FERROFLUIDS: MODELING, NUMERICAL ANALYSIS, AND SCIENTIFIC COMPUTATION

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dc.contributor.advisorNochetto, Ricardo Hen_US
dc.contributor.authorTomas, Ignacioen_US
dc.contributor.departmentApplied Mathematics and Scientific Computationen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.date.accessioned2015-06-26T05:33:58Z
dc.date.available2015-06-26T05:33:58Z
dc.date.issued2015en_US
dc.description.abstractThis dissertation presents some developments in the Numerical Analysis of Partial Differential Equations (PDEs) describing the behavior of ferrofluids. The most widely accepted PDE model for ferrofluids is the Micropolar model proposed by R.E. Rosensweig. The Micropolar Navier-Stokes Equations (MNSE) is a subsystem of PDEs within the Rosensweig model. Being a simplified version of the much bigger system of PDEs proposed by Rosensweig, the MNSE are a natural starting point of this thesis. The MNSE couple linear velocity u, angular velocity w, and pressure p. We propose and analyze a first-order semi-implicit fully-discrete scheme for the MNSE, which decouples the computation of the linear and angular velocities, is unconditionally stable and delivers optimal convergence rates under assumptions analogous to those used for the Navier-Stokes equations. Moving onto the much more complex Rosensweig's model, we provide a definition (approximation) for the effective magnetizing field h, and explain the assumptions behind this definition. Unlike previous definitions available in the literature, this new definition is able to accommodate the effect of external magnetic fields. Using this definition we setup the system of PDEs coupling linear velocity u, pressure p, angular velocity w, magnetization m, and magnetic potential h_d. We show that this system is energy-stable and devise a numerical scheme that mimics the same stability property. We prove that solutions of the numerical scheme always exist and, under certain simplifying assumptions, that the discrete solutions converge. A notable outcome of the analysis of the numerical scheme for the Rosensweig's model is the choice of finite element spaces that allow the construction of an energy-stable scheme. Finally, with the lessons learned from Rosensweig's model, we develop a diffuse-interface model describing the behavior of two-phase ferrofluid flows and present an energy-stable numerical scheme for this model. For a simplified version of this model and the corresponding numerical scheme we prove (in addition to stability) convergence and existence of solutions as by-product . Throughout this dissertation, we will provide numerical experiments, not only to validate mathematical results, but also to help the reader gain a qualitative understanding of the PDE models analyzed in this dissertation (the MNSE, the Rosenweig's model, and the Two-phase model). In addition, we also provide computational experiments to illustrate the potential of these simple models and their ability to capture basic phenomenological features of ferrofluids, such as the Rosensweig instability for the case of the two-phase model. In this respect, we highlight the incisive numerical experiments with the two-phase model illustrating the critical role of the demagnetizing field to reproduce physically realistic behavior of ferrofluids.en_US
dc.identifierhttps://doi.org/10.13016/M2RH0N
dc.identifier.urihttp://hdl.handle.net/1903/16598
dc.language.isoenen_US
dc.subject.pqcontrolledApplied mathematicsen_US
dc.subject.pqcontrolledCondensed matter physicsen_US
dc.subject.pqcontrolledMechanicsen_US
dc.subject.pquncontrolleddemagnetizing fielden_US
dc.subject.pquncontrolledferrofluidsen_US
dc.subject.pquncontrolledfree surfaceen_US
dc.subject.pquncontrolledphase fielden_US
dc.subject.pquncontrolledrosensweig instabilityen_US
dc.subject.pquncontrolledstray fielden_US
dc.titleFERROFLUIDS: MODELING, NUMERICAL ANALYSIS, AND SCIENTIFIC COMPUTATIONen_US
dc.typeDissertationen_US

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