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Dans cette thèse, nous nous intéressons à deux approximations hyperboliques des équations de Navier-Stokes incompressibles en dimensions 2 et 3 d'espace. Dans un premier temps, on considère une perturbation hyperbolique de l'équation de la chaleur, introduite par Cattaneo en 1949, pour remédier au paradoxe de la propagation instantanée de cette équation. En 2004, Brenier, Natalini et Puel remarquent que la même perturbation, qui consiste à rajouter ε∂tt à l'équation, intervient en relaxant les équations d'Euler. En dimension 2, les auteurs montrent que, pour des sonnées régulières et sous certaines hypothèses de petitesse, la solution globale de la perturbation converge vers l'unique solution globale de (NS). En 2007, Paicu et Raugel améliorent les résultats de [BNP] en étendant la théorie à la dimension 3 et en prenant des données beaucoup moins régulières. Nous avons obtenu des résultats de convergence, avec données de régularité quasi-critique, qui complètent et prolongent ceux de [BNP] et [PR]. La seconde approximation que l'on considère est un nouveau modèle hyperbolique à vitesse de propagation finie. Ce modèle est obtenu en pénalisant la contrainte d'incompressibilité dans la perturbation de Cattaneo. Nous démontrons que les résultats d'existence globale et de convergence du précédent modèle sont encore vérifiés pour celui-ci. / In this work, we are interested in two hyperbolic approximations of the 2D and 3D Navier-Stokes equations. The first model we consider comes from Cattaneo's hyperbolic perturbation of the heat equation to obtain a finite speed of propagation equation. Brenier, Natalini and Puel studied the same perturbation as a relaxed version of the 2D Euler equations and proved that the solution to this relaxation converges towards the solution to (NS) with smooth data, provided some smallness assumptions. Later, Paicu and Raugel improved their results, extending the theory to the 3D setting and requiring significantly less regular data. Following [BNP] and [PR], we prove global existence and convergence results with quasi-critical regularity assumptions on the initial data. In the second part, we introduce a new hyperbolic model with finite speed of propagation, obtained by penalizing the incompressibility constraint in Cattaneo's perturbation. We prove that the same global existence and convergence results hold for this model as well as for the first one.

This thesis derives a first approximation model that will describe the flow of fluid in microfluidic devices such as in microchannels, microdiffusers and micronozzles using electrical network modelling. The important parameter that is of concern is the flow rates of these devices. The purpose of this work is to contribute to the physical component of our interactive Virtual Reality (VR)-prototyping tool for MEMS, with emphasis on fast calculations for interactive CAD design. Current calculations are too time consuming and not suitable for interactive CAD with dynamic animations. This work contributes to and fills the need for the development of MEMS dynamic visualisation, showing the movement of fluid within microdevices in time scale. Microfluidic MEMS devices are used in a wide range of applications, such as in chemical analysis, gene expression analysis, electronic cooling system and inkjet printers. Their success lies in their microdimensions, enabling the creation of systems that are considerably minute yet can contain many complex subsystems. With this reduction in size, the advantages of requiring less material for analysis, less power consumption, less wastage and an increase in portability becomes their selling point. Market size is in excess of US$50 billion in 2004, according to a study made by Nexus. New applications are constantly being developed leading to creation of new devices, such as the DNA and the protein chip. Applications are found in pharmaceuticals, diagnostic, biotechnology and the food industry. An example is the outcome of the mapping and sequencing of the human genome DNA in the late 1990's leading to greater understanding of our genetic makeup. Armed with this knowledge, doctors will be able to treat diseases that were deemed untreatable before, such as diabetes or cancer. Among the tools with which that can be achieved include the DNA chip which is used to analyse an individual's genetic makeup and the Gene chip used in the study of cancer. With this burgeoning influx of new devices and an increase in demand for them there is a need for better and more efficient designs. The MEMS design process is time consuming and costly. Many calculations rely on Finite Element Analysis, which has slow and time consuming algorithms, that make interactive CAD unworkable. This is because the iterative algorithms for calculating the animated images showing the ongoing proccess as they occur, are too slow. Faster computers do not solve the void of efficient algorithms, because with faster computer also comes the demand for a fasters response. A 40 - 90 minute FEA calculation will not be replaced by a faster computer in the next decades to an almost instant response. Efficient design tools are required to shorten this process. These interactive CAD tools need to be able to give quick yet accurate results. Current CAD tools involve time consuming numerical analysis technique which requires hours of numerous iterations for the device structure design followed by more calculations to achieve the required output specification. Although there is a need for a detailed analysis, especially in solving for a particular aspect of the design, having a tool to quickly get a first approximation will greatly shorten the guesswork involved in determining the overall requirement. The underlying theory for the fluid flow model is based on traditional continuum theory and the Navier-Stokes equation is used in the derivation of a layered flow model in which the flow region is segmented into layered sections, each having different flow rates. The flow characteristics of each sections are modeled as electrical components in an electrical circuit. Matlab 6.5 (MatlabTM) is used for the modelling aspect and Simulink is used for the simulation.

Approximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing, medical imaging, as well as neural networks. Several applications demand that certain physical properties be fulfilled, such as a function being divergence free. No such class of radial basis functions that reflects these physical properties was known until 1994, when Narcowich and Ward introduced a family of matrix-valued radial basis functions that are divergence free. They also obtained error bounds and stability estimates for interpolation by means of these functions. These divergence-free functions are very smooth, and have unbounded support. In this thesis we introduce a new class of matrix-valued radial basis functions that are divergence free as well as compactly supported. This leads to the possibility of applying fast solvers for inverting interpolation matrices, as these matrices are not only symmetric and positive definite, but also sparse because of this compact support. We develop error bounds and stability estimates which hold for a broad class of functions. We conclude with applications to the numerical solution of the Navier-Stokes equation for certain incompressible fluid flows.

Um problema auxiliar crucial à análise do problema de Stokes Compressível é estudado via a técnica de potenciais de camada dupla em regiões Lipschitz através de um método primeiro utilizado por Verchota e subseqüentemente estendido ao caso parabólico por Brown e Shen. Desse modo, mediante a utilização e cálculo da condição de salto na fronteira é possível estabelecer a existência e unicidade da solução em apropriados espaços funcionais via o estudo de potenciais de camada. / An auxiliary problem crucial to the analysis of the compressible Stokes problem is studied by means of the technique of double layer in Lipschitz regions through a method first used by Verchota and subsequently extended to the parabolic case by Brown and Shen. In this way through the use and calculation of the boundary jump condition it is possible to establish the existence and unicity of the solution in appropriate function spaces via the study of boundary layer potentials.

Um problema auxiliar crucial à análise do problema de Stokes Compressível é estudado via a técnica de potenciais de camada dupla em regiões Lipschitz através de um método primeiro utilizado por Verchota e subseqüentemente estendido ao caso parabólico por Brown e Shen. Desse modo, mediante a utilização e cálculo da condição de salto na fronteira é possível estabelecer a existência e unicidade da solução em apropriados espaços funcionais via o estudo de potenciais de camada. / An auxiliary problem crucial to the analysis of the compressible Stokes problem is studied by means of the technique of double layer in Lipschitz regions through a method first used by Verchota and subsequently extended to the parabolic case by Brown and Shen. In this way through the use and calculation of the boundary jump condition it is possible to establish the existence and unicity of the solution in appropriate function spaces via the study of boundary layer potentials.

Um problema auxiliar crucial à análise do problema de Stokes Compressível é estudado via a técnica de potenciais de camada dupla em regiões Lipschitz através de um método primeiro utilizado por Verchota e subseqüentemente estendido ao caso parabólico por Brown e Shen. Desse modo, mediante a utilização e cálculo da condição de salto na fronteira é possível estabelecer a existência e unicidade da solução em apropriados espaços funcionais via o estudo de potenciais de camada. / An auxiliary problem crucial to the analysis of the compressible Stokes problem is studied by means of the technique of double layer in Lipschitz regions through a method first used by Verchota and subsequently extended to the parabolic case by Brown and Shen. In this way through the use and calculation of the boundary jump condition it is possible to establish the existence and unicity of the solution in appropriate function spaces via the study of boundary layer potentials.

We examine the Rayleigh–Bénard convection as modelled by the Boussinesq equation. Our aim is at deriving bounds for the heat enhancement factor in the vertical direction, the Nusselt number, which reproduce physical scalings. In the first part of the dissertation, we examine the the simpler model when the acceleration of the fluid is neglected (Pr=∞) and prove the non-optimality of the temperature background field method by showing a lower bound for the Nusselt number associated to it. In the second part we consider the full model (Pr<∞) and we prove a new upper bound which improve the existing ones (for large Pr numbers) and catches a transition at Pr~Ra^(1/3).

Turbulence is ubiquitous in the flows of fluids and plasmas. This thesis is devoted to studies of the statistical properties of turbulence in the three-dimensional (3D) Hall magnetohydrodynamic (Hall-MHD) equations, the two-dimensional (2D) MHD equations, the one-dimensional (1D) hyperviscous Burgers equation, and the 3D Navier-Stokes equations. Chapter 1 contains a brief introduction to statistically homogeneous and isotropic turbulence. This is followed by an over-view of the equations we study in the subsequent chapters, the motivation for the studies and a summary of problems we investigate in chapters 2-6. In Chapter 2 we present our study of Hall-MHD turbulence [1]. We show that a shell-model version of the 3D Hall-MHD equations provides a natural theoretical model for investigating the multiscaling behaviors of velocity and magnetic structure functions. We carry out extensive numerical studies of this shell model, obtain the scaling exponents for its structure functions, in both the low-k and high-k power-law ranges of 3D Hall-MHD, and find that the extended-self-similarity procedure is helpful in extracting the multiscaling nature of structure functions in the high-k regime, which otherwise appears to display simple scaling. Our results shed light on intriguing solar-wind measurements. In Chapter 3 we present our study of the inverse-cascade regime in two-dimensional magnetohydrodynamic turbulence [2]. We present a detailed direct numerical simulation (DNS) of statistically steady, homogeneous, isotropic, two-dimensional magnetohydrodynamic (2D MHD) turbulence. Our study concentrates on the inverse cascade of the magnetic vector potential. We examine the dependence of the statistical properties of such turbulence on dissipation and friction coefficients. We extend earlier work significantly by calculating fluid and magnetic spectra, probability distribution functions (PDFs) of the velocity, magnetic, vorticity, current, stream-function, and magnetic-vector-potential fields and their increments. We quantify the deviations of these PDFs from Gaussian ones by computing their flatnesses and hyperflatnesses. We also present PDFs of the Okubo-Weiss parameter, which distinguishes between vortical and extensional flow regions, and its magnetic analog. We show that the hyperflatnesses of PDFs of the increments of the stream-function and the magnetic vector potential exhibit significant scale dependence and we examine the implication of this for the multiscaling of structure functions. We compare our results with those of earlier studies. In Chapter 4 we compare the statistical properties of 2D MHD turbulence for two different energy injection scales. We present systematic DNSs of statistically steady 2D MHD turbulence. Our two DNSs are distinguished by kinj, the wave number at which we inject energy into the system. In our first DNS (run R1), kinj = 2 and, in the second (run R2) kinj = 250. We show that various statistical properties of the turbulent states in the runs R1 and R2 are strikingly different The nature of energy spectrum, probability distribution functions, and topological structures are compared for the two runs R1 and R2 are found to be strikingly different. In Chapter 5 we study the hyperviscous Burgers equation for very high α, order of hyperviscosity [3]. We show, by using direct numerical simulations and theory, how, by increasing α in equations of hydrodynamics, there is a transition from a dissipative to a conservative system. This remarkable result, already conjectured for the asymptotic case α →∞ [U. Frisch et al., Phys. Rev. Lett. 101, 144501 (2008)], is now shown to be true for any large, but finite, value of α greater than a crossover value α crossover. We thus provide a self-consistent picture of how dissipative systems, under certain conditions, start behaving like conservative systems, and hence elucidate the subtle connection between equilibrium statistical mechanics and out-of-equilibrium turbulent flows. In Chapter 6 we show how to use asymptotic-extrapolation and Richardson extrapolation methods to extract the exponents ξ p that characterize the dependence of the order-p moments of the velocity gradients on the Reynolds number Re. To use these extrapolation methods we must have high-precision data for such moments. We obtain these high-precision data by carrying out the most extensive, quadruple precision, pseudospectral DNSs of the Navier-Stokes equation.

Fluid dynamics considers the physics of liquids and gases. This is a branch of classical physics and is totally based on Newton's laws of motion. Nevertheless, the equation of fluid motion, Navier-Stokes equation, becomes very complicated to solve even for very simple configurations. This thesis treats mainly analytical vortex solutions to Navier-Stokes equations. Vorticity is usually concentrated to smaller regions of the flow, sometimes isolated objects, called vortices. If one are able to describe vortex structures exactly, important information about the flow properties are obtained. Initially, the modeling of a conical vortex geometry is considered. The results are compared with wind-tunnel measurements, which have been analyzed in detail. The conical vortex is a very interesting phenomenaon for building engineers because it is responsible for very low pressures on buildings with flat roofs. Secondly, a suggested analytical solution to Navier-Stokes equation for internal flows is presented. This is based on physical argumentation concerning the vorticity production at solid boundaries. Also, to obtain the desired result, Navier-Stokes equation is reformulated and integrated. In addition, a model for required information of vorticity production at boundaries is proposed. The last part of the thesis concerns the examples of vortex models in 2-D and 3-D. In both cases, analysis of the Navier-Stokes equation, leads to the opportunity to construct linear solutions. The 2-D studies are, by the use of diffusive elementary vortices, describing experimentally observed vortex statistics and turbulent energy spectrums in stratified systems and in soapfilms. Finally, in the 3-D analysis, three examples of recent experimentally observed vortex objects are reproduced theoretically. First, coherent structures in a pipe flow is modeled. These vortex structures in the pipe are of interest since they appear for Re in the range where transition to turbulence is expected. The second example considers the motion in a viscous vortex ring. The model, with diffusive properties, describes the experimentally measured velocity field as well as the turbulent energy spectrum. Finally, a streched spiral vortex is analysed. A rather general vortex model that has many degrees of freedom is proposed, which also may be applied in other configurations.