Navier-Stokes Equations in Planar Domains

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.

Contents:

• 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

Readership: Graduate students and researchers in applied mathematics (particularly computational fluid dynamics), partial differential equations, and mathematical physics (specifically nonlinear evolution equations).
Key Features:

• Provides an up-to-date account of recent developments in vorticity theory
• Presents comprehensive treatment of viscous flow problems in flat domains
• Addresses the classical problem: How rapidly do rotating flows evolve in time?
• Includes theoretical and numerical methods
• Presents state-of-the-art streamfunction formalism (theoretical and numerical)