Die vorliegende Arbeit beschaeftigt sich mit dem strukturellen Aufbau von (1-100) M-plane GaN, das mit plasmaunterstuetzter Molekularstrahlepitaxie auf gamma-LiAlO2(100) Substraten gewachsen wurde. Die heteroepitaktische Ausrichtung einerseits, sowie die Mikrostruktur und die Erzeugungsmechanismen der Defekte andererseits, wurde mit der Transmissionselektronenemikroskopie (TEM) systematisch untersucht. Das gamma-LiAlO2 Substrat reagiert heftig im Mikroskop unter Bestrahlung mit hochenergetischen Elektronen. Waehrend dieser Strahlenschaedigung verliert das Material seine urspruengliche kristalline Struktur und vollzieht eine Phasentransformation, die anhand einer Serie von Feinbereichsbeugungsdiagrammen nachgewiesen werden konnte. Die atomare Grenzflaechenstruktur zwischen epitaktisch gewachsenem alpha-GaN(1-100) und tetragonalem gamma-LiAlO2 Substrat ist mittels HRTEM untersucht worden. Die neuartige Epitaxiebeziehung ist mit Elektronenbeugung bestaetigt worden und lautet folgendermassen: (1-100)GaN liegt parallel zu (100)gamma-LiAlO2 und [11-20]GaN ist parallel zu [001]gamma-LiAlO2. Die Realstruktur der M-plane GaN Schichten, die auf (100)gamma-LiAlO2 gewachsen werden, unterscheidet sich erheblich von der in C-plane Orientierung hergestellten Epischichten. Ausfuehrliche TEM Untersuchungen zeigen, dass die M-plane Schichten vor allem intrinsische (I1 und I2) und extrinsische (E) Stapelfehler in der Basalebene enthalten. Der vorherrschende I2 Stapelfehler besitzt keine Komponente des Verschiebungsvektors senkrecht zur Ebene und ist damit nicht geeignet, epitaktische Dehnung entlang der [11-20] Richtung abzubauen. Darueberhinaus ist eine komplexe Grenze in der (10-10) Prismen- flaeche entdeckt worden, die zur Grenzflaeche geneigt verlaeuft. Die Defekte in den M-plane GaN Epischichten werden waehrend der anfaenglichen Keimbildungsphase erzeugt. Atomare Stufen entlang der [001] Richtung auf dem LiAlO2 Substrat fuehren zur Bildung von Stapelfehlern vom Typ I2. / In this work the structure of (1-100)M-plane GaN epitaxially grown on gamma-LiAlO2(100) by using plasmaassisted molecular beam epitaxy (PAMBE) is studied. The heteroepitaxial alignment and the microstructure of M-plane GaN as well as the defect formation in the layer are systematically investigated by using transmission electron microscopy (TEM). The gamma-LiALO2 substrate reacts under irradiation of high-energy electrons in the TEM (200-300 keV).The material looses its original crystalline structure during this process undergoing irradiation damage followed by a phase transformation as it is verified by a series of selected area diffraction patterns taken under constant electron dose. The result is a structural phase transformation from the tetragonal gamma to the trigonal alpha phase. The atomic interface structure of epitaxially grown hexagonal alpha-GaN(1-100) layers on tetragonal gamma-LiAlO2 (100) substrates is investigated by means of HRTEM. The novel epitaxial orientation relationship verified by electron diffraction is given by (1-100)GaN parallel to (100)gamma-LiAlO2 and [11-20]GaN parallel to [001]gamma-LiAlO2. The defect structure of M-plane GaN epilayers grown on gamma-LiAlO2(100) substrates is different to that of C-plane GaN. Our detailed TEM studies reveal that the M-plane layers mainly contain intrinsic I1 and I2 and extrinsic E basal plane stacking faults. The dominant I2 stacking fault has no out-of-plane displacement vector component and is thus not qualified for epitaxial strain relief along the [11-20] axis. Beyond this, a complex type of planar defect is detected in the (10-10) prism plane which is inclined with respect to the interface. The study of nucleation samples shows that the surface morphology is directly correlated to the generation of the dominant planar defects. Atomic steps along the [001] direction in the gamma-LiAlO2 substrate result in the formation of basal plane stacking faults I2.

The author presents some generalized Burgers' equations for incompressible and isothermal flow of viscous non-Newtonian fluids based on the Cross model, the Carreau model, and the Power-Law model and some simple assumptions on the flows. The author numerically solves the traveling wave equations for the Cross model, the Carreau model, the Power-Law model by using industrial data. The author proves existence and uniqueness of solutions to the traveling wave equations of each of the three models. The author also provides numerical estimates of the shock thickness as well as maximum strain $\varepsilon_{11}$ for each of the fluids.

This PhD thesis treats some problems concerning nonlinear differential equations. In the first two papers computer-assisted proofs are used. The differential equations there are rewritten as fixed point problems, and the existence of solutions are proved. The problem in the first paper is one-dimensional; with one boundary condition given by an integral. The problem in the second paper is three-dimensional, and Dirichlet boundary conditions are used. Both problems have their origins in fluid dynamics. Paper III describes an inverse problem for the heat equation. Given the solution, a solution dependent diffusion coefficient is estimated by intervals at a finite set of points. The method includes the construction of set-valued level curves and two-dimensional splines. In paper IV we prove that there exists a unique, globally attracting fixed point for a differential equation system. The differential equation system arises as the number of peers in a peer-to-peer network, which is described by a suitably scaled Markov chain, goes to infinity. In the proof linearization and Dulac's criterion are used.

The structure and dynamics of the smectic-A liquid crystal 8CB (4 cyano-4 octylbiphenyl) when sheared and confined to mesoscale gaps (with crossed cylindrical geometry and mica confining surfaces) were studied using a Surface Forces Apparatus (SFA). Triangular shear patterns with frequencies of 0.01, 0.1, 1.0 and 10 Hz, and amplitudes of 62.5 nm, 625 nm and 6.25 m were applied to samples at gap sizes of 0.5 and 5.0 m. The study was performed at room temperature (20.5C) and at two higher temperatures (22C and 27C). In order to minimize the thermal fluctuations within the test chamber and hence to allow for the rapid re-initialization of test runs, the SFA was modified to allow for quick, precise and remote control of the confining surfaces. The procedure maximized the number of tests that could be undertaken with a single pair of surfaces so that a single gap geometry could be maintained for the duration of the test run. In order to run the SFA remotely, scripts written with a commercial software package, LabVIEW, were used to control of the SFA components, its FECO-monitoring camera and all its peripheral electronic equipment as well. Samples were agitated to disrupt any shear-induced liquid crystal domain alignment from previous testing following each shear test, and methodologies were developed to ascertain the extent of confinement quickly and remotely following agitation. Separate methods were developed for gap sizes at each extreme of the mesoscale regime, where the transition from bulklike structure and dynamics to nano-confinement occurs (between 1 and 10 microns for smectic-A 8CB). The results revealed that the greater amplitude-gap aspect ratio and surface-to-domain contact associated with smaller gaps facilitated reorientation of the domains in the shear direction. Evidence was also presented of domains at the higher end or outside of the mesoscale regime that, while straining and accreting, were unable to reorient and thereby led to an overall increase of viscoelastic response. The effective viscosity was found to obey a simple power law with respect to shear rate, , and the flow behaviour indices, n, slightly in excess of unity indicate shear thickening occurs with large enough shear amplitude, and that the viscosity reached a plateau near unity over shear rates of 0.005 to 500 s-1 within the mesoscale regime. Different K and n values were observed depending on the shear amplitude used. Unlike bulk smectic 8CB, whose domains do not align well in the shear direction with large shear-strain amplitude, at mesoscale levels of confinement large amplitude shearing (up to 12.5 shear strain amplitude) was found to be very effective at aligning domains. In general domain reorientation is found to be much more rapid within the mesoscale regime than has been reported in bulk. Aggressive shearing was found to result in a complete drop in viscoelastic response within seconds, while gentler shearing is found to produce a very gradual increase that persists for more than six hours, with individual shear periods exhibiting frequent and significant deviations from the expected smooth shear path that may be a product of discrete domain reorientations. From these findings, certain traits of the smectic 8CB domain structures under mesoscale confinement were deduced, including how they respond to shear depending on the level of confinement, and how their reorientation due to shear varies not only with shear rate but also independently with shear amplitude. An equation describing the viscosity change as a function of both shear rate and shear amplitude is proposed. The shear amplitude dependence introduces the notion of shearing beyond the proposed smectic 8CB “viscoelastic limit”, which was shown to exhibit behaviour in accordance with Large Amplitude Oscillatory Shear (LAOS) techniques developed for Fourier Transform rheology. The findings provided an understanding of the behavioural changes that occur as one reduces the level of confinement of smectic materials from bulk to nanoconfinement.

Ce travail porte sur l'étude de mésophases lamellaires lyotropes qui contiennent des polymères hydrosolubles confinés dans les lamelles d'eau de l'empilement. L'effet du polymère confiné sur la stabilité de la structure lamellaire est en particulier étudié grâce à une mesure directe des interactions entre les membranes de tensioactif grâce la technique du Surface Force Apparatus (SFA). Les systèmes étudiés sont des fluides complexes et très visqueux ce qui a donné lieu au développement d'une nouvelle méthode pour analyser les données collectées. En l'absence de polymère dissous dans l'empilement lamellaire, la valeur du module de compressibilité élastique mesurée est remarquablement interprétée avec la théorie électrostatique corrigée des corrélations des contre-ions. En présence de polymère, au fur et à mesure que les macromolécules remplacent le contenu en eau, le module élastique de compressibilité de l'empilement lamellaire chute,signature d'une interaction attractive due à la présence des macromolécules. Néanmoins, les développements théoriques proposés ne parviennent pas à interpréter quantitativement cette décroissance. De plus, des comportements très intéressants et inattendus ont été mis en évidence lors de l'approche d'un point critique : la présence de dislocations de très grands vecteurs de Burgers à grande séparation ainsi que des phénomènes d'avalanches. En particulier, des énergies de nucléation de dislocation ont pu être extraites.

The structure and dynamics of the smectic-A liquid crystal 8CB (4 cyano-4 octylbiphenyl) when sheared and confined to mesoscale gaps (with crossed cylindrical geometry and mica confining surfaces) were studied using a Surface Forces Apparatus (SFA). Triangular shear patterns with frequencies of 0.01, 0.1, 1.0 and 10 Hz, and amplitudes of 62.5 nm, 625 nm and 6.25 m were applied to samples at gap sizes of 0.5 and 5.0 m. The study was performed at room temperature (20.5C) and at two higher temperatures (22C and 27C). In order to minimize the thermal fluctuations within the test chamber and hence to allow for the rapid re-initialization of test runs, the SFA was modified to allow for quick, precise and remote control of the confining surfaces. The procedure maximized the number of tests that could be undertaken with a single pair of surfaces so that a single gap geometry could be maintained for the duration of the test run. In order to run the SFA remotely, scripts written with a commercial software package, LabVIEW, were used to control of the SFA components, its FECO-monitoring camera and all its peripheral electronic equipment as well. Samples were agitated to disrupt any shear-induced liquid crystal domain alignment from previous testing following each shear test, and methodologies were developed to ascertain the extent of confinement quickly and remotely following agitation. Separate methods were developed for gap sizes at each extreme of the mesoscale regime, where the transition from bulklike structure and dynamics to nano-confinement occurs (between 1 and 10 microns for smectic-A 8CB). The results revealed that the greater amplitude-gap aspect ratio and surface-to-domain contact associated with smaller gaps facilitated reorientation of the domains in the shear direction. Evidence was also presented of domains at the higher end or outside of the mesoscale regime that, while straining and accreting, were unable to reorient and thereby led to an overall increase of viscoelastic response. The effective viscosity was found to obey a simple power law with respect to shear rate, , and the flow behaviour indices, n, slightly in excess of unity indicate shear thickening occurs with large enough shear amplitude, and that the viscosity reached a plateau near unity over shear rates of 0.005 to 500 s-1 within the mesoscale regime. Different K and n values were observed depending on the shear amplitude used. Unlike bulk smectic 8CB, whose domains do not align well in the shear direction with large shear-strain amplitude, at mesoscale levels of confinement large amplitude shearing (up to 12.5 shear strain amplitude) was found to be very effective at aligning domains. In general domain reorientation is found to be much more rapid within the mesoscale regime than has been reported in bulk. Aggressive shearing was found to result in a complete drop in viscoelastic response within seconds, while gentler shearing is found to produce a very gradual increase that persists for more than six hours, with individual shear periods exhibiting frequent and significant deviations from the expected smooth shear path that may be a product of discrete domain reorientations. From these findings, certain traits of the smectic 8CB domain structures under mesoscale confinement were deduced, including how they respond to shear depending on the level of confinement, and how their reorientation due to shear varies not only with shear rate but also independently with shear amplitude. An equation describing the viscosity change as a function of both shear rate and shear amplitude is proposed. The shear amplitude dependence introduces the notion of shearing beyond the proposed smectic 8CB “viscoelastic limit”, which was shown to exhibit behaviour in accordance with Large Amplitude Oscillatory Shear (LAOS) techniques developed for Fourier Transform rheology. The findings provided an understanding of the behavioural changes that occur as one reduces the level of confinement of smectic materials from bulk to nanoconfinement.

The solution of the Burgers' equation computed by the standard finite element method is degraded by oscillations, which are the manifestation of the Gibbs phenomenon. In this work we study the following numerical me- thods: Discontinuous Galerkin method, stable low order schemes and the flux corrected technique method in order to prevent the undesired Gibbs phenomenon. The focus is on the reduction of severe overshoots and under- shoots and the preservation of the smoothness of the solution. We consider a simple 1D problem on the interval (0, 1) with different initial conditions to demonstrate the properties of the presented methods. The numerical results of individual methods are provided. 1

Turbulence is an open and challenging problem for mathematical approaches, physical modeling and numerical simulations. Numerical solutions contribute significantly to the understand of the nature and effects of turbulence. The focus of this thesis is the development of appropriate numerical methods for the computer simulation of turbulent flows. Many of the existing approaches to turbulence utilize analogies from kinetic theory. Degond & Lemou (J. Math. Fluid Mech., 4, 257-284, 2002) derived a k-✏ type turbulence model completely from kinetic theoretic framework. In the first part of this thesis, a numerical method is developed for the computer simulation based on this model. The Boltzmann equation used in the model has an isotropic, relaxation collision operator. The relaxation time in the collision operator depends on the microscopic turbulent energy, making it difficult to construct an efficient numerical scheme. In order to achieve the desired numerical efficiency, an appropriate change of frame is applied. This introduces a stiff relaxation source term in the equations and the concept of asymptotic preserving schemes is then applied to tackle the stiffness. Some simple numerical tests are introduced to validate the new scheme. In the second part of this thesis, alternative approaches are sought for more efficient numerical techniques. The Lattice Boltzmann Relaxation Scheme (LBRS) is a novel method developed recently by Rohan Deshmukh and S.V. Raghuram Rao for simulating compressible flows. Two different approaches for the construction of implicit sub grid scale -like models as Implicit Large Eddy Simulation (ILES) methods, based on LBRS, are proposed and are tested for Burgers turbulence, or Burgulence. The test cases are solved over a largely varying Reynolds number, demonstrating the efficiency of this new ILES-LBRS approach. In the third part of the thesis, as an approach towards the extension of ILES-LBRS to incompressible flows, an artificial compressibility model of LBRS is proposed. The modified framework, LBRS-ACM is then tested for standard viscous incompressible flow test cases.

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.

<p dir="ltr">The foundation of numerical codes used in engineering analyses of two-phase flows is the two-fluid model (TFM). However, the TFM codes use artificial regularization to remove the high frequency ill-posed instability in the numerical solution. This work demonstrates that incorporating variational inertial coupling terms to the numerical two-fluid model code, makes it more complete, objective and well-posed without the need for any regularization. The variational TFM is implemented in an industrial CFD code and the two-fluid Burgers problem is used to verify the numerical TFM against analytic solutions.</p>