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.
Basic Theory: Introduction; Existence and Uniqueness of Smooth Solutions; Estimates for Smooth Solutions; Extension of the Solution Operator; Measures as Initial Data; Asymptotic Behavior for Large Time; Some Theorems from Functional Analysis; Approximate Solutions: Introduction; Notation; Finite Difference Approximation to Second-Order Boundary-Value Problems; From Hermitian Derivative to the Compact Discrete Biharmonic Operator; Polynomial Approach to the Discrete Biharmonic Operator; Compact Approximation of the Navier - Stokes Equations in Streamfunction Formulation; Fully Discrete Approximation of the Navier - Stokes Equations; Numerical Simulations of the Driven Cavity Problem.