by Matania Ben-Artzi (Hebrew University of Jerusalem, Israel), Jean-Pierre Croisille (Universite Paul Verlaine — Metz, France), & Dalia Fishelov (Tel-Aviv Academic College of Engineering, Israel)

This volume deals with the classical Navier-Stokes system of equations governing the planar flow of incompressible, viscid fluid. It is a first-of-its-kind book, devoted to all aspects of the study of such flows, ranging from theoretical to numerical, including detailed accounts of classical test problems such as “driven cavity” and “double-driven cavity”.

A comprehensive treatment of the mathematical theory developed in the last 15 years is elaborated, heretofore never presented in other books. It gives a detailed account of the modern compact schemes based on a “pure streamfunction” approach. In particular, a complete proof of convergence is given for the full nonlinear problem.

This volume aims to present a variety of numerical test problems. It is therefore well positioned as a reference for both theoretical and applied mathematicians, as well as a text that can be used by graduate students pursuing studies in (pure or applied) mathematics, fluid dynamics and mathematical physics.

• Part I:
  • Introduction and Various Formulations of the System: A Brief Survey of the Classical Energy (Leray) Method
  • The Well-Posedness Theory for the Vorticity in L1 of the Whole Plane
  • The Well-Posedness of the Vorticity in Bounded Domains
  • Asymptotic Behavior (Large Time) of the Solutions in the Whole Plane
  • Stability and Instability of Steady States (in Streamfunction Formulation)
• Part II:
  • A Pure Streamfunction Formulation of the Discretized System in a Rectangle or Periodic (the Stephenson Biharmonic and Its 4th-Order Improvement)
  • Convergence Theorem for the Full Nonlinear System (in Rectangle or Periodic)
  • Numerical Implementation and FFT
  • Some Numerical Results and Discussions of Bifurcations and Approach to Periodic Solutions