Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow
- 2014
- Book
- Authors
- Hamid Bellout
- Frederick Bloom
- Book Series
- Advances in Mathematical Fluid Mechanics
- Publisher
- Springer International Publishing
About this book
The theory of incompressible multipolar viscous fluids is a non-Newtonian model of fluid flow, which incorporates nonlinear viscosity, as well as higher order velocity gradients, and is based on scientific first principles. The Navier-Stokes model of fluid flow is based on the Stokes hypothesis, which a priori simplifies and restricts the relationship between the stress tensor and the velocity. By relaxing the constraints of the Stokes hypothesis, the mathematical theory of multipolar viscous fluids generalizes the standard Navier-Stokes model. The rigorous theory of multipolar viscous fluids is compatible with all known thermodynamical processes and the principle of material frame indifference; this is in contrast with the formulation of most non-Newtonian fluid flow models which result from ad hoc assumptions about the relation between the stress tensor and the velocity. The higher-order boundary conditions, which must be formulated for multipolar viscous flow problems, are a rigorous consequence of the principle of virtual work; this is in stark contrast to the approach employed by authors who have studied the regularizing effects of adding artificial viscosity, in the form of higher order spatial derivatives, to the Navier-Stokes model.
A number of research groups, primarily in the United States, Germany, Eastern Europe, and China, have explored the consequences of multipolar viscous fluid models; these efforts, and those of the authors, which are described in this book, have focused on the solution of problems in the context of specific geometries, on the existence of weak and classical solutions, and on dynamical systems aspects of the theory.
This volume will be a valuable resource for mathematicians interested in solutions to systems of nonlinear partial differential equations, as well as to applied mathematicians, fluid dynamicists, and mechanical engineers with an interest in the problems of fluid mechanics.
Table of Contents
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Frontmatter
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Chapter 1. Incompressible Multipolar Fluid Dynamics
Hamid Bellout, Frederick BloomAbstractThe study of the motions of an incompressible viscous fluid by mathematicians, physicists, and engineers, has been an ongoing enterprise since the publication by G.G. Stokes [Sto] of his classical memoir on the internal friction of fluids in motion in 1849. -
Chapter 2. Plane Poiseuille Flow of Incompressible Bipolar Viscous Fluids
Hamid Bellout, Frederick BloomAbstractIn Sect. 1.4 we introduced the model of an incompressible, nonlinear, bipolar viscous fluid; this model, which is consistent with the basic principles of continuum mechanics and thermodynamics, as delineated in Sect. 1.4, is based on the following constitutive hypotheses for the Cauchy stress tensor τ ij and the first multipolar stress tensor τ ijk -
Chapter 3. Incompressible Bipolar Fluid Dynamics: Examples of Other Flows and Geometries
Hamid Bellout, Frederick BloomAbstractThe mathematical model of a nonlinear, incompressible, bipolar viscous fluid was introduced in Sect. 1.6 and conforms to the constitutive hypotheses for the Cauchy stress tensor τ ij and the first multipolar stress tensor τ ijk -
Chapter 4. General Existence and Uniqueness Theorems for Incompressible Bipolar and Non-Newtonian Fluid Flow
Hamid Bellout, Frederick BloomAbstractIn Sect.1.4 we introduced the equations which govern the motion of a nonlinear, incompressible, bipolar fluid. For a bounded domain in \({\mathbb{R}}^{n}\), n = 2, 3 the appropriate boundary conditions were set forth in Sect.1.4 and, for flows in all of \({\mathbb{R}}^{n}\), the relevant periodic (boundary) conditions were also delineated. -
Chapter 5. Attractors for Incompressible Bipolar and Non-Newtonian Flows: Bounded Domains and Space Periodic Problems
Hamid Bellout, Frederick BloomAbstractFrom the existence and uniqueness theorems established in Chap. 4, both for the initial-boundary value problems, as well as for the space-periodic problems associated with nonlinear, incompressible, bipolar (μ 1 > 0) and non-Newtonian flow (μ 1 = 0), it follows that under appropriate sets of conditions one may show that the solution operator \(\boldsymbol{S}(t)\) yields a nonlinear semigroup; in this chapter we examine the behavior of the orbits of such semigroups as t → ∞. Our interest here is focused on the existence of maximal compact global attractors for bounded domains and space periodic problems. -
Chapter 6. Inertial Manifolds, Orbit Squeezing, and Attractors for Bipolar Flow in Unbounded Channels
Hamid Bellout, Frederick BloomAbstractIn Chap. 5 we discussed, in considerable detail, the existence of maximal compact global attractors for bipolar and non-Newtonian flows associated with either (5.2a,b), (5.3a), (5.4), \(\Omega \subseteq {R}^{n}\), n = 2, 3, a bounded domain, or (5.2a,b), (5.3b), (5.4) where \(\Omega = {[0,L]}^{n}\), n = 2, 3, L> 0. -
Backmatter
- Title
- Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow
- Authors
-
Hamid Bellout
Frederick Bloom
- Copyright Year
- 2014
- Publisher
- Springer International Publishing
- Electronic ISBN
- 978-3-319-00891-2
- Print ISBN
- 978-3-319-00890-5
- DOI
- https://doi.org/10.1007/978-3-319-00891-2
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