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The motivation for this study results from the following practical observation: for the experimenter, the value of the capillary pressure is a priori an unknown of the problem, it corresponds to a spontaneous value resulting a posteriori from a static equilibrium. A general classification of axisymmetric capillary bridge profiles for unknown capillary pressure between a sphere and a plane is addressed in this article. The meridian of the liquid bridge is classified into convex or concave profiles as portions of nodoid or unduloid surfaces based on measured geometrical data with special limit cases as catenoid, cylinder, or sphere. The theoretical work is then validated by an image-processing technique applied to images of real capillary bridges from experiments for small water volumes and small sphere–plate separation distances. The laboratory tests show very good agreement between the theoretical calculations and the associated classification proposed. Even the transition case of a catenoid can be observed.

This paper presents an original approach for measuring the surface tension of a liquid from capillary bridges between monodisperse spherical particles. This method is based on an exact analytical solution of Young-Laplace equation for a capillary doublet between two spherical particles coupled with capillary force measurement. The capillary forces between two equal-sized spherical particles are measured by a micro-balance, while the geometries of the capillary bridge are recorded using a high resolution camera. From the obtained images of capillary bridges, the meridional profiles of capillary bridges are determined by a high-resolution image processing technique and correlated to the measured capillary forces. The accuracy and effectiveness of the proposed approach have been demonstrated by measuring the surface tensions of silicone oil, castor oil and impure water. The developed capillary bridge method, only requiring a small volume of liquid, can be applied to all liquids with high accuracy.

A calculation method for the meridional profile of axisymmetric bridges between two spheres of different size is introduced in this manuscript. From geometrical data of the capillary bridge, such as the neck radius and boundary conditions (filling and contact angle), the shape of the capillary bridge is calculated analytically as a solution of the Young–Laplace equation. Its free surfaces, of constant mean curvature, may be classified into portions of nodoid, unduloid, and other limit cases. Moreover, other properties of the liquid bridge can be computed analytically, such as the associated capillary force exerted on the solid surfaces, liquid volume, mean curvature, and free surface area.

This manuscript focuses on the meridional profile of the axisymmetric capillary bridges built between two unequal-sized solid spheres. We propose an original method of resolution of Young Laplace equation based on inverse problem. The shapes of capillary bridges are classified into different categories depending on the geometrical properties, including neck radius, half-filling angle, wetting angle, and distance between two spheres. Practically, all the physical characteristics of capillary bridges, such as the free surface area of the liquid bridge and the inter-particle force, can be calculated easily. In addition, the experimental data provided in this manuscript are compared with analytical results, by matching the theoretical estimation with the data obtained from real experiments using image processing algorithm.

In this paper, we show how very relevant mathematical works of P.I. Plotnikov, publicized by L.C. Evans and M. Portilheiro, can be used to model the effects of cyclic hysteresis phenomena for flows in unsaturated porous media. We draw particular attention to the example of a model for water–air flows, simplified for purposes of illustration. First, an unstable “spinodal” interval is artificially introduced. Then S.L. Sobolev’s method of dynamic regularization allows associating with the continuity equations additional information in the form of entropy type inequalities. The asymptotic limits of viscous approximate solutions generate effects of irreversibility and the expected clockwise hysteresis loop.

International audience; This article addresses the experimental study of capillary bridge properties with the use of analytical calculation of bridge profile, based on solution of Young–Laplace equation. Using the measurements of some parameters as the contact angle, half-filling angle and the neck radius of the capillary bridge between two spherical particles of radius r, the shape of the bridge is estimated using theoretical solutions of Young–Laplace equation. The corresponding analytical solution is superposed and compared with image data.

We are interested by a nonlinear single lithology diffusion model adapted from ideas originally developed by the Institut Francais du Petrole (IFP). The geological stratigraphic modeling has to describe transports of sediments, erosion and sedimentation processes by taking into account a limited weathering condition; the method by which the history of a sedimentary basin is revealed relies on knowledge of both initial and final data and can be generalized to multiple lithology. For this purpose, we introduce a relaxation time related to a delayed response for establishing equilibrium states; this approach introduces regularizing effects according to the ideas of G.I. Barenblatt - S. Sobolev and J.-L. Lions - O. A. Oleinik. New well-posedness results are presented.

The aim of this paper is to study the properties of Nernst–Planck–Poisson system describing the evolution of ionic concentrations in porous media. In particular, the well-posedness character of the system is proved and some qualitative properties of the global solution are established (energy and entropy laws, stationary states, Boltzmann distribution, influence of an external electrical field). We prove also that a superficial accumulation of ions occurs when an important external electrical field is applied.

International audience; In this study we use the recent analytical model to analyze capillary interactions in liquid bridge between two spherical grains, with fixed volumes of liquid and varying separation distance. Sequences of images of capillary bridges with different parameters are recorded during experimental tests. Geometrical parameters, as contact angle, half-filling angle and neck radius, are determined by image processing. Profiles of examined bridges are approximated as a Delaunay’s roulette and superposed on recorded images. Evolution of associated variables (Laplace pressure, capillary force) is also calculated. Results of theoretical modeling are compared with the experimental ones. They match very accurately for small volumes and/or small separation distances, when influence of gravity is not significant. For larger liquid volumes and/or larger separation distances between grains the influence of the gravity is observed as a distortion (loss of symmetry) of capillary bridge. To avoid this deformation, several test were realized in micro-gravity conditions. For these tests, theoretical results are in good agreement with experimental ones, also for higher liquid volumes and/or separations distances.

International audience; This work addresses the study of capillary bridge properties between two grains, with use of recent analytical model, based on solutions of Young-Laplace equation from an inverse problem. A simple explicit criterion allows to classify the profile of capillary bridge as a surface of revolution with constant mean curvature (Delaunay roulette) using its measured geometrical parameters (gorge radius, contact angle, half-filling angle). Necessary data are obtained from experimental tests, realized on liquid bridges between two equal spherical grains. Sequences of images are recorded at several (fixed) volumes of liquid and different separations distances between the spheres (from contact to rupture), in laboratory and in micro-gravity conditions. For each configuration, an exact parametric representation of the meridian is revealed. Mean bridge curvature, internal pressure and intergranular capillary force are also determined.

In this work, we propose an original resolution of Young–Laplace equation for capillary doublets from an inverse problem. We establish a simple explicit criterion based on the observation of the contact point, the wetting angle and the gorge radius, to classify in an exhaustive way the nature of the surface of revolution. The true shape of the admissible static bridges surface is described by parametric equations; this way of expressing the profile is practical and well efficient for calculating the binding forces, areas and volumes. Moreover, we prove that the inter-particle force may be evaluated on any section of the capillary bridge and constitutes a specific invariant.

The Nernst-Planck-Poisson-Boltzmann system describes the evolution of ionic concentrations and electrocapillarity effects in porous media. The aim of this paper is a theoretical study of various drift-diffusion modellings. The well-posedness of the systems is proved and some qualitative properties of the global solution are shown to be satisfied (energy law, entropy law in the weak sense of Lyapunov functions, stationary states, Maxwellian distribution, influence of an external electric field). Moreover, some explicit solutions are established in the one-dimensional case.

This work focuses on performing the homogenization of the Nernst-Planck-Poisson system, in order to obtain an efficient modelling of the ionic transfer and electrocapillary effects at the macroscopic scale for saturated porous media. The homogenization is performed with the powerful two-scale convergence method, which enables to obtain both the homogenized model and the associated convergence result.

In this work, we analyse the physical consequences of capillary bridges coalescence between spherical particles agglomerates and more particularly the jump of the capillary force. By referring to Murase et al. (Adv Powder Technol 19(4):349–367, 2008) and Rynhart et al. (Res Lett Inf Math Sci 5:119–127, 2003) about bridges adhered to three particles, we analyse the effects of coalescense between three bridges with two grains and a bridge joining three grains. This monographic synthesis intends to explain analytically and geometrically the significant increase of the inter-particle force, a strengthening cohesion effect, experimentally observed, reported and still largely unelucidated to our knowledge in the literature.

This work deals with the study of some stratigraphic models for the formation of geological basins under a maximal erosion rate constrain. It leads to introduce differential inclusions of degenerated hyperbolic-parabolic type , where H is the maximal monotonous graph of the Heaviside function and E is a given non-negative function. Firstly, we present the new and realistic models and an original mathematical formulation, taking into account the weather-limited rate constraint in the conservation law, with a unilateral constraint on the outflow boundary. Then, we give a study of the 1-D case with numerical illustrations.

In this paper, we are interested in lithology diffusion models applied in the field of stratigraphic basin simulations for large-scale depositional transport processes of sediments. Such models describe erosion-sedimentation processes and take into account limited weathering via nonstandard unilateral problems. Various theoretical results, illustrations, and numerical solutions are presented for the monolithologic column case. A new conservation law involved in modeling is formulated, and mathematical tools for solving the problem are described.

In this paper, we consider a 2D mathematical modelling of the vertical compaction effect in a water saturated sedimentary basin. This model is described by the usual conservation laws, Darcy's law, the porosity as a function of the vertical component of the effective stress and the Kozeny-Carman tensor, taking into account fracturation effects. This model leads to study the time discretization of a nonlinear system of partial differential equations. The existence is obtained by a fixed-point argument. The uniqueness proof, by Holmgren's method, leads to work out a linear, strongly coupled, system of partial differential equations and boundary conditions.

On expose les difficultes d'ordre mathematique que posent des modeles recents de sedimentation-erosion de bassins elabores par l'Institut Francais du Petrole et fondes sur la prise en compte de diverses contraintes d'unilateralite. On presente quelques resultats partiels theoriques et des directions de recherche pour la resolution d'un probleme inverse pose par l'etude stratigraphique d'une colonne monolithologique.

Our aim here is to study the thermal diusion phenomenon in a forced convective flow. A system of nonlinear parabolic equations governs the evolution of the mass fractions in multicomponent mixtures. Some existence and uniqueness results are given under suitable conditions on state functions. Then, we present a numerical scheme based on a \mixed nite element" method adapted to a nite volume scheme, of which we give numerical analysis. In a last part, we apply an homogenization technique to the studied equations in order to obtain an ecient modelling of Soret eect and adsorption in a porous medium at a macroscopic scale. R esum e. On etudie un syst eme d' equations paraboliques non lin eaires mod elisant l' evolution des fractions massiques d'un fluide multiconstituant dans un ecoulement convectif forc e sous l'influence d'un gradient thermique. Des r esultats d'existence et d'unicit e sont donn es sous des hypoth eses relatives aux fonctions d' etat des equations. On propose ensuite une m ethode num erique de type \ el ements nis mixtes" aboutissant a un schema \volumes nis" dont on eectue l'analyse et la pr esentation des premiers r esultats. Nous appliquons enn une technique d'homog en eisation aux equations etudi ees dans le but d'obtenir une mod elisation macroscopique d ele des ph enom enes d'adsorption et d'eet Soret en milieu poreux.

On etablit l'existence, l'unicite et la dependance continue par rapport a la donnee initiale des solutions fortes entropiques pour des problemes de Cauchy-Dirichlet associes a la loi scalaire: u t - Δφ(u) -div(v(u)G) = 0. On se place ici dans le cas ou φ est une fonction croissante au sens large et constante sur un ensemble L 1 - non negligeable: la loi scalaire consideree introduit donc une equation hyperbolique-parabolique degeneree, melant des phenomenes de parabolicite et d'hyperbolicite non lineaires.

We study a class of degenerate equations ut − Δφ(u) − div(ν(u) G) = 0 associated to Cauchy-Dirichlet problem. The special feature of the framework is double: the nonlinear equation degenerates into first order hyperbolic type if the unknown value u is less than a critical value, i.e. on an interval of solution values; we are concerned with the existence, stability and uniqueness of strong entropy solutions under appropriate assumptions on the data.

Resume On presente un concept de solution renormalisee pour les equations generales de diffusion-convection en milieu poreux, associees a des conditions de bord melees. L'originalite de cette notion est d'introduire une formulation equivalente a la formulation faible usuelle, qui discrimine implicitement les regions d'hyperbolicite et de parabolicite de l'equation et tient compte d'une condition de raccord « paraboliquehyperbolique; cette approche fournit des resultats nouveaux d'unicite.

The main objective of this work is to present, for practical use, some original results about several qualitative properties of the solutions of a large class of degenerate diffusion-convection equations arising from fluid mechanics. Current interest in models of the simultaneous motion of two immiscible incompressible liquids results from its significance for many applied fields such as, for instance, the theoretical modelling of oil reservoirs where the pores of a threedimensional porous medium contain some hydrocarbon component (oil). In secondary recovery, a second inexpensive fluid (water) is injected into the porous medium in order to push the oil towards the producing wells.

This work deals with the study of a nonlinear degenerate pseudoparabolic problem. Arising from the modelling of sedimentary basin formation, the equation degenerates in order to take implicitly into account a constraint on the time derivative of the unknown. An existence result of a solution with an adapted compactness result and qualitative properties of the solutions are proposed (localization, finite speed of propagation, etc.).

The aim of the paper is an analytical and numerical approach to the pseudo-compositionah black-oil model for simulating a 3-13 isothermal constrained polyphasic flow in porous media, taking into account realistic boundary coneitions. The handling of tite component conservation laws leads to a strongly coupled system including parabolic quasilinear degenerated equations and fxrst-order hyperbolic inequahities: tite introduction of unilateral problems arises from tite nature of the thermodynamical equilibrium functions and of tite governing equations describing tite model: it depends locally whether tite gas-pitase exists (satured oil) or the gas- pitase does not exist (sub-saturated oil), witich leads to a free bound~ry value problem, according to Gibb's laws.

In this thesis, we study two models describing the evolution of a sedimentary basin under an erosion rate constraint. These models are obtained by applying a mass conservation law on the flow of matter, which leads to Darcy's equation or to the equation of Darcy-Barenblatt whichever flow chosen among two possible expressions according to geologists. The equation of Darcy-Barenblatt is obtained from that of Darcy by adding a diffusion term. In addition, the maximum erosion constraint is implicit in the model formulation of Darcy-Barenblatt, but not in that of Darcy in dimension 2. After the presentation of these models in the introduction of the thesis, the first part is devoted to Darcy-Barenblatt's model. We obtained an existence result for a solution through a Schauder- Tikhonov fixed point method. Then, we have showed a regularity result by using the results of Meyers and Necas on elliptic equations with Hölder coefficients, this regularity result is specific to a dimension inferior or equal to 2. The first part concludes by demonstrating a result of uniqueness of the solution. The Darcy model is considered in the second part of the thesis. We obtained a solution of the problem discretized in time ; however in 2-dimensional space, passing to a continuous formulation imposes to study the product of two weak convergences. It raises theoretical difficulties and open problems. In dimension 1, we have solved a Bernoulli evolution problem to obtain a continuous solution in the case of the marine sedimentation.; Dans cette thèse, on étudie deux modèles décrivant l'évolution d'un bassin sédimentaire sous une contrainte sur le taux d'érosion. Ces modèles sont obtenus par l'application de la loi de conservation de masse sur le flux de matières, ce qui conduit à l'équation de Darcy ou à l'équation de Darcy-Barenblatt selon l'expression du flux choisie parmi deux expressions possibles d'après les géologues. L'équation de Darcy-Barenblatt est obtenue de celle de Darcy en ajoutant un terme de diffusion. En outre, la contrainte d'érosion maximale est implicitement contenue dans la formulation du modèle de Darcy-Barenblatt mais pas dans celle de Darcy en dimension 2. Après la présentation de ces modèles dans l'introduction de la thèse, la première partie est consacrée au modèle de Darcy-Barenblatt. On a obtenu un résultat d'existence d'une solution par une méthode de point fixe de Schauder-Tikhonov. Ensuite, on a montré un résultat de régularité en utilisant des résultats de Meyers et de Necas sur les équations elliptiques à coefficients höldériens, ce résultat de régularité est propre à une dimension inférieur ou égale à 2. La première partie se termine par la démonstration d'un résultat d'unicité de la solution. Le modèle de Darcy est étudié dans la deuxième partie de la thèse, on a obtenu une solution du problème discrétisé en temps, mais en dimension 2 d'espace le passage à la formulation continue fait apparaitre des produits de deux convergences faibles et soulève des difficultés théoriques non surmontées. En dimension 1, on a obtenu une solution continue pour le cas de la sédimentation marine en résolvant un problème à frontière libre de type Bernoulli d'évolution.

Interaction between a plasma and a magnetic field appears and has an important role in various domains such as thermonuclear fusion by magnetic confinement or astrophysical plasmas for example. In evolution, these interactions are described by the equations of magnetohydrodynamics (MHD). At equilibrium, the MHD equations reduce to the magnetostatic equations involving the magnetic field and the kinetic pressure of the plasma. The magnetostatic equations form a system of 3D non linear partial differential equations involving a magnetic field and a kinetic plasma pressure. When the pressure is supposed negligible, the magnetic field is known as Beltrami field. In a first time, we propose to solve numerically the Beltrami fields problem using a fixed point iterative algorithm associated with finite element methods. This iterative strategy is extended in a second time to the computation of magnetostatic configurations with pressure.In the sequel, we interest in the approximation of ideal MHD equations. This system forms a nonlinear hyperbolic conservation law. We propose to use a finite volume approach, in which fluxes are calculated by a Roe's method on a tetrahedral mesh. Fluxes of the magnetic field are modified in order to satisfy the constraint of divergence free imposed on it.The proposed methods have been implemented in two new three dimensional codes called TETRAFFF for equilibrium, and TETRAMHD for MHD. The obtained numerical results confirm the high performance of these methods.; L'étude des interactions entre un plasma et un champ magnétique joue un rôle important dans différents domaines tels que la fusion thermonucléaire par confinement magnétique, les plasmas astrophysiques. En évolution, ces interactions sont décrites par les équations de la magnétohydrodynamique (MHD). A l'équilibre, les équations de la MHD se réduisent à celles de la magnétostatique.Les équations de la magnétostatique forment un système d'équations aux dérivées partielles non linéaires en dimension 3 faisant intervenir le champ magnétique et la pression cinétique du plasma. Quand on néglige la pression, le champ magnétique est alors dit de Beltrami. Nous proposons de résoudre numériquement les équations régissant les champs de Beltrami par un algorithme itératif de type point fixe associé à des méthodes d'éléments finis. Cette stratégie itérative est étendue au cas des configurations d'équilibres avec pression.On s'intéresse ensuite à l'approximation des équations de la MHD idéale instationnaires. Il s'agit d'un système de loi de conservation hyperbolique non linéaire. Nous proposons une approche de type volumes finis dans laquelle les flux sont calculés par une méthode de Roe sur un maillage tétraédrique et où les flux du champ magnétique sont modifiés afin de satisfaire la contrainte de divergence nulle qui lui est imposée. Les méthodes proposées ont été implantées dans deux nouveaux codes tridimensionnels TETRAFFF pour les équilibres, et TETRAMHD pour la MHD. Les résulats numériques obtenus par ces codes montrent la performance des méthodes employées.

Interaction between a plasma and a magnetic field appears and has an important role in various domains such as thermonuclear fusion by magnetic confinement or astrophysical plasmas for example. In evolution, these interactions are described by the equations of magnetohydrodynamics (MHD). At equilibrium, the MHD equations reduce to the magnetostatic equations involving the magnetic field and the kinetic pressure of the plasma. The magnetostatic equations form a system of 3D non linear partial differential equations involving a magnetic field and a kinetic plasma pressure. When the pressure is supposed negligible, the magnetic field is known as Beltrami field. In a first time, we propose to solve numerically the Beltrami fields problem using a fixed point iterative algorithm associated with finite element methods. This iterative strategy is extended in a second time to the computation of magnetostatic configurations with pressure.In the sequel, we interest in the approximation of ideal MHD equations. This system forms a nonlinear hyperbolic conservation law. We propose to use a finite volume approach, in which fluxes are calculated by a Roe's method on a tetrahedral mesh. Fluxes of the magnetic field are modified in order to satisfy the constraint of divergence free imposed on it.The proposed methods have been implemented in two new three dimensional codes called TETRAFFF for equilibrium, and TETRAMHD for MHD. The obtained numerical results confirm the high performance of these methods.; L'étude des interactions entre un plasma et un champ magnétique joue un rôle important dans différents domaines tels que la fusion thermonucléaire par confinement magnétique, les plasmas astrophysiques. En évolution, ces interactions sont décrites par les équations de la magnétohydrodynamique (MHD). A l'équilibre, les équations de la MHD se réduisent à celles de la magnétostatique.Les équations de la magnétostatique forment un système d'équations aux dérivées partielles non linéaires en dimension 3 faisant intervenir le champ magnétique et la pression cinétique du plasma. Quand on néglige la pression, le champ magnétique est alors dit de Beltrami. Nous proposons de résoudre numériquement les équations régissant les champs de Beltrami par un algorithme itératif de type point fixe associé à des méthodes d'éléments finis. Cette stratégie itérative est étendue au cas des configurations d'équilibres avec pression.On s'intéresse ensuite à l'approximation des équations de la MHD idéale instationnaires. Il s'agit d'un système de loi de conservation hyperbolique non linéaire. Nous proposons une approche de type volumes finis dans laquelle les flux sont calculés par une méthode de Roe sur un maillage tétraédrique et où les flux du champ magnétique sont modifiés afin de satisfaire la contrainte de divergence nulle qui lui est imposée. Les méthodes proposées ont été implantées dans deux nouveaux codes tridimensionnels TETRAFFF pour les équilibres, et TETRAMHD pour la MHD. Les résulats numériques obtenus par ces codes montrent la performance des méthodes employées.

Ce travail regroupe un ensemble de résultats concernant essentiellement l'analyse mathématique de problèmes d'obstacles intérieurs pour une classe d'opérateurs quasi linéaires hyperboliques du premier ordre ou paraboliques du second ordre.Dans un premier chapitre on s'intéresse aux problèmes d'obstacles pour des opérateurs du premier ordre. On donne un résultat général d'unicité et des résultats d'existence (par pénalisation) dans le cas d'obstacle constants ou dépendant des variables de temps et d'espace et selon la régularité des données. Puis on établit des propriétés de sensibilité et de comportement par rapport à la condition d'obstacle. Enfin on propose une approximation numérique de la solution par une méthode de Time-Splitting.Au second chapitre au considère les problèmes d'obstacles pour des opérateur du second ordre faiblement ou fortement dégénérés. Dans le premier cas, par utilisation de la méthode de viscosité artificielle et celle de pénalisation on prouve l'existence puis l'unicité d'une solution faible caractérisée par une inéquation variationnelle. Dans le second cas, la solution faible est caractérisée par une formulation d'entropique.Dans le troisième chapitre on expose de nouvelles thématiques de recherche, principalement le problème du couplage hyperbolique/parabolique avec « raccord » le long d'une interface commune. On donne une formulation faible à l'aide d'une inégalité d'entropie dans tout le domaine d'étude. On établit un résultat d'unicité dans le cas où l'interface est incluse dans le lieu des caractéristiques sortantes pour l'opérateur hyperbolique. L'existence est obtenue par ajout d'un terme de viscosité sur la zone d'hyperbolicité.

Ce travail regroupe un ensemble de résultats concernant essentiellement l'analyse mathématique de problèmes d'obstacles intérieurs pour une classe d'opérateurs quasi linéaires hyperboliques du premier ordre ou paraboliques du second ordre.Dans un premier chapitre on s'intéresse aux problèmes d'obstacles pour des opérateurs du premier ordre. On donne un résultat général d'unicité et des résultats d'existence (par pénalisation) dans le cas d'obstacle constants ou dépendant des variables de temps et d'espace et selon la régularité des données. Puis on établit des propriétés de sensibilité et de comportement par rapport à la condition d'obstacle. Enfin on propose une approximation numérique de la solution par une méthode de Time-Splitting.Au second chapitre au considère les problèmes d'obstacles pour des opérateur du second ordre faiblement ou fortement dégénérés. Dans le premier cas, par utilisation de la méthode de viscosité artificielle et celle de pénalisation on prouve l'existence puis l'unicité d'une solution faible caractérisée par une inéquation variationnelle. Dans le second cas, la solution faible est caractérisée par une formulation d'entropique.Dans le troisième chapitre on expose de nouvelles thématiques de recherche, principalement le problème du couplage hyperbolique/parabolique avec « raccord » le long d'une interface commune. On donne une formulation faible à l'aide d'une inégalité d'entropie dans tout le domaine d'étude. On établit un résultat d'unicité dans le cas où l'interface est incluse dans le lieu des caractéristiques sortantes pour l'opérateur hyperbolique. L'existence est obtenue par ajout d'un terme de viscosité sur la zone d'hyperbolicité.