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Kim, Tae Eun. "Quasi-solution Approach to Nonlinear Integro-differential Equations: Applications to 2-D Vortex Patch Problems." The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1499793039477532.
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Handel, Andreas. "Limits of Localized Control in Extended Nonlinear Systems." Diss., Georgia Institute of Technology, 2004. http://hdl.handle.net/1853/5025.
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We investigate the limits of localized linear control
in spatially extended, nonlinear systems.
Spatially extended, nonlinear systems can be found in virtually
every field of engineering and science.
An important category of such systems are fluid flows.
Fluid flows play an important role in many commercial applications,
for instance in the chemical, pharmaceutical and food-processing industries.
Other important fluid flows include
air- or water flows around cars, planes or ships.
In all these systems, it is highly desirable to control the flow of the
respective fluid.
For instance control of the air flow around an airplane or car
leads to better fuel-economy and reduced noise production.
Usually, it is impossible to apply control everywhere.
Consider an airplane: It would not be feasibly to cover the
whole body of the plane with control units.
Instead, one can place the control units at localized regions,
such as points along the edge of the wings,
spaced as far apart from each other as possible.
These considerations lead to an important question:
For a given system, what is the minimum number of localized controllers
that still ensures successful control?
Too few controllers will not achieve control,
while using too many leads to unnecessary expenses and wastes resources.
To answer this question, we study localized control
in a class of model equations.
These model equations are good representations
of many real fluid flows.
Using these equations,
we show how one can design localized control that renders the system stable.
We study the properties of the control
and derive several expressions that allow
us to determine the limits of successful control.
We show how the number of controllers
that are needed for successful control
depends on the size and type of the system, as well as the way control is
implemented. We find that especially the nonlinearities and the amount
of noise present in the system play a crucial role.
This analysis allows us to determine under which circumstances
a given number of controllers can successfully stabilize a given system.
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Paddick, Matthew. "Stabilité de couches limites et d'ondes solitaires en mécanique des fluides." Thesis, Rennes 1, 2014. http://www.theses.fr/2014REN1S049/document.
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La présente thèse traite de deux questions de stabilité en mécanique des fluides. Les deux premiers résultats de la thèse sont consacrés au problème de la limite non-visqueuse pour les équations de Navier-Stokes. Il s'agit de déterminer si une famille de solutions de Navier-Stokes dans un demi-espace avec une condition de Navier au bord converge vers une solution du modèle non visqueux, l'équation d'Euler, lorsque les paramètres de viscosité tendent vers zéro. Dans un premier temps, on considère le modèle incompressible 2D. Nous obtenons la convergence dans L2 des solutions faibles de Navier-Stokes vers une solution forte d'Euler, et une instabilité dans L∞ en temps très court pour certaines données initiales qui sont des solutions stationnaires de l'équation d'Euler. Ces résultats ne sont pas contradictoires, et on construit un exemple de donnée initiale permettant de voir se réaliser les deux phénomènes simultanément dans le cadre périodique. Dans un second temps, on s'intéresse au modèle compressible isentropique (température constante) en 3D. On démontre l'existence de solutions dans des espaces de Sobolev conormaux sur un temps qui ne dépend pas de la viscosité lorsque celle-ci devient très petite, et on obtient la convergence forte de ces solutions vers une solution de l'équation d'Euler sur ce temps uniforme par des arguments de compacité. Le troisième résultat de cette thèse traite d'un problème de stabilité d'ondes solitaires. Précisément, on considère un fluide isentropique et non visqueux avec capillarité interne, régi par le modèle d'Euler-Korteweg, et on montre l'instabilité transverse non-linéaire de solitons, c'est-à-dire que des perturbations 2D initialement petites d'une solution sous forme d'onde progressive 1D peuvent s'éloigner de manière importante de celle-ci<br>This thesis deals with a couple of stability problems in fluid mechanics. In the first two parts, we work on the inviscid limit problem for Navier-Stokes equations. We look to show whether or not a sequence of solutions to Navier-Stokes in a half-space with a Navier slip condition on the boundary converges towards a solution of the inviscid model, the Euler equation, when the viscosity parameters vanish. First, we consider the 2D incompressible model. We obtain convergence in L2 of weak solutions of Navier-Stokes towards a strong solution of Euler, as well as the instability in L∞ in a very short time of some initial data chosen as stationary solutions to the Euler equation. These results are not contradictory, and we construct initial data that allows both phenomena to occur simultaneously in the periodic setting. Second, we look at the 3D isentropic (constant temperature) compressible equations. We show that solutions exist in conormal Sobolev spaces for a time that does not depend on the viscosity when this is small, and we get strong convergence towards a solution of the Euler equation on this uniform time of existence by compactness arguments. In the third part of the thesis, we work on a solitary wave stability problem. To be precise, we consider an isentropic, compressible, inviscid fluid with internal capillarity, governed by the Euler-Korteweg equations, and we show the transverse nonlinear instability of solitons, that is that initially small 2D perturbations of a 1D travelling wave solution can end up far from it
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Kasnakoglu, Cosku. "Reduced order modeling, nonlinear analysis and control methods for flow control problems." Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1195629380.
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Sweet, Erik. "ANALYTICAL AND NUMERICAL SOLUTIONS OF DIFFERENTIALEQUATIONS ARISING IN FLUID FLOW AND HEAT TRANSFER PROBLEMS." Doctoral diss., University of Central Florida, 2009. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/2585.
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The solutions of nonlinear ordinary or partial differential equations are important in the study of fluid flow and heat transfer. In this thesis we apply the Homotopy Analysis Method (HAM) and obtain solutions for several fluid flow and heat transfer problems. In chapter 1, a brief introduction to the history of homotopies and embeddings, along with some examples, are given. The application of homotopies and an introduction to the solutions procedure of differential equations (used in the thesis) are provided. In the chapters that follow, we apply HAM to a variety of problems to highlight its use and versatility in solving a range of nonlinear problems arising in fluid flow. In chapter 2, a viscous fluid flow problem is considered to illustrate the application of HAM. In chapter 3, we explore the solution of a non-Newtonian fluid flow and provide a proof for the existence of solutions. In addition, chapter 3 sheds light on the versatility and the ease of the application of the Homotopy Analysis Method, and its capability in handling non-linearity (of rational powers). In chapter 4, we apply HAM to the case in which the fluid is flowing along stretching surfaces by taking into the effects of "slip" and suction or injection at the surface. In chapter 5 we apply HAM to a Magneto-hydrodynamic fluid (MHD) flow in two dimensions. Here we allow for the fluid to flow between two plates which are allowed to move together or apart. Also, by considering the effects of suction or injection at the surface, we investigate the effects of changes in the fluid density on the velocity field. Furthermore, the effect of the magnetic field is considered. Chapter 6 deals with MHD fluid flow over a sphere. This problem gave us the first opportunity to apply HAM to a coupled system of nonlinear differential equations. In chapter 7, we study the fluid flow between two infinite stretching disks. Here we solve a fourth order nonlinear ordinary differential equation. In chapter 8, we apply HAM to a nonlinear system of coupled partial differential equations known as the Drinfeld Sokolov equations and bring out the effects of the physical parameters on the traveling wave solutions. Finally, in chapter 9, we present prospects for future work.<br>Ph.D.<br>Department of Mathematics<br>Sciences<br>Mathematics PhD
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Philipowski, Robert. "Stochastic interacting particle systems and nonlinear partial differential equations from fluid mechanics." [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=986005622.
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Manna, Utpal. "Harmonic and stochastic analysis aspects of the fluid dynamics equations." Laramie, Wyo. : University of Wyoming, 2007. http://proquest.umi.com/pqdweb?did=1414120661&sid=1&Fmt=2&clientId=18949&RQT=309&VName=PQD.
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Qiao, Zhonghua. "Numerical solution for nonlinear Poisson-Boltzmann equations and numerical simulations for spike dynamics." HKBU Institutional Repository, 2006. http://repository.hkbu.edu.hk/etd_ra/727.
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Lam, Chun-kit, and 林晉傑. "The dynamics of wave propagation in an inhomogeneous medium: the complex Ginzburg-Landau model." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2008. http://hub.hku.hk/bib/B40887881.
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Lam, Chun-kit. "The dynamics of wave propagation in an inhomogeneous medium the complex Ginzburg-Landau model /." Click to view the E-thesis via HKUTO, 2008. http://sunzi.lib.hku.hk/hkuto/record/B40887881.
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García, López Claudia. "Patterns in partial differential equations arising from fluid mechanics." Thesis, Rennes 1, 2020. http://www.theses.fr/2020REN1S028.
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Cette thèse est consacrée à l’émergence de solutions périodiques en temps pour des modèles hamiltoniens issus de la mécanique des fluides. Dans la première partie, nous explorons dans le plan les solutions en mouvement rigide (rotation ou translation pures) avec des distributions uniformes ou non pour des modèles standards comme les équations d’Euler incompressibles ou l’équation de surface quasi–géostrophique généralisée. Dans la deuxième partie, nous menons une étude analogue pour le système quasi–géostrophique en 3D. L’étude de ce modèle montre une remarquable richesse par rapport aux modèles 2D que ce soit par rapport l’ensemble des solutions stationnaires ou la diversité des problèmes spectraux associés. Dans la dernière partie, nous discutons quelques travaux en cours de cette thèse<br>This dissertation is centered around the existence of time–periodic solutions for Hamiltonian models that arise from Fluid Mechanics. In the first part, we explore relative equilibria taking the form of rigid motion (pure rotations or translations) in the plane with uniform and non uniform distributions for standard models like the incompressible Euler equations or the generalized quasi-geostrophic equation. In the second part, we focus on a similar study for the 3D quasi-geostrophic system. The study of this model shows a remarkable diversity compared to the 2D models due to the existence of a large set of stationary solutions or the variety of the associated spectral problems. In the last part, we show some works in progress of this dissertation, and also some conclusions and perspectives
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Seadler, Bradley T. "Signed-Measure Valued Stochastic Partial Differential Equations with Applications in 2D Fluid Dynamics." Case Western Reserve University School of Graduate Studies / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=case1333062148.
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Blockley, Edward William. "Nonlinear solutions of the amplitude equations governing fluid flow in rotating spherical geometries." Thesis, University of Exeter, 2008. http://hdl.handle.net/10036/41950.
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We are interested in the onset of instability of the axisymmetric flow between two concentric spherical shells that differentially rotate about a common axis in the narrow-gap limit. The expected mode of instability takes the form of roughly square axisymmetric Taylor vortices which arise in the vicinity of the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radii. At the heart of the difficulties faced is the presence of phase mixing in the system, characterised by a non-zero frequency gradient at the equator and the tendency for vortices located off the equator to oscillate. This mechanism serves to enhance viscous dissipation in the fluid with the effect that the amplitude of any initial disturbance generated at onset is ultimately driven to zero. In this thesis we study a complex Ginzburg-Landau equation derived from the weakly nonlinear analysis of Harris, Bassom and Soward [D. Harris, A. P. Bassom, A. M. Soward, Global bifurcation to travelling waves with application to narrow gap spherical Couette flow, Physica D 177 (2003) p. 122-174] (referred to as HBS) to govern the amplitude modulation of Taylor vortex disturbances in the vicinity of the equator. This equation was developed in a regime that requires the angular velocities of the bounding spheres to be very close. When the spherical shells do not co-rotate, it has the remarkable property that the linearised form of the equation has no non-trivial neutral modes. Furthermore no steady solutions to the nonlinear equation have been found. Despite these challenges Bassom and Soward [A. P. Bassom, A. M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid Mech. 499 (2004) p. 277-314] (referred to as BS) identified solutions to the equation in the form of pulse-trains. These pulse-trains consist of oscillatory finite amplitude solutions expressed in terms of a single complex amplitude localised as a pulse about the origin. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and the whole pulse-train is modulated under an envelope and drifts away from the equator at a relatively slow speed. The survival of the pulse-train depends upon the nonlinear mutual-interaction of close neighbours; as the absence of steady solutions suggests, self-interaction is inadequate. Though we report new solutions to the HBS co-rotation model the primary focus in this work is the physically more interesting case when the shell velocities are far from close. More specifically we concentrate on the investigation of BS-style pulse-train solutions and, in the first part of this thesis, develop a generic framework for the identification and classification of pulse-train solutions. Motivated by relaxation oscillations identified by Cole [S. J. Cole, Nonlinear rapidly rotating spherical convection, Ph.D. thesis, University of Exeter (2004)] whilst studying the related problem of thermal convection in a rapidly rotating self-gravitating sphere, we extend the HBS equation in the second part of this work. A model system is developed which captures many of the essential features exhibited by Cole's, much more complicated, system of equations. We successfully reproduce relaxation oscillations in this extended HBS model and document the solution as it undergoes a series of interesting bifurcations.
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Fogelklou, Oswald. "Computer-Assisted Proofs and Other Methods for Problems Regarding Nonlinear Differential Equations." Doctoral thesis, Uppsala universitet, Analys och tillämpad matematik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-161314.
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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.
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Beauregard, Matthew Alan. "Nonlinear Dynamics of Elastic Filaments Conveying a Fluid and Numerical Applications to the Static Kirchhoff Equations." Diss., The University of Arizona, 2008. http://hdl.handle.net/10150/194164.
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Two problems in the study of elastic filaments are considered.First, a reliable numerical algorithm is developed that candetermine the shape of a static elastic rod under a variety ofconditions. In this algorithm the governing equations are writtenentirely in terms of local coordinates and are discretized usingfinite differences. The algorithm has two significant advantages:firstly, it can be implemented for a wide variety of the boundaryconditions and, secondly, it enables the user to work with generalconstitutive relationships with only minor changes to thealgorithm. In the second problem a model is presented describingthe dynamics of an elastic tube conveying a fluid. First weanalyze instabilities that are present in a straight rod or tubeunder tension subject to increasing twist in the absence of afluid. As the twist is increased beyond a critical value, thefilament undergoes a twist-to-writhe bifurcation. A multiplescales expansion is used to derive nonlinear amplitude equationsto examine the dynamics of the elastic rod beyond the bifurcationthreshold. This problem is then reinvestigated for an elastic tubeconveying a fluid to study the effect of fluid flow on thetwist-to-writhe instability. A linear stability analysisdemonstrates that for an infinite rod the twist-to-writhethreshold is lowered by the presence of a fluid flow. Amplitudeequations are then derived from which the delay of bifurcation dueto finite tube length is determined. It is shown that the delayedbifurcation threshold depends delicately on the length of the tubeand that it can be either raised or lowered relative to thefluid-free case. The amplitude equations derived for the case of aconstant average fluid flux are compared to the case where theflux depends on the curvature. In this latter case it is shownthat inclusion of curvature results in small changes in some ofthe coefficients in the amplitude equations and has only a smalleffect on the post-bifurcation dynamics.
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Raju, Anil Roy Christopher J. "Discretization error estimation using the method of nearby problems one-dimensional cases/." Auburn, Ala., 2005. http://repo.lib.auburn.edu/2005%20Fall/Thesis/RAJU_ANIL_41.pdf.
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Yang, Zhiyun. "A Cartesian grid method for elliptic boundary value problems in irregular regions /." Thesis, Connect to this title online; UW restricted, 1996. http://hdl.handle.net/1773/6759.
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Nguyen, Thi. "On the Evolution of Virulence." CSUSB ScholarWorks, 2014. https://scholarworks.lib.csusb.edu/etd/91.
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The goal of this thesis is to study the dynamics behind the evolution of virulence. We examine first the underlying mechanics of linear systems of ordinary differential equations by investigating the classification of fixed points in these systems, then applying these techniques to nonlinear systems. We then seek to establish the validity of a system that models the population dynamics of uninfected and infected hosts---first with one parasite strain, then n strains. We define the basic reproductive ratio of a parasite, and study its relationship to the evolution of virulence. Lastly, we investigate the mathematics behind superinfection.
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Madurasinghe, M. A. D. "Splashless ship bows and waveless sterns /." Title page, summary, contents and general introduction only, 1986. http://web4.library.adelaide.edu.au/theses/09PH/09phm183.pdf.
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Abell, K. A. "Analysis and computation of the dynamics of spatially discrete phase transition equations." Thesis, University of Sussex, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.344064.
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Dadashi, Shirin. "Modeling and Approximation of Nonlinear Dynamics of Flapping Flight." Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/78224.
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The first and most imperative step when designing a biologically inspired robot is to identify the underlying mechanics of the system or animal of interest. It is most common, perhaps, that this process generates a set of coupled nonlinear ordinary or partial differential equations. For this class of systems, the models derived from morphology of the skeleton are usually very high dimensional, nonlinear, and complex. This is particularly true if joint and link flexibility are included in the model. In addition to complexities that arise from morphology of the animal, some of the external forces that influence the dynamics of animal motion are very hard to model. A very well-established example of these forces is the unsteady aerodynamic forces applied to the wings and the body of insects, birds, and bats. These forces result from the interaction of the flapping motion of the wing and the surround- ing air. These forces generate lift and drag during flapping flight regime. As a result, they play a significant role in the description of the physics that underlies such systems. In this research we focus on dynamic and kinematic models that govern the motion of ground based robots that emulate flapping flight. The restriction to ground based biologically inspired robotic systems is predicated on two observations. First, it has become increasingly popular to design and fabricate bio-inspired robots for wind tunnel studies. Second, by restricting the robotic systems to be anchored in an inertial frame, the robotic equations of motion are well understood, and we can focus attention on flapping wing aerodynamics for such nonlinear systems. We study nonlinear modeling, identification, and control problems that feature the above complexities. This document summarizes research progress and plans that focuses on two key aspects of modeling, identification, and control of nonlinear dynamics associated with flapping flight.<br>Ph. D.
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Hohenegger, Christel. "Small Scale Stochastic Dynamics For Particle Image Velocimetry Applications." Diss., Georgia Institute of Technology, 2006. http://hdl.handle.net/1853/10464.
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Fluid velocities and Brownian effects at nanoscales in the near-wall region of microchannels can be experimentally measured in an image plane parallel to the wall using, for example, evanescent wave illumination technique combined with particle image velocimetry [R. Sadr extit{et al.}, J. Fluid. Mech. 506, 357-367 (2004)]. The depth of field of this technique being difficult to modify, reconstruction of the out-of-plane dependence of the in-plane velocity profile remains extremely challenging. Tracer particles are not only carried by the flow, but they undergo random fluctuation imposed by the proximity of the wall. We study such a system under a particle based stochastic approach (Langevin) and a probabilistic approach (Fokker-Planck). The Langevin description leads to a coupled system of stochastic differential equations. Because the simulated data will be used to test a statistical hypothesis, we pay particular attention to the strong order of convergence of the scheme developing an appropriate Milstein scheme of strong order of convergence 1. Based on the probability density function of mean in-plane displacements, a statistical solution to the problem of the reconstruction of the out-of-plane dependence of the velocity profile is proposed. We developed a maximum likelihood algorithm which determines the most likely values for the velocity profile based on simulated perfect particle position, simulated perfect mean displacements and simulated observed mean displacements. Effects of Brownian motion on the approximation of the mean displacements are briefly discussed. A matched particle is a particle that starts and ends in the same image window after a measurement time. AS soon as the computation and observation domain are not the same, the distribution of the out-of-plane distances sampled by matched particles during the measurement time is not uniform. The combination of a forward and a backward solution of the one dimensional Fokker-Planck equation is used to determine this probability density function. The non-uniformity of the resulting distribution is believed to induce a bias in the determination of slip length and is quantified for relevant experimental parameters.
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Ferrara, Joseph. "A Study of Nonlinear Dynamics in Mathematical Biology." UNF Digital Commons, 2013. http://digitalcommons.unf.edu/etd/448.
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We first discuss some fundamental results such as equilibria, linearization, and stability of nonlinear dynamical systems arising in mathematical modeling. Next we study the dynamics in planar systems such as limit cycles, the Poincaré-Bendixson theorem, and some of its useful consequences. We then study the interaction between two and three different cell populations, and perform stability and bifurcation analysis on the systems. We also analyze the impact of immunotherapy on the tumor cell population numerically.
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Holmquist, Sonia. "AN EXAMINATION OF THE EFFECTIVENESS OF THE ADOMIAN DECOMPOSITION METHOD IN FLUID DYNAMIC APPLICATIONS." Doctoral diss., University of Central Florida, 2007. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/2524.
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Since its introduction in the 1980's, the Adomian Decomposition Method (ADM) has proven to be an efficient and reliable method for solving many types of problems. Originally developed to solve nonlinear functional equations, the ADM has since been used for a wide range of equation types (like boundary value problems, integral equations, equations arising in flow of incompressible and compressible fluids etc...). This work is devoted to an evaluation of the effectiveness of this method when used for fluid dynamic applications. In particular, the ADM has been applied to the Blasius equation, the Falkner-Skan equation, and the Orr-Sommerfeld equation. This study is divided into five Chapters and an Appendix. The first chapter is devoted to an introduction of the Adomian Decomposition method (ADM) with simple illustrations. The Second Chapter is devoted to the application of the ADM to generalized Blasius Equation and our result is compared to other published results when the parameter values are appropriately set. Chapter 3 presents the solution generated for the Falkner-Skan equation. Finally, the Orr-Sommerfeld equation is dealt with in the fourth Chapter. Chapter 5 is devoted to the findings and recommendations based on this study. The Appendix contains details of the solutions considered as well as an alternate solution for the generalized Blasius Equation using Bender's delta-perturbation method.<br>Ph.D.<br>Department of Mathematics<br>Sciences<br>Mathematics PhD
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Bates, Dana Michelle. "On a free boundary problem for ideal, viscous and heat conducting gas flow." Diss., University of Iowa, 2016. https://ir.uiowa.edu/etd/2180.
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We consider the flow of an ideal gas with internal friction and heat conduction in a layer between a fixed plane and an upper free boundary. We describe the top free surface as the graph of a time dependent function. This forces us to exclude breaking waves on the surface. For this and other reasons we need to confine ourselves to flow close to a motionless equilibrium state which is fairly easy to compute. The full equations of motion, in contrast to that, are quite difficult to solve. As we are close to an equilibrium, a linear system of equations can be used to approximate the behavior of the nonlinear system.
Analytic, strongly continuous semigroups defined on a suitable Banach space X are used to determine the behavior of the linear problem. A strongly continuous semigroup is a family of bounded linear operators {T(t)} on X where 0 ≤ t < infinity satisfying the following conditions.
1. T(s+t)=T(s)T(t) for all s,t ≥ 0
2. T(0)=E, the identity mapping.
3. For each x ∈ X, T(t)x is continuous in t on [0,infinity).
Then there exists an operator A known as the infinitesimal generator of such that T(t)=exp (tA). Thus, an analytic semigroup can be viewed as a generalization of the exponential function.
Some estimates about the decay rates are derived using this theory. We then prove the existence of long term solutions for small initial values. It ought to be emphasized that the decay is not an exponential one which engenders significant difficulties in the transition to nonlinear stability.
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Ingraham, Daniel. "External Verification Analysis: A Code-Independent Approach to Verifying Unsteady Partial Differential Equation Solvers." University of Toledo / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1430491745.
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Shcherbakov, Victor. "Localised Radial Basis Function Methods for Partial Differential Equations." Doctoral thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-332715.
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Radial basis function methods exhibit several very attractive properties such as a high order convergence of the approximated solution and flexibility to the domain geometry. However the method in its classical formulation becomes impractical for problems with relatively large numbers of degrees of freedom due to the ill-conditioning and dense structure of coefficient matrix. To overcome the latter issue we employ a localisation technique, namely a partition of unity method, while the former issue was previously addressed by several authors and was of less concern in this thesis. In this thesis we develop radial basis function partition of unity methods for partial differential equations arising in financial mathematics and glaciology. In the applications of financial mathematics we focus on pricing multi-asset equity and credit derivatives whose models involve several stochastic factors. We demonstrate that localised radial basis function methods are very effective and well-suited for financial applications thanks to the high order approximation properties that allow for the reduction of storage and computational requirements, which is crucial in multi-dimensional problems to cope with the curse of dimensionality. In the glaciology application we in the first place make use of the meshfree nature of the methods and their flexibility with respect to the irregular geometries of ice sheets and glaciers. Also, we exploit the fact that radial basis function methods are stated in strong form, which is advantageous for approximating velocity fields of non-Newtonian viscous liquids such as ice, since it allows to avoid a full coefficient matrix reassembly within the nonlinear iteration. In addition to the applied problems we develop a least squares radial basis function partition of unity method that is robust with respect to the node layout. The method allows for scaling to problem sizes of a few hundred thousand nodes without encountering the issue of large condition numbers of the coefficient matrix. This property is enabled by the possibility to control the coefficient matrix condition number by the rate of oversampling and the mode of refinement.
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Perin, Maxime. "Hamiltonian fluid reductions of kinetic equations in plasma physics." Thesis, Aix-Marseille, 2016. http://www.theses.fr/2016AIXM4050/document.
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La réduction fluide des équations cinétiques est un procédé couramment utilisé en physique des plasmas qui a pour objectif de remplacer la fonction de distribution définie dans l'espace des phases par des grandeurs fluides comme la densité et la pression. Cette réduction diminue la complexité du système initial. En contrepartie, la réduction fluide s'accompagne de la nécessité d'effectuer une fermeture sur les moments d'ordre supérieur. Celle-ci est souvent construite ad hoc en se basant sur des arguments physiques (e.g., quantités conservées, existance d'un théorème H, ...). Dans ce manuscrit, on propose un procédé de réduction qui permet de préserver la structure hamiltonienne du modèle cinétique parent. Ceci est important pour assurer qu'aucune dissipation d'origine non physique est introduite dans le modèle fluide, le munissant ainsi d'une structure hamiltonienne dont l'origine peut être suivie jusqu'à celle de la dynamique microscopique des particules. On utilise cette méthode pour construire des modèles fluides non-adiabatiques pour les trois premiers moments de la fonction de distribution associée à l'équation de Vlasov-Poisson à une dimension, i.e., la densité, la vitesse fluide et la pression. Les résultats sont ensuite étendus pour inclure la dynamique du flux de chaleur en considérant des fermetures construites à partir de l'analyse dimensionnelle. On montre également, pour un nombre arbitraire de champs, la relation existant avec le modèle water-bags. L'extension à des dimensions supérieures est étudiée dans le cadre de l'équation drift-cinétique ainsi que de l'équation de Vlasov-Poisson à trois dimensions<br>Fluid reduction of kinetic equations is a ubiquitous procedure in plasma physics which aims to replace the distribution function defined in phase space with more concrete fluid quantities defined solely in configuration space such as the density, the fluid velocity and the pressure. This reduction lowers the complexity of the initial system, leading to a gain of physical insight into the phenomena under investigation as well as a significant decrease of the cost of numerical simulations. On the other hand, in order for the fluid reduction to be complete, one needs to perform a closure on the higher order fluid moments. The choice of the closure usually relies on some ad hoc physical arguments (e.g., conserved quantities, existence of an H-theorem, ...). In this manuscript, we present a reduction procedure that preserves the Hamiltonian structure of the parent kinetic model. This is important in order to ensure that no non-physical dissipation is introduced in the resulting fluid model, providing it with a geometric structure that can be traced back to the microscopic dynamics of the particles. We use this procedure to derive non-adiabatic fluid models for the first three fluid moments of the distribution function of the one dimensional Vlasov-Poisson equation, namely the density, the fluid velocity and the pressure. The results are extended to include the dynamics of the heat-flux by considering a closure based on dimensional analysis. For an arbitrary number of fields, we demonstrate the relationship with the water-bags model. Finally, the extension to higher dimensions is investigated through the drift-kinetic equation and the three dimensional Vlasov-Poisson equation
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Ellis, Truman Everett. "High Order Finite Elements for Lagrangian Computational Fluid Dynamics." DigitalCommons@CalPoly, 2010. https://digitalcommons.calpoly.edu/theses/282.
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A general finite element method is presented to solve the Euler equations in a Lagrangian reference frame. This FEM framework allows for separate arbitrarily high order representation of kinematic and thermodynamic variables. An accompanying hydrodynamics code written in Matlab is presented as a test-bed to experiment with various basis function choices. A wide range of basis function pairs are postulated and a few choices are developed further, including the bi-quadratic Q2-Q1d and Q2-Q2d elements. These are compared with a corresponding pair of low order bi-linear elements, traditional Q1-Q0 and sub-zonal pressure Q1-Q1d. Several test problems are considered including static convergence tests, the acoustic wave hourglass test, the Sod shocktube, the Noh implosion problem, the Saltzman piston, and the Sedov explosion problem. High order methods are found to offer faster convergence properties, the ability to represent curved zones, sharper shock capturing, and reduced shock-mesh interaction. They also allow for the straightforward calculation of thermodynamic gradients (for multi-physics calculations) and second derivatives of velocity (for monotonic slope limiters), and are more computationally efficient. The issue of shock ringing remains unresolved, but the method of hyperviscosity has been identified as a promising means of addressing this. Overall, the curvilinear finite elements presented in this thesis show promise for integration in a full hydrodynamics code and warrant further consideration.
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Bilyeu, David L. "A HIGHER-ORDER CONSERVATION ELEMENT SOLUTION ELEMENT METHOD FOR SOLVING HYPERBOLIC DIFFERENTIAL EQUATIONS ON UNSTRUCTURED MESHES." The Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1396877409.
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Zhao, Kun. "Initial-boundary value problems in fluid dynamics modeling." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/31778.
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Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2010.<br>Committee Chair: Pan, Ronghua; Committee Member: Chow, Shui-Nee; Committee Member: Dieci, Luca; Committee Member: Gangbo, Wilfrid; Committee Member: Yeung, Pui-Kuen. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Leto, Jonathan. "SOLITARY WAVE FAMILIES IN TWO NON-INTEGRABLE MODELS USING REVERSIBLE SYSTEMS THEORY." Master's thesis, University of Central Florida, 2008. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/3504.
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In this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized Pochhammer-Chree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned above, solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions of both models here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves for each model, we find a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. For the microstructure equation, the new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work, including the dynamics of each family of solitary waves using exponential asymptotics techniques, are also mentioned.<br>M.S.<br>Department of Mathematics<br>Sciences<br>Mathematical Science MS
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Lindgren, Joseph B. "Orbital Stability Results for Soliton Solutions to Nonlinear Schrödinger Equations with External Potentials." UKnowledge, 2017. http://uknowledge.uky.edu/math_etds/46.
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For certain nonlinear Schroedinger equations there exist solutions which are called solitary waves. Addition of a potential $V$ changes the dynamics, but for small enough $||V||_{L^\infty}$ we can still obtain stability (and approximately Newtonian motion of the solitary wave's center of mass) for soliton-like solutions up to a finite time that depends on the size and scale of the potential $V$. Our method is an adaptation of the well-known Lyapunov method.
For the sake of completeness, we also prove long-time stability of traveling solitons in the case $V=0$.
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Pirilla, Patrick Brian. "On the Trajectories of Particles in Solitary Waves." Youngstown State University / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1311100628.
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Shu, Yupeng. "Numerical Solutions of Generalized Burgers' Equations for Some Incompressible Non-Newtonian Fluids." ScholarWorks@UNO, 2015. http://scholarworks.uno.edu/td/2051.
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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.
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Benedito, Antone dos Santos. "Multi-stage population models applied to insect dynamics." Botucatu, 2020. http://hdl.handle.net/11449/192335.
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Orientador: Cláudia Pio Ferreira<br>Abstract: This thesis presents two manuscripts previously sent to publication in scientific journals. In the first manuscript, a delay differential equation model is developed to study the dynamics of two Aedes aegypti mosquito populations: infected by the intracellular bacteria Wolbachia and non-infected (wild) individuals. All the steady states of the system are determined, namely extinction of both populations, extinction of the infected population and persistence of the non-infected one, and coexistence. Their local stability is analyzed, including Hopf bifurcation, which promotes periodic solutions around the nontrivial equilibrium points. Finally, one investigates the global asymptotic stability of the trivial solution. In the second manuscript, after rearing soybean looper Chrysodeixis includens in laboratory conditions, thermal requirements for this insect-pest are estimated, from linear and nonlinear regression models, as well as the intrinsic growth rate. This parameter depends on the life-history traits and can provide a measure of population viability of the species.<br>Doutor
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Iwamura, Chihiro, and chihiro_iwamura@ybb ne jp. "A fast solver for large systems of linear equations for finite element analysis on unstructured meshes." Swinburne University of Technology, 2004. http://adt.lib.swin.edu.au./public/adt-VSWT20051020.091538.
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The objective of this thesis is to develop a more efficient solver for a large system of linear equations arising from finite element discretization on unstructured tetrahedral meshes for a scalar elliptic partial differential equation of second order for pressure in a commercial computational fluid dynamics (CFD) simulation.
Segregated solution methods (or pressure correction type methods) are a widely used approach to obtain solutions of Navier-Stokes equations during numerical simulation by many commercial CFD codes. At each time step, these simulations usually require the approximate solution of a series of scalar equations for velocity, pressure and temperature. Even if the simulation does not require high-accuracy approximations, the large systems of linear equations for pressure may not be efficiently solved. The matrices of these systems of linear equations of real-life industry problems often strongly violate weak diagonal dominance and the numerical simulation often requires solutions of very large systems with over a few hundred thousands degrees of freedom. These conditions produce very ill-conditioned systems of linear equations. Therefore, it is very difficult to solve such systems of linear equations efficiently using most currently available common iterative solvers.
A survey of solvers for systems of linear equations was undertaken to determine the preferred solution methodology. An algebraic multigrid preconditioned conjugate gradient (AMGPCG) method solver was chosen for these problems. This solver uses the algebraic multigrid (AMG) cycle as a preconditioner for the conjugate gradient (CG) method. The disadvantages of the conventional AMG method are an expensive setup time and large memory requirements, particularly for three dimensional problems. The disadvantage of an expensive setup time needs to be overcome because the simulation usually requires only low-accuracy approximations for pressure. Also it is important to overcome the disadvantage of the large memory requirements for use in commercial software.
In this work, an efficient AMGPCG solver is developed by overcoming the disadvantages of the conventional AMG method. The robustness of AMGPCG is shown theoretically so that the solver is always convergent. Optimum or close to optimum rates of convergence behavior for the solver are shown numerically so that the number of necessary iterations to obtain the estimated solution is approximately independent of mesh resolution. Furthermore, numerical experiments of solving pressure for some industry problems were carried out and compared with other efficient solvers including a fast commercial AMGPCG solver (SAMG, release 20b1). It was found that the developed AMGPCG solver was the fastest among these solvers for solving these problems and its algorithm has been numerically proven to be efficient. In addition, the memory requirement is at an acceptable level for commercial CFD codes.
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Murali, Vasanth Kumar. "Code verification using the method of manufactured solutions." Master's thesis, Mississippi State : Mississippi State University, 2002. http://library.msstate.edu/etd/show.asp?etd=etd-11112002-121649.
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Chaudhuri, Nilasis [Verfasser], Etienne [Akademischer Betreuer] Emmrich, Eduard [Akademischer Betreuer] Feireisl, Eduard [Gutachter] Feireisl, Etienne [Gutachter] Emmrich, and Antonin [Gutachter] Novotny. "Qualitative properties of solutions to partial differential equations arising in fluid dynamics / Nilasis Chaudhuri ; Gutachter: Eduard Feireisl, Etienne Emmrich, Antonin Novotny ; Etienne Emmrich, Eduard Feireisl." Berlin : Technische Universität Berlin, 2021. http://d-nb.info/1239177267/34.
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Mason, Kevin Richard. "Development of numerical schemes to improve the efficiency of CFD simulation of high speed viscous aerodynamic flows." Thesis, Swansea University, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.678434.
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Villamizar, Roa Elder Jesus. "Soluções ultra fracas, fracas, brandas e fortes para equações do tipo Navier-Stokes." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306129.
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Orientadores: Marko Antonio Rojas Medar, Maria Angeles Rodriguez Bellido<br>Tese (doutorado) - Universidade Estadual de Campinas. Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-04T14:33:11Z (GMT). No. of bitstreams: 1
VillamizarRoa_ElderJesus_D.pdf: 1860214 bytes, checksum: 00b73d5c74c2d58828f10d232dbe32fb (MD5)
Previous issue date: 2005<br>Resumo: Abordamos vários problemas relativos à existência, unicidade, regularidade e estabilidade de alguns sistemas de equações do tipo Navier-Stokes; Estudamos a existência de soluções ultra fracas para as equações de Boussinesq estacionárias, com condições de fronteira ouco regulares, do tipo Dirichlet para a velocidade e condições mistas do tipo Dirichlet e Neumann para a temperatura. Seguidamente provamos a existência de soluções tempo-periódicas brandas e fortes em domínios não limitados para as equações de Boussinesq de evolução em espaços de Lorentz. Via a teoria de semigrupos provamos resultados de existência global, comportamento assintótico e estabilidade das soluções para as equações de fluidos micropolares. Posteriormente consideramos uma classe de equações não lineares estacionárias abstratas em um espaço de Hilbert separável e mostramos algumas propriedades qualitativas relativas a esse modelo, entre elas, uma propriedade sobre a cardinalidade do conjunto das soluções do modelo abstrato em questão e uma propriedade de dependência contínua das soluções com respeito aos dados do problema. Finalmente provamos a existência de soluções fortes para as equações de Navier-Stokes com densidade variável em domínios espaciais não limitados<br>Abstract: The main objective of this work is to study the number of solutions of polynomial equations over finite fields. For that we used basic results on Character sums and on the number of solutions of a Quadratic Form. This approach uses elementary techniques even considering the increasing on computations. Therefore this method allowed us to study and determine formulae for the number of solutions of certain polynomial equations well known, without the need of more sophisticated tools. Among the applications of the obtained formulae, we have some examples of plane algebraic curves which number of rational points achieve the Weil bound, that is, maximal curves which are of great interest in code theory. In addition, other examples were obtained of projective manifolds over finite fields which number of points achieve the Weil-Deligne bound<br>Doutorado<br>Equações Diferenciais Parciais<br>Doutor em Matemática
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Myers, Jeremy. "Computational Fluid Dynamics in a Terminal Alveolated Bronchiole Duct with Expanding Walls: Proof-of-Concept in OpenFOAM." VCU Scholars Compass, 2017. http://scholarscompass.vcu.edu/etd/5011.
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Mathematical Biology has found recent success applying Computational Fluid Dynamics (CFD) to model airflow in the human lung. Detailed modeling of flow patterns in the alveoli, where the oxygen-carbon dioxide gas exchange occurs, has provided data that is useful in treating illnesses and designing drug-delivery systems. Unfortunately, many CFD software packages have high licensing fees that are out of reach for independent researchers. This thesis uses three open-source software packages, Gmsh, OpenFOAM, and ParaView, to design a mesh, create a simulation, and visualize the results of an idealized terminal alveolar sac model. This model successfully demonstrates that OpenFOAM can be used to model airflow in the acinar region of the lung under biologically relevant conditions.
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Schrecker, Matthew. "Hyperbolic problems in fluids and relativity." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:2dec2eb9-4253-4625-a071-0c19d0c1f76d.
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In this thesis, we present a collection of newly obtained results concerning the existence of vanishing viscosity solutions to the one-dimensional compressible Euler equations of gas dynamics, with and without geometric structure. We demonstrate the existence of such vanishing viscosity solutions, which we show to be entropy solutions, to the transonic nozzle problem and spherically symmetric Euler equations in Chapter 4, in both cases under the simple and natural assumption of relative finite-energy. In Chapter 5, we show that the viscous solutions of the one-dimensional compressible Navier-Stokes equations converge, as the viscosity tends to zero, to an entropy solution of the Euler equations, again under the assumption of relative finite-energy. In so doing, we develop a compactness framework for the solutions and approximate solutions to the Euler equations under the assumption of a physical pressure law. Finally, in Chapter 6, we consider the Euler equations in special relativity, and show the existence of bounded entropy solutions to these equations. In the process, we also construct fundamental solutions to the entropy equations and develop a compactness framework for the solutions and approximate solutions to the relativistic Euler equations.
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Lao, Kun Leng. "Multigrid algorithm based on cyclic reduction for convection diffusion equations." Thesis, University of Macau, 2010. http://umaclib3.umac.mo/record=b2148274.
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Gadelha, Hermes. "Mathematical modelling of human sperm motility." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:34a11669-5d14-470b-b10b-361cf3688a30.
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The propulsion mechanics driving the movement of living cells constitutes one of the most incredible engineering works of nature. Active cell motility via the controlled movement of a flagellum beating is among the phylogentically oldest forms of motility, and has been retained in higher level organisms for spermatozoa transport. Despite this ubiquity and importance, the details of how each structural component within the flagellum is orchestrated to generate bending waves, or even the elastic material response from the sperm flagellum, is far from fully understood. By using microbiomechanical modelling and simulation, we develop bio-inspired mathematical models to allow the exploration of sperm motility and the material response of the sperm flagellum. We successfully construct a simple biomathematical model for the human sperm movement by taking into account the sperm cell and its interaction with surrounding fluid, through resistive-force theory, in addition to the geometrically non-linear response of the flagellum elastic structure. When the surrounding fluid is viscous enough, the model predicts that the sperm flagellum may buckle, leading to profound changes in both the waveforms and the swimming cell trajectories. Furthermore, we show that the tapering of the ultrastructural components found in mammalian spermatozoa is essential for sperm migration in high viscosity medium. By reinforcing the flagellum in regions where high tension is expected this flagellar accessory complex is able to prevent tension-driven elastic instabilities that compromise the spermatozoa progressive motility. We equally construct a mathematical model to describe the structural effect of passive link proteins found in flagellar axonemes, providing, for the first time, an explicit mathematical demonstration of the counterbend phenomenon as a generic property of the axoneme, or any cross-linked filament bundle. Furthermore, we analyse the differences between the elastic cross-link shear and pure material shear resistance. We show that pure material shearing effects from Cosserat rod theory or, equivalently, Timoshenko beam theory or are fundamentally different from elastic cross-link induced shear found in filament bundles, such as the axoneme. Finally, we demonstrate that mechanics and modelling can be utilised to evaluate bulk material properties, such as bending stiffness, shear modulus and interfilament sliding resistance from flagellar axonemes its constituent elements, such as microtubules.
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Pittayakanchit, Weerapat. "The Global Stability of the Solution to the Morse Potential in a Catastrophic Regime." Scholarship @ Claremont, 2016. http://scholarship.claremont.edu/hmc_theses/72.
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Swarms of animals exhibit aggregations whose behavior is a challenge for mathematicians to understand. We analyze this behavior numerically and analytically by using the pairwise interaction model known as the Morse potential. Our goal is to prove the global stability of the candidate local minimizer in 1D found in A Primer of Swarm Equilibria. Using the calculus of variations and eigenvalues analysis, we conclude that the candidate local minimizer is a global minimum with respect to all solution smaller than its support. In addition, we manage to extend the global stability condition to any solutions whose support has a single component. We are still examining the local minimizers with multiple components to determine whether the candidate solution is the minimum-energy configuration.
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Stevens, Ben. "Short-time structural stability of compressible vortex sheets with surface tension." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:378713da-cd05-4b9a-856d-bee2b0fb47ce.
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The main purpose of this work is to prove short-time structural stability of compressible vortex sheets with surface tension. The main result can be summarised as follows. Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We assume the fluids are modelled by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density in each fluid such that the sound speed is positive. Then, for a short time, which may depend on the initial configuration, there exists a unique solution of the equations with the same structure, that is, two fluids with density bounded below flowing smoothly past each other, where the surface tension force across the common interface balances the pressure jump. The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al in the setting of a compressible liquid-vacuum interface. Although already considered by Shkoller et al, we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.
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Marano, Susan Aileen. "Smarticles: A Method for Identifying and Correcting Instability and Error Caused by Explicit Integration Techniques in Physically Based Simulations." DigitalCommons@CalPoly, 2014. https://digitalcommons.calpoly.edu/theses/1304.
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Using an explicit integration method in physically based animations has many advantages including conceptual and computational simplicity, however, it re- quires small time steps to ensure low numerical instability. Simulations with large numbers of individually interacting components such as cloth, hair, and fluid models, are limited by the sections of particles most susceptible to error. This results in the need for smaller time steps than required for the majority of the system. These sections can be diverse and dynamic, quickly changing in size and location based on forces in the system. Identifying and handling these trou- blesome sections could allow for a larger time step to be selected, while preventing a breakdown in the simulation.
This thesis presents Smarticles (smart particles), a method of individually de- tecting particles exhibiting signs of instability and stabilizing them with minimal adverse effects to visual accuracy. As a result, higher levels of error introduced from large time steps can be tolerated with minimal overhead. Two separate approaches to Smarticles were implemented. They attempt to find oscillating particles by analyzing a particle’s (1) past behavior and (2) behavior with re- spect to its neighbors along a strand. Both versions of Smarticles attempt to correct unstable particles using velocity dampening. Smarticles was applied to a two dimensional hair simulation modeled as a continuum using smooth particle hydrodynamic. Hair strands are formed by linking particles together using one of two methods: position based dynamics or mass-spring forces.
Both versions of Smarticles, as well as a control of normal particles, were directly compared and evaluated based on stability and visual fluidity. Hair particles were exposed to various forms of external forces under increasing time step lengths. Testing showed that both versions of Smarticles working together allowed an average increase of 18.62% in the time step length for hair linked with position based dynamics. In addition, Smarticles was able to significantly reduce visible instability at even larger time steps. While these results suggest Smarticles is successful, the method used to correct particle instability may jeopardize other important aspects of the simulation. A more accurate correction method would likely need to be developed to make Smarticles an advantageous method.
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Burtea, Cosmin. "Méthodes d'analyse de Fourier en hydrodynamique : des mascarets aux fluides avec capillarité." Thesis, Paris Est, 2017. http://www.theses.fr/2017PESC1047/document.
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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<br>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
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van, Damme Myron. "Modelling embankment breaching due to overflow." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:57d0c96f-1a8a-4e00-82a4-585f3d337ad6.
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Correct modelling of embankment breach formation is essential for an accurate assessment of the associated flood risk. Modelling breach formation due to overflow requires a thorough understanding of the geotechnical processes in unsaturated soils as well as erosion processes under supercritical flow conditions. This thesis describes 1D slope stability analysis performed for unsaturated soils whose moisture content changes with time. The analysis performed shows that sediment-laden gravity flows play an important role in the erosion behaviour of embankments. The thesis also describes a practical, fast breach model based on a simplified description of the physical processes that can be used in modelling and decision support frameworks for flooding. To predict the breach hydrograph, the rapid model distinguishes between breach formation due to headcut erosion and surface erosion in the case of failure due to overflow. The model also predicts the breach hydrograph in the case of failure due to piping. The assumptions with respect to breach flow modelling are reviewed, and result in a new set of breadth-integrated Navier-Stokes equations, that account for wall shear stresses and a variable breadth geometry. The vertical 2D flow field described by the equations can be used to calculate accurately the stresses on the embankment during the early stages of breach formation. Pressure-correction methods are given for solving the 2D Navier-Stokes equations for a variable breadth, and good agreement is found when validating the flow model against analytical solutions.