Renardy, MichaelOverview
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Most widely held works by
Michael Renardy
An introduction to partial differential equations
by Michael Renardy
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Book
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37 editions published between 1992 and 2011 in English and Undetermined and held by 1,073 WorldCat member libraries worldwide "Partial differential equations (PDEs) are fundamental to the modeling of natural phenomena, arising in every field of science. Consequently, the desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians; it has inspired such diverse fields as complex function theory, functional analysis, and algebraic topology. Like algebra, topology, and rational mechanics, PDEs are a core area of mathematics." "This book aims to provide the background necessary to initiate work on a Ph.D. thesis in PDEs for beginning graduate students. Prerequisites include a truly advanced calculus course and basic complex variables. Lebesgue integration is needed only in Chapter 10, and the necessary tools from functional analysis are developed within the course. The book can be used to teach a variety of different courses."BOOK JACKET
Mathematical analysis of viscoelastic flows
by Michael Renardy
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Book
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11 editions published in 2000 in English and held by 238 WorldCat member libraries worldwide This monograph is based on a series of lectures presented at the 1999 NSFCBMS Regional Research Conference on Mathematical Analysis of Viscoelastic Flows. It begins with an introduction to phenomena observed in viscoelastic flows, the formulation of mathematical equations to model such flows, and the behavior of various models in simple flows. It also discusses the asymptotics of the high Weissenberg limit, the analysis of flow instabilities, the equations of viscoelastic flows, jets and filaments and their breakup, as well as several other topics
Viscoelasticity and rheology
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8 editions published in 1985 in English and held by 214 WorldCat member libraries worldwide
Mathematical problems in viscoelasticity
by Michael Renardy
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9 editions published in 1987 in English and Undetermined and held by 209 WorldCat member libraries worldwide
Quasiperiodische VerzweigungsloÌˆsungen rotationssymmetrischer Probleme
by Michael Renardy
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Book
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3 editions published in 1980 in German and held by 21 WorldCat member libraries worldwide
LaxWendroff Methods for Hyperbolic History Value Problems
by Mathematics Research Center (United States. Army)
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Book
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2 editions published in 1982 in English and held by 2 WorldCat member libraries worldwide The motion of viscoelastic materials can be modelled by partial integrodifferential equations. For several model problems, recent investigations have been concerned with the question whether or not these equations allow the development of shocks. This paper is concerned with LaxWendroff methods for a class of hyperbolic history value problems. These problems have the feature that globally (in time) smooth solutions exist if the data are sufficiently small and that solutions develop singularities for large data. The authors prove (second order) convergence of the LaxWendroff method for smooth solutions and investigate numerically the dependence on the initial data. They demonstrate the occurrence of shock type singularities and compare the results to the quasilinear wave equation (without Volterra term)
Special issue on rheological fluid dynamics
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Book
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1 edition published in 1993 in English and held by 2 WorldCat member libraries worldwide
Existence of Slow Steady Flows of Viscoelastic Fluids with Differential Constitutive Equations
by Michael Renardy
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Book
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3 editions published in 1984 in English and held by 2 WorldCat member libraries worldwide Questions of existence and uniqueness of steady flows of viscoelastic fluids have thus far not been understood, even for slow flows perturbing rest. This paper provides an existence result for slow flows with no in and outflow boundaries. The fluid is assumed to be described by constitutive equations of a differential nature. The method used to prove existence is constructive and in fact very close to procedures used in numerical calculations
Singularly Perturbed Hyperbolic Evolution Problems with Infinite Delay and an Application to Polymer Rheology
by Mathematics Research Center (United States. Army)
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Book
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2 editions published in 1982 in English and held by 2 WorldCat member libraries worldwide We prove an existence theorem locally in time for quasilinear hyperbolic equations, in which the coefficients are allowed to depend on the history of the dependent variable. Singular perturbations, which change the type of the equation to parabolic, are included, and continuous dependence of the solutions on the perturbation parameter is shown. It is demonstrated that, for a substantial number of constitutive models suggested in the literature, the stretching of filaments of polymeric liquids is described by equations of the kind under study here. (Author)
Perturbation of a multiple eigenvalue in the Benard problem for two fluid layers
by Mathematics Research Center (United States. Army)
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Book
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3 editions published in 1984 in English and held by 2 WorldCat member libraries worldwide In a recent paper, Renardy and Joseph study the Bernard problem for two layers of different fluids lying on top of each other and bounded by walls. Their study shows that, in contrast to the Benard problem for one fluid, the onset of instability can be oscillatory. The number of parameters involved in the problem is large, and there is yet no comprehensive picture of when the instability is oscillatory and when it is not. The study of limiting cases, accessible by perturbation methods, may be helpful in this respect. In this document, an analysis is given for the case when the properties of the two fluids are nearly equal and the fluids are allowed to slip at the boundaries. (Author)
Inflow Boundary Conditions for Steady Flows of Viscoelastic Fluids with Differential Constitutive Laws
by Michael Renardy
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Book
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2 editions published in 1986 in Undetermined and English and held by 2 WorldCat member libraries worldwide Steady flows of viscoelastic fluids can not be uniquely determined by imposing boundary conditions only for the velocities as in the Newtonian case. The reason for this is that the fluids have memory, and therefore the flow inside the domain is affected by what happened before the fluid entered the domain. This leads to the need for extra boundary conditions at an inflow boundary. The nature of these inflow boundary conditions has not been analyzed previously, and it is certainly dependent on the constitutive law. In this paper, we look at the special case of differential constitutive relations with a single relaxation mode. We consider steady transverse flows across a strip which are small perturbations of a flow with constant velocity. It turns out that in this case two extra inflow boundary conditions are required in two dimensions, and four in three dimensions. This is what would be expected from an analysis of characteristics, but it contradicts the belief of many rheologists that it is possible to prescribe the extra stress at an inflow boundary. The problem studied here is of potential relevance for numerical simulations of steady flows. Many of the flows currently simulated are on infinite domains. Numerically, these domains are truncated, and on the inflow boundary of the truncated domain people usually prescribe the extra stress. According to the analysis in this paper, this is an overdetermined problem, and therefore errors must be expected from this procedure unless the artificial boundaries are chosen far enough out. Keywords: Upper Convected Maxwell Model
A Local Existence and Uniqueness Theorem for a KBKZFluid
by Mathematics Research Center (United States. Army)
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Book
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2 editions published in 1983 in English and held by 2 WorldCat member libraries worldwide The existence theory for models of viscoelastic fluids has so far not been very well developed, in particular in three dimensional situations. Here, the author proves an existence theorem for a particular class of models, suggested by Kaye and Bernstein, Kearsley and Zapas. This theory is based on a postulated analogy with hyperelasticity. It is assumed that the fluid occupies all of space. Abstracts methods developed originally for quasilinear hyperbolic systems can be used to prove the wellposedness of the initial value problem
Hyperbolicity and Change of Type in the Flow of Viscoelastic Fluids
by United States
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Book
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1 edition published in 1984 in English and held by 1 WorldCat member library worldwide The equations governing the flow of viscoelastic liquids are classified according to the symbol of their differential operators. Propagation of singularities is discussed and conditions for a change of type are investigated. The vorticity equation for steady flow can change type when a critical condition involving speed and stresses is satisfied. This leads to a partitioning of the field of flow into subcritical and supercritical regions, as in the problem of transonic flow. (Author)
Rheological fluid dynamics
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Book
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1 edition published in 1993 in English and held by 1 WorldCat member library worldwide
On the domain space for constitutive laws in linear viscoelasticity
by Mathematics Research Center (United States. Army)
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Book
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1 edition published in 1982 in English and held by 1 WorldCat member library worldwide
The Numerical Solution of a Class of Quasilinear Parabolic Volterra Equations Arising in Polymer Rheology
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Book
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1 edition published in 1983 in English and held by 1 WorldCat member library worldwide
A Finite Difference Study of the Stretching and BreakUp of Filaments of Polymer Solutions
by Mathematics Research Center (United States. Army)
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Book
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2 editions published in 1984 in English and held by 1 WorldCat member library worldwide The stretching and breakup of a viscoelastic filament pulled by a constant weight is studied numerically by a finite difference method. The results show the following tendencies: 1. Newtonian filaments, even in the absence of surface tension, show a rapid increase in elongation at one particular point (these break there). 2. The addition of a viscoelastic polymer prevents or at least delays the breakup, even if it makes only a small difference to shear viscosity. 3. Surface tension accelerates breakup, but even in the presence of surface tension elasticity has a stabilizing effect. (Author)
Rheological fluid dynamics, part 1
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Book
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1 edition published in 1993 in English and held by 1 WorldCat member library worldwide
A first graduate course in partial differential equations
by Michael Renardy
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Book
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1 edition published in 1993 in English and held by 1 WorldCat member library worldwide
A Center Manifold theorem for hyperbolic PDEs
by Michael Renardy
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Book
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1 edition published in 1991 in English and held by 1 WorldCat member library worldwide more
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Renardy, M.
Renardy, M. (Michael)
Renardy, Michael
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