Theory and Applications of Viscous Fluid Flows

Abstract

5R6. Theory and Applications of Viscous Fluid Flows. - RK Zeytounian (12 Rue Saint-Fiacre, Paris, 75002, France). Springer-Verlag, Berlin. 2004. 488 pp. ISBN 3-540-44013-5. $109.00.Reviewed by MF Platzer (Dept of Aeronaut and Astronaut, Naval Postgraduate Sch, Code AA/PL, Monterey CA 93943-5000).Starting with the derivation of the Navier-Stokes equations for viscous heat-conducting fluids the author proceeds to discuss various forms of these equations, including the special cases of compressible isentropic viscous flow of polytropic gases and viscous incompressible fluid flow. He then discusses the Orr-Sommerfeld theory for the plane Poiseuille flow as well as other basic flow cases, such as steady flow through an arbitrary cylinder, annular flow between concentric cylinders, Benard thermal convection flow, Benard-Marangoni flow induced by tangential gradients of variable surface tension, flow due to a rotating disc, and Rayleigh flow caused by an impulsively started flat plate. The next three chapters are devoted to the very large and very low Reynolds number limits and to the low Mach number incompressible limit. In the chapter on very large Reynolds number flow the author discusses the application of the method of matched asymptotic expansions to the two-dimensional steady flat-plate flow problem and delineates the relationship of the unsteady Navier-Stokes equations to the inviscid Euler, the Prandtl boundary layer, the one-dimensional gas dynamics and the Rayleigh compressible flow equations. He also discusses the triple deck concept, laminar flow separation on a circular cylinder, and the three-dimensional boundary layer equations. In the chapter on very low Reynolds numbers, the unsteady-state matched Stokes-Oseen solution for the flow past a sphere and the flow over an impulsively started circular cylinder are discussed, followed by a consideration of the Stokes and Oseen steady-state compressible flow equations and the asymptotic analysis for small Reynolds number flows on a rotating disc. In the next chapter on low Mach number incompressible limit, the author discusses subtleties involved in analyzing unsteady weakly compressible flows; flow in a bounded cavity and through large aspect ratio channels. He then provides further examples by analyzing the acoustic streaming effect caused by an oscillating circular cylinder, the incompressible flow past a rotating and translating cylinder, the Ekman and Stewartson layers on rotating cylinders, and the Benard-Marangoni thermo-capillary instability problem due to heating of a horizontal viscous liquid from below. Also presented are some aspects of non-adiabatic viscous atmospheric flows and a few other topics, such as the entrainment of a viscous fluid in a two-dimensional cavity and the laminar boundary layer separation phenomenon near the leading-edge region of an airfoil and on an impulsively started cylinder. In this regard, he emphasizes the need for the simultaneous solution of the boundary layer and inviscid flow equations in order to remove the singularity at the separation point, as implemented in the viscous-inviscid interaction procedures. The next two chapters are devoted to a discussion of the existence, regularity and uniqueness of solutions for the viscous incompressible and compressible flow equations and the stability theory of fluid motion. In particular, the Guiraud-Zeytounian asymptotic approach to nonlinear hydrodynamic stability is elucidated and applied to the Rayleigh-Benard convection problem, followed by an analysis of the Benard-Marangoni thermo-capillary instability problem and the Couette-Taylor viscous flow between two rotating cylinders. The final chapter of Theory and Applications of Viscous Fluid Flows presents the finite-dimensional dynamical systems approach to turbulence by reviewing the Landau-Hopf, Ruelle-Takens-Newhouse, Feigenbaum and Pomeau-Manneville transition scenarios to turbulence. The book is ended by giving examples of strange attractors occurring in various fluid flows, such as in viscous isobaric wave motions or in the flow of an incompressible but thermally conducting liquid down a vertical plane (the Benard-Marangoni problem for a free-falling vertical film). It is evident from this brief summary that the author’s emphasis is on the mathematical aspects of the viscous flow equations and their various asymptotic limit cases and analytical solution methods. His choice of topics and flow problems is meant to provide young researchers in fluid mechanics, applied mathematics and theoretical physics with an up-to-date presentation of some key problems in the analysis of viscous fluid flows. Although the author intentionally limited himself to a select few topics, teachers of advanced viscous flow courses and researchers in this field will welcome this book for its thorough review of current work and the listing of 1156 relevant papers. In my judgment, it meets the stated objective of bridging the gap between standard undergraduate texts in fluid mechanics and specialized monographs.