This report describes the development of a software for computing viscous incompressiblemultiphase ows. The software does this with solving the coupled non-linear Navier-Stokes(Fluid) and the Cahn-Hilliard (Phase-Field) equations using a Finite Element Method. Thereason for the development is to produce a simulation tool, which eventually is capable ofsimulating the ow of uids inside the OptusAir aeronator manufactured by the Sorubincompany. The solving software developed is built on the ParMetis, PETSc and OpenMPIframeworks. Our primary benchmark has been a geometry resembling the OptusAir product,a cylinder with a rotating bottom. We have made comparisons between simulation resultsand the theory of a free surface in a uniform rotating ow.This thesis shows that the shape of the interface between two uids coincides with theoryto some extent, while the approximate boundary conditions prevent it from coinciding fully.

For the latter half of the last century renormalization methods have played an important part in tackling problems in fundamental physics and in providing a deeper understanding of systems with many interacting scales or degrees of freedom with strong coupling. The study of turbulence is no exception, and this thesis presents an investigation of renormalization techniques available in the study of the statistical theory of homogeneous and isotropic turbulence. The thesis consists of two parts which assess the two main renormalization approaches available in modeling turbulence. In particular we will be focusing on the renormalization procedures developed by McComb and others. The first part of this thesis will discuss Renormalization Group (RG) approaches to turbulence, with a focus on applications to reduce the degrees of freedom in a large-eddy simulation. The RG methods as applied to classical dynamical systems will be reviewed in the context of the Navier-Stokes equations describing fluid flow. This will be followed by introducing a functional based formalism of a conditional average first introduced by McComb, Roberts and Watt [Phys. Rev A 45, 3507 (1992)] as a tool for averaging out degrees of freedom needed in an RG calculation. This conditional average is then used in a formal RG calculation applied to the Navier-Stokes equations, originally done by McComb and Watt [Phys. Rev. A 46, 4797 (1992)], and later revised by Mc- Comb and Johnston [Physica A 292, 346 (2001)]. A correction to the summing of the time-integral detailed in the latter work is shown to introduce an extra viscous life-time term to the denominator of the increment to the renormalized viscosity and is shown to have a negligible effect in the numerical calculations. We follow this study by outlining some problems with the previous approach. In particular it is shown that a cross-term representing the interaction between high and low wavenumber modes which was neglected in the previous studies on the grounds that it does not contribute to energy dissipation, does in fact contribute significantly. A heuristic method is then put forward to include the effects of this term in the RG calculation. This leads to results which agree qualitatively with numerical calculations of eddy-viscosities. We finish this part of the thesis with an application of the RG method to the modeling of a passive scalar advected by a turbulent velocity field. The second part of this thesis will begin by reviewing Eulerian renormalized perturbation theory attempts in closing the infinite moment hierarchy introduced by averaging the Navier-Stokes equations. This is followed by presenting a new formulation of the local energy transfer theory (LET) of McComb et. al. [J. Fluid Mech. 245, 279 (1992)] which resolves some problems of previous derivations. In particular we show by the introduction of time-ordering that some previous problems with the exponential representation of the correlator can be overcome. Furthermore, we show that the singularity in the LET propagator equation cancels by way of a counter-term. We end this study by introducing a single-time Markovian closure based on LET which, unlike other Markovian closures, does not rely on any arbitrary parameters being introduced in the theory.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. February 16, 2015. / The Adomian decomposition method is an effective procedure for the analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions, closure approximations, perturbation theory, or restrictive assumptions on stochasticity. Our aim here is to apply the Adomian decomposition method to steady two-dimensional blood flow through a constricted artery in the presence of a uniform transverse magnetic field. Blood flow is the study of measuring blood pressure and determining flow through arteries. Blood flow is assumed to be Newtonian and is governed by the equation of continuity and the momentum balanced equation (which are known as the Navier-Stokes equations). This model is consistent with the principles of ferro-hydrodynamics and magnetohydrodynamics and takes into account both magnetization and electrical conductivity of blood. We apply the Adomian decomposition method to the equations governing blood flow through arteries in the presence of an external transverse magnetic field. The results show that the e ect of a uniform external transverse magnetic field applied to blood flow through arteries favors the physiological condition of blood. The motion of blood in stenosed arteries can be regulated by applying a magnetic field externally and increasing/decreasing the intensity of the applied field.

A finite difference based solution method is derived for the velocity-pressure formulation of the two-dimensional incompressible Navier-Stokes equations. The method is proven stable using the energy method, facilitated by SBP operators, for characteristic and Dirichlet boundary condition implemented using the SAT technique. The numerical experiments show the utility of high-order finite difference methods as well as emphasize the problem of pressure boundary conditions. Furthermore, we demonstrate that a discretely divergence free solution can be obtained by use of the projection method.

In this thesis, we study a two fluid system which describes the motion of two charged particles in a strict neutral incompressible plasma. We study the well-posdness of the system in both space dimensions two and three. Regardless of the size of the initial data, we prove the global well-posedness of the Cauchy problem when the space dimension is two. In space dimension three, we construct global weak-solutions, and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently small. We also study the stability of the solution around zero given that the initial data is small and has sufficient regularity. It turns out that our system is a system of regularity-loss and the L2 norm of lower derivatives of the solution decays. At last, this two fluid system can be used to derive the classic MHD at least formally. Arsenio, Ibrahim and Masmoudi (2015) proved that the two fluid system converges to MHD under some constraints. We showed numerically that the two fluid system converges to MHD with no such constraint and found the approximate converge rate. / Graduate

Este trabalho consiste na simulação de escoamentos aerodinâmicos sobre configurações tipo foguete, para os casos de um corpo isolado e de uma geometria multicorpo, utilizando as equações de Navier-Stokes com média de Reynolds, com aproximação de camada fina. Estas equações são implementadas em diferenças finitas, utilizando o algoritmo implícito de fatorização aproximada de Beam e Warming. O fenômeno de turbulência é modelado com o modelo algébrico de viscosidade de vórtice, de duas camadas, de Baldwin e Lomax. Os resultados obtidos são comparados com dados experimentais e a concordância é muito boa. Contornos de pressão e densidade são também apresentados para se comprovar a validade física da simulação. Finalmente, apresentam-se conclusões e idéias para trabalhos futuros.

Chu Shun Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 63-65). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Introduction --- p.3 / Chapter 1 --- Preliminaries --- p.8 / Chapter 2 --- Singular Integrals --- p.15 / Chapter 2.1 --- Marcinkiewicz Integral --- p.15 / Chapter 2.2 --- Decomposition in cubes of open sets in Rn --- p.17 / Chapter 2.3 --- Interpolation Theorem for Lp --- p.18 / Chapter 2.4 --- Singular Integrals on homogeneous of degree 0 --- p.25 / Chapter 3 --- Solutions to the Euler Equations --- p.36 / Chapter 3.1 --- Existence and Uniqueness of smooth solutions for Euler Equations --- p.36 / Chapter 3.2 --- Rate of Convergence and Decay in Time --- p.43 / Chapter 3.2.1 --- Rate of Convergence --- p.43 / Chapter 3.2.2 --- Lp Decay for Solutions of the Navier-Stokes Equations --- p.46 / Chapter 3.3 --- Weak Solution to the Euler Equations --- p.48 / Chapter 3.3.1 --- Weak Solution to the Velocity Formulation --- p.49 / Chapter 3.3.2 --- Weak Solution to the Vorticity Formulation --- p.52 / Bibliography --- p.63

Chan Man-fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 40-41). / Abstracts in English and Chinese. / Acknowledgement --- p.i / Abstract --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Equation of motion and main results --- p.3 / Chapter 3 --- The space K(x) --- p.9 / Chapter 4 --- Proof of the main theorem --- p.17 / Chapter 4.1 --- The passage to the limit as ε →0 --- p.18 / Chapter 4.2 --- The passage to the limit as δ→ 0 --- p.26 / Chapter 4.3 --- Properties of the solution --- p.29 / Chapter 5 --- Conclusion and comments on future works --- p.36 / Appendix --- p.38 / Bibliography --- p.40

In chapter 1, we focus on the noncharacteristic boundary layers for the parabolic regularization of quasi-linear hyperbolic problems, where the viscosity matrix is positive definite, with the zero Dirichlet boundary conditions. We adapt the method developed by Grenier and Gues [?] where the center-stable manifold theorem is used to prove the existence and exponential decay property of the leading boundary layer profile under suitable conditions on the boundary x = 0. With this boundary condition we prove the well-posedness of the initial boundary value problem of the inviscid flow. Then we prove the stability of the boundary layer by an energy estimate, where exponential decay property of the boundary layer profile plays an important role. Finally, we can specify the limit of the viscous solutions to the corresponding inviscid solution. / In chapter 2, we consider the noncharacteristic one-dimensional compressible full Navier-Stokes equations for the ideal gas with outflow boundary condition on the velocity and suitable initial conditions, which make all the three characteristics to the corresponding Euler equations negative up to some local time, especially on the boundary. By the aymptotic analysis, we derive an algebraic-differential equation for the leading boundary layer functions. The center-stable manifold theorem helps to prove the existence and exponential decay property of the leading boundary layer function. The outflow boundary condition makes it possible to estimate the normal derivatives. Combining this with the tangential derivative estimate, we can recover the H1 estimate of the error term. Thus we establish the stability of the boundary layers which satisfy an algebraic-differential equation in this case. With this stability result, we obtain the relation between the solutions to Navier-Stokes and Euler equations. / In chapter 3, we concentrate on the existence and nonlinear stability of the totally characteristic boundary layer for the quasi-linear equations with positive definite viscosity matrix under the assumption that the boundary matrix vanishes identically on the boundary x = 0. We carry out a weighted estimate to the boundary layer equations---Prandtl type equations to get the regularity and the far field behavior of the solutions. This allows us to perform a weighted energy estimate for the error equation to prove the stability of the boundary layers. The stability result finally implies the asymptotic limit of the viscous solutions. / In this thesis we study three kinds of asymptotic limiting behavior of the solutions to the initial boundary value problem of one-dimensional quasilinear equations with viscosity by carrying out the boundary layer analysis. / Wang, Jing. / Adviser: Zhouping Xin. / Source: Dissertation Abstracts International, Volume: 71-01, Section: B, page: 0407. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 107-112). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese.

Finally, we prove that weak solutions to the compressible Navier-Stokes equations with the Navier boundary condition stabilize to static equilibrium states under a fair condition. / First, we show that the most general class of weak solutions to one-dimensional full compressible Navier-Stokes equations do not exhibit vacuum states in a finite time provided that no vacuum is present initially with the minimum physical assumptions on the data. Moreover, two initially non interacting vacuum regions will never meet each other in the future. / Secondly, we construct the local classical solutions to the compressible Navier-Stokes equations for initial vacuum far fields. In this case, we describe the blow-up phenomena of two-dimensional compact support smooth spherically symmetric solutions. When the far field of the initial state is away from vacuum, we obtain the global classical solutions and show the large time blow-up behavior of the gradient of the density. / This thesis deals with some important problems of compressible Navier-Stokes equations, including the well-posedness of the Cauchy problem, the regularity of the weak solutions constructed by Lions and Feireisl, and the dynamics of vacuum states, etc.. / Luo, Zhen. / Adviser: Zhouping Xin. / Source: Dissertation Abstracts International, Volume: 72-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 152-161). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.