Fundamental Directions in Mathematical Fluid Mechanics

Table of Contents

1. Frontmatter

2. An Introduction to the Navier-Stokes Initial-Boundary Value Problem

   Giovanni P. Galdi

   Abstract

   The equations of motion of an incompressible, Newtonian fluid — usually called Navier-Stokes equations — have been written almost one hundred eighty years ago. In fact, they were proposed in 1822 by the French engineer C.M.L.H. Navier upon the basis of a suitable molecular model. It is interesting to observe, however, that the law of interaction between the molecules postulated by Navier were shortly recognized to be totally inconsistent from the physical point of view for several materials and, in particular, for liquids. It was only more than twenty years later that the same equations were rederived by the twenty-six year old G. H. Stokes 1845 in a quite general way, by means of the theory of continua.

3. Spectral Approximation of Navier-Stokes Equations

   P. Gervasio, A. Quarteroni, F. Saleri

   Abstract

   We review some basic aspects of spectral methods and their application to the numerical solution of Navier-Stokes equations for viscous incompressible flows.

4. Simple Proofs of Bifurcation Theorems

   John G. Heywood, Wayne Nagata

   Abstract

   We give short, elementary, constructive proofs of the basic theorems concerning the bifurcation of equilibrium and periodic solutions, from a trivial solution of an ordinary differential equation, as a parameter in the equation passes through a critical value. We begin by considering a finite dimensional equation, formulated to be analogous to an abstract Navier-Stokes equation in Hilbert space. Later in the paper we consider generalizations, including generalizations to partial differential equations, and especially to the Navier-Stokes equations.

5. On The Steady Transport Equation

   John G. Heywood, Mariarosaria Padula

   Abstract

   We give a simple approach to the theory of the steady transport equation, providing some results that are needed in the theory of steady compressible viscous flow. In particular, we prove the assertions that are collected together in Lemma 3 of our accompanying paper [11]. Our methods are based on a construction of solutions by Galerkin approximation and on duality arguments with the adjoint equation. We conclude with some heuristic observations based on the theory of characteristics, explaining, particularly, the need for successively stronger restrictions on the size of the velocity field, in order to prove successively more regularity of the solution.

6. On the Existence and Uniqueness Theory for Steady Compressible Viscous Flow

   John G. Heywood, Mariarosaria Padula

   Abstract

   We give a simplified approach to the existence theory for compressible viscous flow, based on a new iterative scheme. This paper introduces the scheme in the context of steady isothermal flow in a bounded domain with homogeneous boundary conditions. The solution that is obtained is shown to be locally unique, using estimates similar to those used in proving its existence. A second direct approach to uniqueness is also investigated. The analysis presented here is based on estimates for the velocity and density in W ^3,2 × W ^2,2. It appears to be possible to give a similar analysis based on estimates in W ^2,p × W ^1,p, for p > n.

7. Finite Element Methods for the Incompressible Navier-Stokes Equations

   Rolf Rannacher

   Abstract

   These notes are based on lectures given in a Short Course on Theoretical and Numerical Fluid Mechanics in Vancouver, British Columbia, Canada, July 27–28, 1996, and at several other places since then. They provide an introduction to recent developments in the numerical solution of the Navier-Stokes equations by the finite element method. The material is presented in eight sections:

   1.

   Introduction: Computational aspects of laminar flows

    

   2.

   Models of viscous flow

    

   3.

   Spatial discretization by finite elements

    

   4.

   Time discretization and linearization

    

   5.

   Solution of algebraic systems

    

   6.

   A review of theoretical analysis

    

   7.

   Error control and mesh adaptation

    

   8.

   Extension to weakly compressible flows

    

   Theoretical analysis is offered to support the construction of numerical methods, and often computational examples are used to illustrate theoretical results. The variational setting of the finite element Galerkin method provides the theoretical framework. The goal is to guide the development of more efficient and accurate numerical tools for computing viscous flows. A number of open theoretical problems will be formulated, and many references are made to the relevant literature.