for the outer boundary and the mutual interface

Le mélanome est un cancer dont la mortalité augmente rapidement avec le temps. Afin d'assurer une détection précoce, des campagnes de sensibilisation ont été menées donnant des critères morphologiques pour le distinguer des grains de beauté. Mais, l'origine des différences d'aspects entre lésions bénignes et malignes reste inconnue. L'objectif est ici de relier les effets des modifications génétiques à l'aspect des tumeurs, en utilisant des outils venus de la physique macroscopique. Les mélanomes ont l'avantage d'être facilement observables et fins, ce qui en font un système idéal. Ce travail commence par rappeler les aspects physiologiques des cancers de la peau. On explique le fonctionnement de la peau saine, puis nous décrivons les différents types de lésions cutanées, et enfin nous donnons un bref aperçu des différents chemins génétiques connus menant au mélanome. Ensuite, nous faisons un rappel des différents modèles mathématiques du cancer. Nous nous attardons sur l'utilisation de la théorie des mélanges comme base théorique de mise en équation des tumeurs. Nous l'appliquons ensuite dans un modèle simplifié à deux phases en deux dimensions. Puis, nous analysons ces équations. Une étude des composantes spatiales montre la possibilité d'un processus de séparation de phases : la décomposition spinodale. L'étude temporelle permet de montrer que ces équations contiennent les ingrédients nécessaires à décrire plusieurs types de mélanomes observés in vivo. Nous terminons par l'étude des effets de la troisième dimension jusqu'alors mis de côté dans le modèle. Nous mettons en équation des mélanomes évoluant sur un épiderme ondulé, au niveau des mains et des pieds. / Melanoma is a cancer whose mortality grows rapidly with time. In order to insure an early diagnosis, advertising campaigns have emphasized the importance of morphological criteria in order to distinguish moles from melanoma. But, the origins of those criteria are still poorly understood. Our goal is to understand the link between genetic modifications and melanoma patterns using physical tools. As melanoma is easily observable and thin, this makes it an ideal system. This work begins by recalling the physiological aspect of skin cancer. Healthy skin is thoroughly described, then cancerous lesions are depictesd, and melanoma genetic pathways are briefly discussed. Then, continuous mathematical models of cancer are reviewed. We show how mixture theory is used to put cancer into equations. Then, this framework is simplified in a two phases 2D model.Those equations are analysed. The spatial study shows the possibility of a phase separation process: the spinodal decomposition. And, the time study shows thet this model contains the ingredients necessary to describe several melanoma types seen in vivo.Focussing finally on the third dimension. Melanoma evolving on a wavy epidermis (hands and feet skin) are studied. We explain how melanoma patterns should follow the skin ridges (fingerprints).

Neste trabalho é apresentada uma nova modelagem matemática para a descrição do escoamento de um líquido incompressível através de um meio poroso rígido homogêneo e isotrópico, a partir do ponto de vista da Teoria Contínua de Misturas. O fenômeno é tratado como o movimento de uma mistura composta por três constituintes contínuos: o primeiro representando a matriz porosa, o segundo representando o líquido e o terceiro representando um gás de baixíssima densidade. O modelo proposto possibilita uma descrição matemática realista do fenômeno de transição insaturado/saturado a partir de uma combinação entre um sistema de equações diferenciais parciais e uma desigualdade. A desigualdade representa uma limitação geométrica oriunda da incompressibilidade do líquido e da rigidez do meio poroso. Alguns casos particulares são simulados e os resultados comparados com resultados clássicos, mostrando as consequências de não levar em conta as restrições inerentes ao problema. / This work is concerned with a new mathematical modelling for describing the flow of an incompressible fluid (a liquid) through a rigid, homogeneous and isotropic porous medium, from a Continuum Mixtures point of view. The phenomenon is regarded as the motion of a mixture composed by three overlaping continuous constituents: the first one representing the porous matrix, the second one representing the liquid and the third one representing a (very) low density gas. The proposed mathematical modelling allows a realistic mathematical description for the unsaturated/saturated transition process by means of a combination between a system of partial differential equations and an inequality. This inequality represents a geometrical constraint arising from the liquid incompressibility merged with the porous matrix rigidity. The simulation of some interesting particular cases is carried out presenting a comparison between the obtained results and the classical ones, showing the consequences of disregarding the constraints associated to the phenomenon.

Neste trabalho é apresentada uma nova modelagem matemática para a descrição do escoamento de um líquido incompressível através de um meio poroso rígido homogêneo e isotrópico, a partir do ponto de vista da Teoria Contínua de Misturas. O fenômeno é tratado como o movimento de uma mistura composta por três constituintes contínuos: o primeiro representando a matriz porosa, o segundo representando o líquido e o terceiro representando um gás de baixíssima densidade. O modelo proposto possibilita uma descrição matemática realista do fenômeno de transição insaturado/saturado a partir de uma combinação entre um sistema de equações diferenciais parciais e uma desigualdade. A desigualdade representa uma limitação geométrica oriunda da incompressibilidade do líquido e da rigidez do meio poroso. Alguns casos particulares são simulados e os resultados comparados com resultados clássicos, mostrando as consequências de não levar em conta as restrições inerentes ao problema. / This work is concerned with a new mathematical modelling for describing the flow of an incompressible fluid (a liquid) through a rigid, homogeneous and isotropic porous medium, from a Continuum Mixtures point of view. The phenomenon is regarded as the motion of a mixture composed by three overlaping continuous constituents: the first one representing the porous matrix, the second one representing the liquid and the third one representing a (very) low density gas. The proposed mathematical modelling allows a realistic mathematical description for the unsaturated/saturated transition process by means of a combination between a system of partial differential equations and an inequality. This inequality represents a geometrical constraint arising from the liquid incompressibility merged with the porous matrix rigidity. The simulation of some interesting particular cases is carried out presenting a comparison between the obtained results and the classical ones, showing the consequences of disregarding the constraints associated to the phenomenon.