Dissertations / Theses on the topic 'Capillarity equation'

2012/2013
We investigate by different techniques, the solvability of a class of capillarity-type problems, in a bounded N-dimensional domain. Since our approach is variational, the natural context where this problem has to be settled is the space of bounded variation functions. Solutions of our equation are defined as subcritical points of the associated action functional.
We first introduce a lower and upper solution method in the space of bounded variation functions. We prove the existence of solutions in the case where the lower solution is smaller than the upper solution. A solution, bracketed by the given lower and upper solutions, is obtained as a local minimizer of the associated functional without any assumption on the boundedness of the right-hand side of the equation. In this context we also prove order stability results for the minimum and the maximum solution lying between the given lower and upper solutions. Next we develop an asymmetric version of the Poincaré inequality in the space of bounded variation functions. Several properties of the curve C are then derived and basically relying on these results, we discuss the solvability of the capillarity-type problem, assuming a suitable control on the interaction of the supremum and the infimum of the function at the right-hand side with the curve C. Non-existence and multiplicity results are investigated as well. The one-dimensional case, which sometimes presents a different behaviour, is also discussed. In particular, we provide an existence result which recovers the case of non-ordered lower and upper solutions.
XXV Ciclo
1985

The contact angle defined by Young's equation depends on the ratio between solid and liquid surface energies. Young's contact angle is constant for a given system, and cannot explain the stability of fluid droplets in capillary tubes. Within this framework, large variations in contact angle and explained aassuming surface roughness, heterogeneity or contamination. This research explores the static and dynamic behavior of fluid droplets within capillary tubes and the variations in contact angle among interacting menisci. Various cases are considered including wetting and non-wetting gluids, droplets in inclined capillary tubes or subjected to a pressure difference, within one-dimensional and three-dimensional capillary systems, and under static or dynamic conditions (either harmonic fluid pressure or tube oscillation). The research approach is based on complementary analytical modeling (total energy formulation) and experimental techniques (microscopic observations). The evolution of meniscus curvatures and droplet displacements are studied in all cases. Analytical and experimental results show that droplets can be stable within capillary tubes even under the influence of an external force, the resulting contact angles are not constant, and bariations from Young's contact angle aare extensively justified as menisci interaction. Menisci introduce stiffness, therefore two immiscible Newtonian fluids behave as a Maxwellian fluid, and droplets can exhibit resonance or relaxation spectral features.

The thesis numerically models and investigates the effect of a variable water table on the soil moisture content. The modelling is done using COMSOL and Richards' equation. The temporal variation plots can be used to find the capillarity of the soil and its impact on other phenomenon such as vapor intrusion and infiltration.

Lo scopo della presente tesi è quello di discutere i risultati ottenuti durante i miei studi di dottorato, principalmente rivolti a problemi non locali. Per prima cosa ci occupiamo di principi di continuazione unica forte ed espansioni asintotiche locali in determinati punti del bordo per soluzioni di due diverse classi di equazioni ellittiche. In particolare, partiamo con lo studio di una classe di equazioni ellittiche frazionarie in un dominio limitato sotto una condizione al contorno di Dirichlet omogenea esterna. Per fare ciò, sfruttiamo la procedura di estensione di Caffarelli-Silvestre, grazie alla quale il problema non locale può essere riformulato in modo equivalente come problema locale in una dimensione in più, generando un problema con condizioni miste. Dopodichè, utilizziamo un'idea classica di Garofalo e Lin per ottenere una condizione di raddoppio tramite una formula di monotonia per la funzione di Almgren. Per superare le difficoltà legate alla perdita di regolarità in corrispondenza della transizione tra le regioni di Dirichlet e di Neumann, introduciamo una nuova tecnica basata su un argomento di approssimazione, che ci permette di derivare una cosiddetta identità di tipo Pohozaev necessaria per stimare la derivata della funzione di Almgren . Otteniamo così un risultato di continuazione unica forte nel contesto locale, che viene a sua volta combinato con l’analisi di blow-up per dedurre espansioni asintotiche locali e, di conseguenza, una continuazione unica forte anche nel contesto non locale. Inoltre forniamo anche un risultato di continuazione unica forte dal bordo di una fessura per le soluzioni di una classe specifica di equazioni ellittiche del secondo ordine in un dominio limitato aperto con una frattura, su cui è assegnata una condizione al contorno di Dirichlet omogenea. Questo problema locale è correlato a un caso particolare dello studio descritto sopra, in virtù di una forte connessione tra questo tipo di problemi e i problemi al contorno con condizioni miste. Nella presente dissertazione, trattiamo anche una teoria della capillarità non locale. In particolare, consideriamo nuclei di interazione più generali che sono possibilmente anisotropici e non necessariamente invarianti rispetto allo stesso riscalamento. In particolare, la perdita di invarianza è modellata utilizzando due diversi esponenti frazionari per tenere conto della possibilità che il contenitore e l'ambiente presentino caratteristiche diverse rispetto alle interazioni delle particelle. Determiniamo inoltre una legge di Young non locale per l'angolo di contatto tra la gocciolina e la superficie del contenitore e discutiamo la solvibilità e l’unicità della soluzione dell'equazione corrispondente in termini di nuclei di interazione e del relativo coefficiente di adesione.
The aim of the present thesis is to discuss the results obtained during my PhD studies, mainly devoted to nonlocal issues. We first deal with strong unique continuation principles and local asymptotic expansions at certain boundary points for solutions of two different classes of elliptic equations. We start the investigation by a class of fractional elliptic equations in a bounded domain under some outer homogeneous Dirichlet boundary condition. To do this, we exploit the Caffarelli-Silvestre extension procedure, which allows us to get an equivalent formulation of the nonlocal problem as a local problem in one dimension more, consisting in a mixed Dirichlet-Neumann boundary value problem. Then, we use a classical idea by Garofalo and Lin to obtain a doubling-type condition via a monotonicity formula for a suitable Almgren-type frequency function. To overcome the difficulties related to the lack of regularity at the Dirichlet-Neumann junction, we introduce a new technique based on an approximation argument, which leads us to derive a so-called Pohozaev-type identity needed to estimate the derivative of the Almgren function. Thus we gain a strong unique continuation result in the local context, which is in turn combined with blow-up arguments to deduce local asymptotics and, consequently, a strong unique continuation result in the nonlocal setting as well. We also provide a strong unique continuation result from the edge of a crack for the solutions to a specific class of second order elliptic equations in an open bounded domain with a fracture, on which a homogeneous Dirichlet boundary condition is prescribed, in the presence of potentials satisfying either a negligibility condition with respect to the inverse-square weight or some suitable integrability properties. This local problem is related to a particular case of the setting described above, by virtue of a strong connection between this type of problems and the mixed Dirichlet-Neumann boundary value problems. We also treat a capillarity theory of nonlocal type. In our setting, we consider more general interaction kernels that are possibly anisotropic and not necessarily invariant under scaling. In particular, the lack of scale invariance is modeled via two different fractional exponents in order to take into account the possibility that the container and the environment present different features with respect to particle interactions. We determine a nonlocal Young's law for the contact angle between the droplet and the surface of the container and discuss the unique solvability of the corresponding equation in terms of the interaction kernels and of the relative adhesion coefficient.

Three-dimensional gravity-capillary steady waves on water of finite-depth, which are uniformly translating in a horizontal propagation direction and periodic in a transverse direction, are considered. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is the time-like variable. The existence of the solutions of the system is determined by two non-dimensional constants: the Bond number b and λ (the inverse of the square of the Froude number). The property of Sobolev spaces and the spectral analysis show that the spectrum of the linear part consists of isolated eigenvalues of finite algebraic multiplicity and the number of purely imaginary eigenvalues are finite. The distribution of eigenvalues is described by b and λ. Assume that C₁ is the curve in (b,λ)-plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and that the intersection point of the curve C₁ with the line λ=1 is (b₀,1) where b₀>0. Two cases (b₀,1) and (b,λ) â C₁ where 0< b< b₀ are investigated. A center-manifold reduction technique and a normal form analysis are applied to show that for each case the dynamical system can be reduced to a system of ordinary differential equations with finite dimensions. The dominant system for the case (b₀,1) is coupled Schrödinger-KdV equations while it is a Schrödinger equation for another case (b,λ) â C₁. Then, from the existence of the homoclinic orbit connecting to the two-dimensional periodic solution (called generalized solitary wave) for the dominant system, it is obtained that such generalized solitary wave solution persists for the original system by using the perturbation method and adjusting some appropriate constants.
Ph. D.

Capillaries in tumours are often severely buckled (in a plane perpendicular to the axis) and / or chaotic in their direction. We develop a model of these phenomena using nonlinear solid mechanics. Our model focusses on the immediate surrounding of a capillary. The vessel and surrounding tissue are modelled as concentric annulii. The growth is dependent on the concentration of a nutrient (oxygen) diffusing from the vessel into the tumour interstitium. The stress is modelled using a multiplicative decomposition of the deformation gradient F=F_e F_g. The stress is determined by substituting the elastic deformation gradient F_e (which gives the deformation gradient from the hypothetical configuration to the current configuration) into a hyperelastic constitutive model as per classical solid mechanics. We use a Blatz-Ko model, parameterised using uniaxial compression experiments. The entire system is in quasi-static equilibrium, with the divergence of the stress tensor equal to zero. We determine the onset of buckling using a linear stability analysis. We then investigate the postbuckling behaviour by introducing higher order perturbations in the deformation and growth before using the Fredholm Alternative to obtain the magnitude of the buckle. Our results demonstrate that the growth-induced stresses are sufficient for the capillary to buckle in the absence of external loading and / or constraints. Planar buckling usually occurs after 2-5 times the cellular proliferation timescale. Buckles with axial variation almost always go unstable after planar buckles. Buckles of fine wavelength are initially preferred by the system, but over time buckles of large wavelength become energetically more favourable. The tumoural hoop stress T_{ThetaTheta} is the most invariant (Eulerian) variable at the time of buckling: it is typically of the order of the tumoural Young's Modulus when this occurs.

Dans la première partie de cette thèse on étudie les systèmes abcd qui ont été dérivés par J.L. Bona, M. Chen et J.-C. Saut en 2002. Ces systèmes sont des modèles approximant le problème d'ondes hydrodynamiques dans le régime de Boussinesq, à savoir, des vagues de faible amplitude et de grande longueur d'onde. Dans les deux premiers chapitres on considère le problème d'existence en temps long à savoir la construction de solutions pour les systèmes abcd qui ont leur temps d'existence minoré par $1/varepsilon$ où $varepsilon$ est le rapport entre une amplitude typique du vague et la profondeur du canal. Dans un premier temps on considère des données initiales appartenant aux espaces de Sobolev qui sont inclus dans l'espace des fonctions continues qui s'annulent à l'infini. D'un point de vue physique cette situatuion correspond à des vagues sont localisées en espace. Le point clé est la construction d'une fonctionnelle non linéaire d'énergie qui contrôle certaines normes de Sobolev sur un intervalle de temps long. Pour y arriver, on travaille avec des équations localisées en fréquence. Cette approche nous permet d'obtenir des résultats d'existence en temps long en demandant moins de régularité sur les données initiales. Un deuxième avantage de notre méthode est que l'on peut traiter d'une manière unifiée presque tous les cas correspondant aux différentes valeurs des paramètres abcd. Dans le deuxième chapitre on montre des résultats d'existence en temps long pour le cas des données ayant un comportement non trivial à l'infini.Ce type des données est relevant pour l'étude de la propagation des mascarets. L'idée qui est à la base de ces résultats est de considérer un découpage convenable de la donnée initiale en hautes et basses fréquences. Dans le troisième chapitre on emploie des schémas de volumes finis afin de construire des solutions numériques. On utilise ensuite nos schémas pour étudier l'interaction d'ondes progressives.La deuxième partie de ce manuscrit est consacrée à l'étude des problèmes de régularité optimale pour le système de Navier-Stokes qui régi l'évolution d'un fluide incompressible, inhomogène et pour le système Navier-Stokes-Korteweg utilisé pour prendre en compte les effets de capillarité. Plus précisément, on montre que ces systèmes sont bien-posés dans leurs espaces critiques, à savoir, les espaces quiont la même invariance par changement d'échelle que les systèmes eux-mêmes. Pour pouvoir démontrer ce type de résultats on a besoin d'établir de nouvelles estimations pour un problème de type Stokes avec des coefficients variables
The first part of the present thesis deals with the so -called abcd systems which were derived by J.L. Bona, M. Chen and J.-C. Saut back in 2002. These systems are approximation models for the waterwaves problem in the Boussinesq regime, that is, waves of small amplitude and long wavelength. In the first two chapters we address the long time existence problem which consists in constructing solutions for the Cauchy problem associated to the abcd systems and prove that the maximal time of existence is bounded from below by some physically relevant quantity. First, we consider the case of initial data belonging to some Sobolev spaces imbedded in the space of continuous functions which vanish at infinity. Physically, this corresponds to spatially localized waves. The key ingredient is to construct a nonlinear energy functional which controls appropriate Sobolev norms on the desired time scales. This is accomplished by working with spectrally localized equations. The two important features of our method is that we require lower regularity levels in order to develop a long time existence theory and we may treat in an uni ed manner most of the cases corresponding to the di erent values of the parameters. In the second chapter, we prove the long time existence results for the case of data thatdoes not necessarily vanish at in nity. This is especially useful if one has in mind bore propagation. One of the key ideas of the proof is to consider a well-adapted high-low frequency decomposition of the initial data. In the third chapter, we propose infinite volume schemes in order to construct numerical solutions. We use these schemes in order to study traveling waves interaction.The second part of this manuscript, is devoted to the study of optimal regularity issues for the incompressible inhomogeneous Navier-Stokes system and the Navier-Stokes-Korteweg system used in order to take in account capillarity effects. More precisely, we prove that these systems are well-posed in their truly critical spaces i.e. the spaces that have the same scale invariance as the system itself. Inorder to achieve this we derive new estimates for a Stoke-like problem with time independent variable coefficients

This work is a theoretical contribution to the study of thermo-hydrodynamic instabilities in fluids submitted to surface-tension (Marangoni) and buoyancy (Rayleigh) effects in layered (Benard) configurations. The driving constraint consists in a thermal (or a concentrational) gradient orthogonal to the plane of the layer(s).

Linear, weakly nonlinear as well as strongly nonlinear analyses are carried out, with emphasis on high Prandtl (or Schmidt) number fluids, although some results are also given for low-Prandtl number liquid metals. Attention is mostly devoted to the mechanisms responsible for the onset of complex spatio-temporal behaviours in these systems, as well as to the theoretical explanation of some existing experimental results.

As far as linear stability analyses (of the diffusive reference state) are concerned, a number of different effects are studied, such as Benard convection in two layers coupled at an interface (for which a general classification of instability modes is proposed), surface deformation effects and phase-change effects (non-equilibrium evaporation). Moreover, a number of different monotonous and oscillatory instability modes (leading respectively to patterns and waves in the nonlinear regime) are identified. In the case of oscillatory modes in a liquid layer with deformable interface heated from above, our analysis generalises and clarifies earlier works on the subject. A new Rayleigh-Marangoni oscillatory mode is also described for a liquid layer with an undeformable interface heated from above (coupling between internal and surface waves).

Weakly nonlinear analyses are then presented, first for monotonous modes in a 3D system. Emphasis is placed on the derivation of amplitude (Ginzburg-Landau) equations, with universal structure determined by the general symmetry properties of the physical system considered. These equations are thus valid outside the context of hydrodynamic instabilities, although they generally depend on a certain number of numerical coefficients which are calculated for the specific convective systems studied. The nonlinear competitions of patterns such as convective rolls, hexagons and squares is studied, showing the preference for hexagons with upflow at the centre in the surface-tension-driven case (and moderate Prandtl number), and of rolls in the buoyancy-induced case.

A transition to square patterns recently observed in experiments is also explained by amplitude equation analysis. The role of several fluid properties and of heat transfer conditions at the free interface is examined, for one-layer and two-layer systems. We also analyse modulation effects (spatial variation of the envelope of the patterns) in hexagonal patterns, leading to the description of secondary instabilities of supercritical hexagons (Busse balloon) in terms of phase diffusion equations, and of pentagon-heptagon defects in the hexagonal structures. In the frame of a general non-variational system of amplitude equations, we show that the pentagon-heptagon defects are generally not motionless, and may even lead to complex spatio-temporal dynamics (via a process of multiplication of defects in hexagonal structures).

The onset of waves is also studied in weakly nonlinear 2D situations. The competition between travelling and standing waves is first analysed in a two-layer Rayleigh-Benard system (competition between thermal and mechanical coupling of the layers), in the vicinity of special values of the parameters for which a multiple (Takens-Bogdanov) bifurcation occurs. The behaviours in the vicinity of this point are numerically explored. Then, the interaction between waves and steady patterns with different wavenumbers is analysed. Spatially quasiperiodic (mixed) states are found to be stable in some range when the interaction between waves and patterns is non-resonant, while several transitions to chaotic dynamics (among which an infinite sequence of homoclinic bifurcations) occur when it is resonant. Some of these results have quite general validity, because they are shown to be entirely determined by quadratic interactions in amplitude equations.

Finally, models of strongly nonlinear surface-tension-driven convection are derived and analysed, which are thought to be representative of the transitions to thermal turbulence occurring at very high driving gradient. The role of the fastest growing modes (intrinsic length scale) is discussed, as well as scalings of steady regimes and their secondary instabilities (due to instability of the thermal boundary layer), leading to chaotic spatio-temporal dynamics whose preliminary analysis (energy spectrum) reveals features characteristic of hydrodynamic turbulence. Some of the (2D and 3D) results presented are in qualitative agreement with experiments (interfacial turbulence).

Doctorat en sciences appliquées
info:eu-repo/semantics/nonPublished

On s'intéresse à l'écoulement d'un mélange d'eau et d'huile dans une matrice poreuse supposée hétérogène, et plus particulièrement apposition de différentes sous-matrices poreuses supposées homogènes. Si la modélisation et l'analyse des écoulements diphasiques dans des milieux poreux homogènes a fait l'objet de nombreuses études préalables, ce travail s'intéresse aux phénomènes liés aux forces provenant de la pression capillaire au niveau des interfaces entre des milieux différents.
Dans un premier temps, on suppose que l'on peut connecter les pressions au niveau des interfaces. Cela nécessite des hypothèses sur les profils de pression capillaire, afin que les raccords soient possibles. On démontre l'existence d'une solution faible du problème parabolique dégénéré obtenu par convergence d'une famille de solutions approchées obtenues à l'aide d'un schéma Volumes Finis. L'unicité est garantie, sous hypothèse sur les dégénérescence, par une méthode de dédoublement de variable aboutissant à un principe de contraction $L^1$.
La modélisation ne garantit pas forcément que le raccord des pressions capillaires aux interfaces soit possible. Dans le chapitre 3, on donne une condition de raccord graphique des pressions capillaires aux interfaces qui permet de traiter des cas beaucoup plus généraux. On montre que de le problème avec raccords graphiques admet une solution. Un résultat d'unicité et de contraction $L^1$ est donné dans le cas unidimensionnel.
Dans le chapitre 4, on montre la convergence d'une approximation Volumes Finis vers l'unique solution du problème unidimensionnel. Ce résultat utilise une borne uniforme sur les flux discrets, analogie discrète de la preuve dans le cas continue faite au chapitre précédent.
On étudie dans les chapitres 5 et 6 la limite des solutions lorsque la dépendance de la pression capillaire par rapport à l'inconnue saturation devient très faible, et que la pression capillaire ne dépend plus que du sous milieux poreux homogène. Il apparaît alors des phénomènes différents selon l'orientation des forces de gravité et de capillarité. Soit la solution su problème est la solution entropique d'une équation hyperbolique à flux discontinus, soit une solution faible, entropique à l'intérieur des sous-domaines homogènes, et laissant apparaître un choc non classique à l'interface.

The purpose of this thesis is to prove the existence of a unique solution to a system of partial differential equations which models the flow of a compressible barotropic fluid under periodic boundary conditions. The equations come from modifying the compressible Navier-Stokes equations. The proof utilizes the method of successive approximations. We will define an iteration scheme based on solving a linearized version of the equations. Then convergence of the sequence of approximate solutions to a unique solution of the nonlinear system will be proven. The main new result of this thesis is that the density data is at a given point in the spatial domain over a time interval instead of an initial density over the entire spatial domain. Further applications of the mathematical model are fluid flow problems where the data such as concentration of a solute or temperature of the fluid is known at a given point. Future research could use boundary conditions which are not periodic.

Les travaux de cette thèse portent sur des méthodes de volumes finis sur maillages quelconque pour la discrétisation de problèmes d'évolution non linéaires modélisant le transport de contaminants en milieu poreux et les écoulements diphasiques.Au Chapitre 1, nous étudions une famille de schémas numériques pour la discrétisation d'une équation parabolique dégénérée de convection-reaction-diffusion modélisant le transport de contaminants dans un milieu poreux qui peut être hétérogène et anisotrope. La discrétisation du terme de diffusion est basée sur une famille de méthodes qui regroupe les schémas de volumes finis hybrides, de différences finies mimétiques et de volumes finis mixtes. Le terme de convection est traité à l'aide d'une famille de méthodes qui s'appuient sur les inconnues hybrides associées aux interfaces du maillage. Cette famille contient à la fois les schémas centré et amont. Les schémas que nous étudions permettent une discrétisation localement conservative des termes d'ordre un et d'ordre deux sur des maillages arbitraires en dimensions d'espace deux et trois. Nous démontrons qu'il existe une solution unique du problème discret qui converge vers la solution du problème continu et nous présentons des résultats numériques en dimensions d'espace deux et trois, en nous appuyant sur des maillages adaptatifs.Au Chapitre 2, nous proposons un schéma de volumes finis hybrides pour la discrétisation d'un problème d'écoulement diphasique incompressible et immiscible en milieu poreux. On suppose que ce problème a la forme d'une équation parabolique dégénérée de convection-diffusion en saturation couplée à une équation uniformément elliptique en pression. On considère un schéma implicite en temps, où les flux diffusifs sont discrétisés par la méthode des volumes finis hybride, ce qui permet de pouvoir traiter le cas d'un tenseur de perméabilité anisotrope et hétérogène sur un maillage très général, et l'on s'appuie sur un schéma de Godunov pour la discrétisation des flux convectifs, qui peuvent être non monotones et discontinus par rapport aux variables spatiales. On démontre l'existence d'une solution discrète, dont une sous-suite converge vers une solution faible du problème continu. On présente finalement des cas test bidimensionnels.Le Chapitre 3 porte sur un problème d'écoulement diphasique, dans lequel la courbe de pression capillaire admet des discontinuité spatiales. Plus précisément on suppose que l'écoulement prend place dans deux régions du sol aux propriétés très différentes, et l'on suppose que la loi de pression capillaire est discontinue en espace à la frontière entre les deux régions, si bien que la saturation de l'huile et la pression globale sont discontinues à travers cette frontière avec des conditions de raccord non linéaires à l'interface. On discrétise le problème à l'aide d'un schéma, qui coïncide avec un schéma de volumes finis standard dans chacune des deux régions, et on démontre la convergence d'une solution approchée vers une solution faible du problème continu. Les test numériques présentés à la fin du chapitre montrent que le schéma permet de reproduire le phénomène de piégeage de la phase huile.

Ce travail présente quelques contributions à la modélisation
et à la simulation de genèse de bassins sédimentaires.
Nous présentons tout d'abord les modèles mathématiques et
les schémas numériques mis en oeuvre à l'Institut Français
du Pétrole dans le cadre du projet Temis. Cette première partie
est illustrée à l'aide de tests numériques portant sur des bassins 1D/2D.
Nous étudions ensuite le schéma amont des pétroliers utilisé pour la résolution des équations de Darcy et nous établissons des résultats mathématiques nouveaux
dans le cas d'un écoulement de type Dead-Oil.
Nous montrons également comment construire un schéma à nombre
de Péclet variable en présence de pression capillaire.
Là encore, nous effectuons une étude mathématique
détaillée et nous montrons la convergence du schéma
dans un cas simplifié. Des tests numériques réalisés
sur un problème modèle montrent que l'utilisation d'un nombre
de Péclet variable améliore la précision des calculs.
Enfin nous considérons dans une dernière partie
un modèle d'écoulement où les changements de lithologie et
les changements de courbes de pression capillaire sont liés.
Nous précisons la condition physique que doivent vérifier
les solutions en saturation aux interfaces de changement de roche et
nous en déduisons une formulation faible originale.
L'existence d'une solution à ce problème est obtenue
par convergence d'un schéma volumes finis.
Des exemples numériques montrent l'influence de la condition
d'interface sur le passage ou la retenue des hydrocarbures.

Understanding and characterising the diffusive transport of capillary oxygen and nutrients in striated muscles is key to assessing angiogenesis and investigating the efficacy of experimental and therapeutic interventions for numerous pathological conditions, such as chronic ischaemia. In articular, the influence of both muscle tissue and microvascular heterogeneities on capillary oxygen supply is poorly understood. The objective of this thesis is to develop mathematical and computational modelling frameworks for the purpose of extending and generalising the current use of histology in estimating the regions of tissue supplied by individual capillaries to facilitate the exploration of functional capillary oxygen supply in striated muscles. In particular, we aim to investigate the balance between local capillary supply of oxygen and oxygen demand in the presence of various anatomical and functional heterogeneities, by capturing tissue details from histological imaging and estimating or predicting regions of capillary supply. Our computational method throughout is based on a finite element framework that captures the anatomical details of tissue cross sections. In Chapter 1 we introduce the problem. In Chapter 2 we develop a theoretical model to describe oxygen transport from capillaries to uniform muscle tissues (e.g. cardiac muscle). Transport is then explored in terms of oxygen levels and capillary supply regions. In Chapter 3 we extend this modelling framework to explore the influence of the surrounding tissue by accounting for the spatial anisotropies of fibre oxygen demand and diffusivity and the heterogeneity in fibre size and shape, as exemplified by mixed muscle tissues (e.g. skeletal muscle). We additionally explore the effects of diffusion through the interstitium, facilitated--diffusion by myoglobin, and Michaelis--Menten kinetics of tissue oxygen consumption. In Chapter 4, a further extension is pursued to account for intracellular heterogeneities in mitochondrial distribution and diffusive parameters. As a demonstration of the potential of the models derived in Chapters 2--4, in Chapter 5 we simulate oxygen transport in myocardial tissue biopsies from rats with either impaired angiogenesis or impaired arteriolar perfusion. Quantitative predictions are made to help explain and support experimental measurements of cardiac performance and metabolism. In the final chapter we summarize the main results and indicate directions for further work.

The primary objective of this research is to develop a mathematical framework that could be used to model or predict the rate of fluid absorption and release in fibrous sheets made up of solid or porous fibers. In the first step, a two-scale two-phase modeling methodology is developed for studying fluid release from saturated/unsaturated thin fibrous media made up of solid fibers when brought in contact with a moving solid surface. Our macroscale model is based on the Richards’ equation for two-phase fluid transport in porous media. The required constitutive relationships, capillary pressure and relative permeability as functions of the medium’s saturation, are obtained through microscale modeling. Here, a mass convection boundary condition is considered to model the fluid transport at the boundary in contact with the target surface. The mass convection coefficient plays a significant role in determining the release rate of fluid. Moreover the release rate depends on the properties of the fluid, fibrous sheet, the target surface as well as the speed of the relative motion, and remains to be determined experimentally. Obtaining functional relationships for relative permeability and capillary pressure is only possible through experimentation or expensive microscale simulations, and needs to be repeated for different media having different fiber diameters, thicknesses, or porosities. In this concern, we conducted series of 3-D microscale simulations in order to investigate the effect of the aforementioned parameters on the relative permeability and capillary pressure of fibrous porous sheets. The results of our parameter study are utilized to develop general expressions for kr(S) and Pc(S). Furthermore, these general expressions can be easily included in macroscale fluid transport equations to predict the rate of fluid release from partially saturated fibrous sheets in a time and cost-effective manner. Moreover, the ability of the model has been extended to simulate the radial spreading of liquids in thin fibrous sheets. By simulating different fibrous sheets with identical parameters but different in-plane fiber orientations has revealed that the rate of fluid spread increases with increasing the in-plane alignment of the fibers. Additionally, we have developed a semi-analytical modeling approach that can be used to predict the fluid absorption and release characteristics of multi-layered composite fabric made up of porous (swelling) and soild (non-swelling) fibrous sheets. The sheets capillary pressure and relative permeability are obtained via a combination of numerical simulations and experiment. In particular, the capillary pressure for swelling media is obtained via height rise experiments. The relative permeability expressions are obtained from the analytical expressions previously developed with the 3-D microscale simulations, which are also in agreement with experimental correlations from the literature. To extend the ability of the model, we have developed a diffusion-controlled boundary treatment to simulate fluid release from partially-saturated fabrics onto surfaces with different hydrophilicy. Using a custom made test rig, experimental data is obtained for the release of liquid from partially saturated PET and Rayon nonwoven sheets at different speeds, and on two different surfaces. It is demonstrated that the new semi-empirical model redeveloped in this work can predict the rate of fluid release from wet nonwoven sheets as a function of time.

When traditional linearised theory is used to study free-surface flows past a surface-piercing object or over an obstruction in a stream, the geometry of the object is usually lost, having been assumed small in one or several of its dimensions. In order to preserve the nonlinear nature of the geometry, asymptotic expansions in the low-Froude or low-Bond limits can be derived, but here, the solution invariably predicts a waveless free-surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In this thesis, we will apply exponential asymptotics to the study of two new problems involving nonlinear geometries. In the first, we examine the case of free-surface flow over a step including the effects of both gravity and surface tension. Here, we shall see that the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects, leads to the discovery of a remarkable new set of solutions. In the second problem, we study the waves produced by bluff-bodied ships in low-Froude flows. We will derive the analytical form of the exponentially small waves for a wide range of hull geometries, including single-cornered and multi-cornered ships, and then provide comparisons with numerical computations. A particularly significant result is our confirmation of the thirty-year old conjecture by Vanden-Broeck & Tuck (1977) regarding the impossibility of waveless single-cornered ships.

Dans cette thèse, on étudie les écoulements oscillatoires en milieux poreux (non saturés ou partiellement saturés) dus à des oscillations tidales des niveaux d'eau dans des milieux ouverts adjacents aux milieux poreux. L'étude est centrée sur le cas des plages de sable en hydrodynamique côtière, mais les applications concernent, potentiellement et plus généralement, les problèmes d'oscillation et de variation temporelle des niveaux d'eau dans des systèmes couplés, lorsque ceux-ci mettent en jeu des interactions entre les écoulements de sub-surface (milieux poreux) et les eaux de surface (milieux ouverts) : plages naturelles et artificielles; digues portuaires; barrages en terre; berges de fleuves; estuaires. Le forçage tidal des écoulements souterrains est représenté et modélisé ici, tant expérimentalement que numériquement, par une oscillation quasi-statique du niveau d'eau dans un réservoir externe ouvert, connecté au domaine poreux. On s'intéresse plus particulièrement aux écoulements verticaux forcés par une pression oscillatoire imposée au bas d'une colonne de sol. Sur le plan expérimental, ce type de forçage est obtenu par une machine à marée équipée d'un arbre rotatif. Au total, on utilise dans ce travail trois types d'approches (expérimentale, numérique, analytique), l'objectif étant d'étudier le mouvement vertical de la surface "libre" et l'écoulement non saturé sus-jacent, de façon à prendre en compte aussi bien les pertes de charge dans la zone saturée que les gradients de pression capillaire dans la zone non saturée. […]
In this thesis, we study hydrodynamic oscillations in porous bodies (unsaturated or partially saturated), due to tidal oscillations of water levels in adjacent open water bodies. The focus is on beach hydrodynamics, but potential applications concern, more generally, time varying and oscillating water levels in coupled systems involving subsurface / open water interactions (natural and artificial beaches, harbor dykes, earth dams, river banks, estuaries). The tidal forcing of groundwater is represented and modeled (both experimentally and numerically) by quasi-static oscillations of water levels in an open water reservoir connected to the porous medium. Specifically, we focus on vertical water movements forced by an oscillating pressure imposed at the bottom of a soil column. Experimentally, a rotating tide machine is used to achieve this forcing. Overall, we use three types of methods (experimental, numerical, analytical) to study the vertical motion of the groundwater table and the unsaturated flow above it, taking into account the vertical head drop in the saturated zone as well as capillary pressure gradients in the unsaturated zone. Laboratory experiments are conducted on vertical sand columns, with a tide machine to force water table oscillations, and with porous cup tensiometers to measure both positive pressures and suctions along the column (among other measurement methods). Numerical simulations of oscillatory water flow are implemented with the BIGFLOW 3D code (implicit finite volumes, with conjugate gradients for the matrix solver and modified Picard iterations for the nonlinear problem). In addition, an automatic calibration based on a genetic optimization algorithm is implemented for a given tidal frequency, to obtain the hydrodynamic parameters of the experimental soil. Calibrated simulations are then compared to experimental results for other non calibrated frequencies. Finally, a family of quasi-analytical multi-front solutions is developed for the tidal oscillation problem, as an extension of the Green-Ampt piston flow approximation, leading to nonlinear, non-autonomous systems of Ordinary Differential Equations with initial conditions (dynamical systems). The multi-front solutions are tested by comparing them with a refined finite volume solution of the Richards equation. Multi-front solutions are at least 100 times faster, and the match is quite good even for a loamy soil with strong capillary effects (the number of fronts required is small, no more than N≈ to 20 at most). A large set of multi-front simulations is then produced in order to analyze water table and flux fluctuations for a broad range of forcing frequencies. The results, analyzed in terms of means and amplitudes of hydrodynamic variables, indicate the existence, for each soil, of a characteristic frequency separating low frequency / high frequency flow regimes in the porous system

In this thesis, we explore two important aspects of study of differential equations: analytical and computational aspects. We first consider a partial differential equation model for a static liquid surface (capillary surface). We prove through mathematical analyses that the solution of this mathematical model (the Laplace-Young equation) in a cusp domain can be bounded or unbounded depending on the boundary conditions. By utilizing the knowledge we have obtained about the singular behaviour of the solution through mathematical analysis, we then construct a numerical methodology to accurately approximate unbounded solutions of the Laplace-Young equation. Using this accurate numerical methodology, we explore some remaining open problems on singular solutions of the Laplace-Young equation. Lastly, we consider ordinary differential equation models used in the pharmaceutical industry and develop a numerical method for estimating model parameters from incomplete experimental data. With our numerical method, the parameter estimation can be done significantly faster and more robustly than with conventional methods.

by Ho Wing Kin.
Thesis submitted in: July 1997.
Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.
Includes bibliographical references (leaves 89-92).
Chapter 1 --- Introduction --- p.5
Chapter 1.1 --- The Euler-Lagrange Equation for Capillary Surfaces --- p.6
Chapter 1.2 --- The Capillary Tube --- p.12
Chapter 2 --- Comparison Principles --- p.16
Chapter 2.1 --- The General Comparison Principle --- p.16
Chapter 2.2 --- Applications --- p.24
Chapter 3 --- Existence Criteria --- p.30
Chapter 3.1 --- Necessary Conditions --- p.31
Chapter 3.2 --- Sufficient Conditions --- p.35
Chapter 4 --- Uniqueness --- p.53
Chapter 4.1 --- General Bounded Domains Case --- p.53
Chapter 4.2 --- Infinite Strip Case --- p.56
Chapter 5 --- Gradient Estimate of Surfaces of Constant Mean Curvature --- p.66
Chapter 5.1 --- The Gradient Estimate --- p.67
Chapter 5.2 --- Behavior as R→ 1 --- p.70
Chapter 5.3 --- Existence of the Comparison Surfaces --- p.76
Chapter 5.4 --- Ro is Best Possible --- p.82
Appendix Mean Curvature --- p.85

Unified separation science not only describes the separation process of each technique, but is also instrumental to the further development of separation science. Previous developments of the theory have focused on macroscopic (often average) properties of separation systems: the average analyte migration rate, the steady state, resolution, sensitivity, precision, etc. On the other hand, microscopic/instantaneous behaviors are essential to understand complex phenomena, such as dynamic complexation, sweeping/stacking of sample analytes, and buffer depletion. Computer simulation is one of the best ways to visualize the instantaneous behaviors of chemical/physical systems. The simulation model of dynamic complexation capillary electrophoresis (SimDCCE) is based on the differential mass transfer equation, the governing principle of analyte migration in all separation techniques. SimDCCE is highly efficient, and is the first to demonstrate the affinity interactions in capillary electrophoresis (CE) in real time or faster. SimDCCE is one big step towards the ultimate goal: the unified computer simulation of all separation techniques. Using SimDCCE, a thorough study of affinity capillary electrophoresis (ACE) mechanisms was carried out. The regression methods for determining binding constants from ACE experiments require the assumption of instant establishment of the steady state condition which was examined in a variety of scenarios in the six cases defined by the order of the mobilities of the analyte, additive, and complex. The enumeration algorithm built upon computer simulation was developed to provide a fast and accurate alternative for determining binding constants when the assumption is invalid. The enumeration approach is equally applicable to binding studies using techniques such as NMR, chromatography, and optical methods. The second part of my research is the application of CE in other research fields, including biochemistry, anesthesia, and forensic chemistry. A systematic optimization of exhaustive electrokinetic injection and sweeping processes was carried out to improve the reproducibility and sensitivity for the detection of amphetamine and its derivatives.
Science, Faculty of
Chemistry, Department of
Graduate

碩士
國立臺灣大學
數學研究所
96
We consider the minimal surface equation in an infinite sector domain with given capillary boundary conditions.First, we give a necessary and sufficient conditions for the existence of the linear solution. Second, we study the behavior of the solutions of the minimal surface equation at the origin and at the infinite by using the blow up and the sip in process. Finally, we claim that the solution is linear on the boundary and conclude that it is a plane.