Boundary Integral and Singularity Methods for Linearized Viscous Flow (C. Pozrikidis)

(4) Mathematical problems arising in differential modelsfor viscoelastic fluids, by C. Guillop6 and J.-C. Saut (29 pages). The characteristic property of viscoelastic fluids is that the stress depends not only on the current velocity field, but on the history of the motion. One way to model such a history dependence is by a system of ordinary differential equations along particle trajectories which relate stress and velocity gradient. Such models are referred to as "differential" as opposed to "integral" models that express the stress in terms of one or several integrals along particle trajectories. The paper reviews the current state of the mathematical theory of the equations resuiting from differential models. The topics discussed include loss of evolution and change of type, existence of steady flows, inflow boundary conditions, existence for initial value problems, and flow stability. (5) Weak solutionsfor thermoconvectiveflows ofBoussinesq-Stefan type, by J.-E Rodrigues (24 pages). This paper considers a solid-liquid interface problem with a phase transition occurring at the interface. The equations are the heat equation in each phase and the Navier-Stokes equation in the fluid phase. A generalized Newtonian constitutive law with temperature dependent viscosity is allowed for the fluid. The existence of steadystate solutions is established under appropriate assumptions. (6) Boundaryand initial-boundary valueproblemsfor the Navier-Stokes equations in domains with noncompact boundaries, by V. A. Solonnikov (46 pages). In comparison to flows in interior or exterior domains, there are two new issues when the boundary extends to infinity. First, in addition to the usual initial and boundary conditions there needs to be some prescription of fluxes or pressure drops when the flow domain has several "exits to infinity. " Secondly, the solutions of interest often have infinite energy integrals. The paper discusses a recently developed technique of integral estimates to deal with this problem. These estimates are called "Saint Venant’s type" because the method was first used in the study of Saint Venant’s principle in elasticity. (7) Stability problems in electrohydrodynamics, ferrohydrodynamics and thermoelectric magnetohydrohynamics, by B. Straughan (30 pages). This paper studies the stability of a number of problems that involve the coupling of fluid mechanics with electrodynamic and thermal effects. Both linear and nonlinear stability are considered. The main technique for the nonlinear case is the energy method. (8) Mathematical results for compressible flows, by A. Valli (37 pages). Results on the existence of smooth solutions for compressible fluid mechanics are reviewed. Both inviscid and Newtonian fluids are considered. The paper discusses results on initial value problems as well as steady and time periodic flows. The book contains the following contributed papers: O. Ban, Une solution numrique des quations de Navier-Stokes stationnaires; L. Consiglieri, Stationary solutions for a Binghamflow with nonlocalfriction; S. Fabre, E’tude de la stabilit du couplage des quations d’Euler et Maxwell ? t tree dimension d’espace; E James, Diphasic equilibrium and chemical engineering; M. R. Levitin, Vibrations ofa viscous compressible fluid in bounded and unbounded domains; Yu.I. Skrynnikov, Shock wave in resonant despersion media; A. Skvortsov, Solitary vorticesna new exact solution ofhydrodynamic equations. The book will be of interest to researchers in partial differential equations who have an interest in fluid mechanics. It differs in a pleasant way from the flood of proceedings volumes that look like they are written with a xerox machine and a stapler. The reader will find wellwritten expository articles on various areas of current research. They will be of value to anyone who would like to work in one of these areas or who would simply like to find out about the current state of the art.