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Neste trabalho estudamos a existência de soluções de equações do tipo Navier-Stokes com retardo na força externa e no termo n~ao linear. Usando a teoria de semigrupos estudamos a existência de soluções para um problema da forma \'d. SUP. dt u(t) - v\'delta\'u(t) + (F(t, \'u IND.t\'). abla)u(t) + abla p = g(t, \'u IND.t\'), em \'OMEGA\' x (0, T), div u(t) = 0 em \'OMEGA\' x (0, T), u(0, x) = \'u POT.0 (x) x PERTENCE a \' OMEGA\', u(t, x) = 0 t > 0, X \'PERTENCE A\' \' PARTIAL\' \'OMEGA\', u(t, x) =\\phi (t, x) t \'PERTENCE A\' (- \'INFINITO\', 0) x \'PERTENCE A\' \'OMEGA\', onde F9t, \'uIND.t) = INT.IND.t SUP. -\' INFINITO\' \' ALFA1(s-t)u(s)ds + u(t-r), g(t, \'u IND.t\') = INT. SUP. t IND. - INFINITO \'BETA\' (s-t)u(s)ds. Similarmente, usando a tecnica de aproximac~oes de Galerkin, estudamos o problema anterior com F(.) e g(.) dadas por f(t; \'u INDS.t\') = u(t-r(t)); e g(t; \'u IND.t\') = G(u(t-\'rô\' (t))), para alguma função G apropriada. Neste caso, também estudamos a estabilidade de soluções estacionarias / In this work we stuy the existence of solutions for a Navier-Stokes typt equations with delay in the external force and in the nonlinear term. Using the semi-group theory we study the existence of solution for a problem in the form \'d. SUP. dt u(t) - v\'delta\'u(t) + (F(t, \'u IND.t\'). abla)u(t) + abla p = g(t, \'u IND.t\'), ijn \'OMEGA\' x (0, T), div u(t) = 0 in \'OMEGA\' x (0, T), u(0, x) = \'u POT.0 (x) x \'IT BELONGS \' OMEGA\', u(t, x) = 0 t > 0, X \'IT BELONGS\' \'PARTIAL\' \'OMEGA\', u(t, x) =\\phi (t, x) t \'IT BELONGS\' (- \'INFINITY\', 0) x \'IT BELONGS\' \'OMEGA\', where F(t, \'u .t) = INT.IND.t SUP. -\' INFINITY\' \' ALFA(s-t)u(s)ds + u(t-r), g(t, \'u IND.t\') = INT. SUP. t IND. - INFINITY \'BETA\' (s-t)u(s)ds. On another hand using the Galerkin appreoximations method we study the same with F(.) e g(.) given by f(t; \'u INDS.t\') = u(t-r(t)); and g(t; \'u IND.t\') = G(u(t-\'rô\' (t))), for some G appropriated. In thiis case, we study also the stability of stanionary solutions

Orientador: Lucas Catão de Freitas Ferreira / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T15:06:30Z (GMT). No. of bitstreams: 1 Alves_BrunoFerreira_M.pdf: 943114 bytes, checksum: 70e3cdb3ee60ae3ed29a79cfeded3f59 (MD5) Previous issue date: 2012 / Resumo: Estudamos as equações de Navier-Stokes (NS) em...Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: We study the Navier-Stokes equations (NS) in...Note: The complete abstract is available with the full electronic document / Mestrado / Matematica / Mestre em Matemática

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The equation of motion for the flows of incompressible Newtonian fluids (Navier Stokes equations) under no-slip boundary conditions have been studied deeply from many perspectives. The questions of existence and uniqueness of both classical and weak solutions have received more than a fair share of attention. In this study the same problem for non-Newtonian fluids of second grade has been studied from the point of view of weak solutions and classical solutions for non-homogeneous boundary data, i.e., dynamical boundary conditions in regions with permeable boundaries. We consider the situation where a container is immersed in a larger fluid body and the boundary admits fluid particles moving across it in the direction of the normal. In this study we give alternative approaches through formulations of' dynamics at the boundary', the idea being that the normal component of velocity at the boundary is viewed as an unknown function which satisfies a differential equation intricately coupled to the flow in the region 'enclosed' by the boundary. We describe two mathematical models denoted by Problem PI and Problem P2. These models lead to dynamics at a permeable boundary, and a kinematical boundary condition for normal flow through the boundary. These conditions take into account the curvature of the boundary which enforces certain stresses. We then show with the help of the energy method that for fluids of second grade, the dynamics at the boundary and the boundary condition lead to conditional stability of the rest state for Problem P1 and Problem P2. We also prove uniqueness of classical solutions for the two models. The existence of a weak solution for this system of evolution equations is proved only for Problem P2 with the help of the Faedo-Galerkin method with a special basis. In this case the special basis is formed by eigenfunctions. The existence proof of at least one classical solution, local in time is established by means of a version of the Fixed-point Theorem of Bohnenblust and Karlin, and the Ascoli-Arzela Theorem. / Thesis (PhD (Applied Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

A Dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, for the degree of Master of Science. School of Computer Science and Applied Mathematics. November 7, 2016. / The equations for the flow of a thin lm of molten glass on a bath of molten tin are extended to the case in which the viscosity of the molten glass depends on the temperature. The continuity equation for an incompressible fluid, the Navier-Stokes equation and the energy balance equation are written in the lubrication (thin fluid lm) approximation. The kinematic boundary condition and the boundary conditions for the normal and tangential stress and the normal heat flux are derived on the upper and lower surfaces of the glass ribbon. It is found for the lubrication approximation that only one equation is obtained for four unknowns which are the two horizontal velocity components, the absolute temperature difference and the thickness of the molten glass rib- bon. The remaining three equations are obtained by taking the calculation to the next order in the square of the ratio of the thickness to length of the glass ribbon. The kinematic edge condition and the edge conditions for the normal and tangential stress and the normal heat flux are derived. The four edge conditions and the boundary conditions at the inlet and outlet give the boundary conditions for the four partial differential equations. It is not the aim of the dissertation to solve the boundary value problem which has been derived, either numerically or analytically. / LG2017

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A generalized Conjugate Gradient like method is used to solve the linear systems of equations formed at each time-integration step of the unsteady, two-dimensional, compressible Navier-Stokes equations of fluid flow. The Navier-Stokes equations are cast in an implicit, upwind finite-volume, flux split formulation. Preconditioning techniques are employed with the Conjugate Gradient like method to enhance the stability and convergence rate of the overall iterative method. The superiority of the new solver is established by comparisons with a conventional Line GaussSeidel Relaxation (LGSR) solver. Comparisons are based on 'number of iterations required to converge to a steady-state solution' and 'total CPU time required for convergence'. Three test cases representing widely varying flow physics are chosen to investigate the performance of the solvers. Computational test results for very low speed (incompressible flow over a backward facing step at Mach 0.1), transonic flow (trailing edge flow in a transonic turbine cascade) and hypersonic flow (shockon- shock interactions on a cylindrical leading edge at Mach 6.0) are presented. For the 1vfach 0.1 case, speed-up factors of 30 (in terms of iterations) and 20 (in terms of CPU time) are found in favor of the new solver when compared with the LGSR solver. The corresponding speed-up factors for the transonic flow case are 20 and 18, respectively. The hypersonic case shows relatively lower speed-up factors of 5 and 4, respectively. This study reveals that preconditioning can greatly enhance the range of applicability and improve the performance of Conjugate Gradient like methods. / Ph. D.