1st Edition

Mathematical Models in Boundary Layer Theory

By O.A. Oleinik, V.N. Samokhin Copyright 1999
    528 Pages
    by Chapman & Hall

    Since Prandtl first suggested it in 1904, boundary layer theory has become a fundamental aspect of fluid dynamics. Although a vast literature exists for theoretical and experimental aspects of the theory, for the most part, mathematical studies can be found only in separate, scattered articles. Mathematical Models in Boundary Layer Theory offers the first systematic exposition of the mathematical methods and main results of the theory.

    Beginning with the basics, the authors detail the techniques and results that reveal the nature of the equations that govern the flow within boundary layers and ultimately describe the laws underlying the motion of fluids with small viscosity. They investigate the questions of existence and uniqueness of solutions, the stability of solutions with respect to perturbations, and the qualitative behavior of solutions and their asymptotics. Of particular importance for applications, they present methods for an approximate solution of the Prandtl system and a subsequent evaluation of the rate of convergence of the approximations to the exact solution.

    Written by the world's foremost experts on the subject, Mathematical Models in Boundary Layer Theory provides the opportunity to explore its mathematical studies and their importance to the nonlinear theory of viscous and electrically conducting flows, the theory of heat and mass transfer, and the dynamics of reactive and muliphase media. With the theory's importance to a wide variety of applications, applied mathematicians-especially those in fluid dynamics-along with engineers of aeronautical and ship design will undoubtedly welcome this authoritative, state-of-the-art treatise.

    The Navier-Stokes Equations and Prandtl
    Derivation of the Prandtl System
    Solution of the Boundary Layer System as the First Approximation to Asymptotic Solution of the Navier-Stokes Equations near the Boundary
    Separation of the Boundary Layer
    Setting of the Main Problems for the Equations of Boundary Layer
    Boundary Layer Equations for Non-Newtonian Fluids
    Boundary Layers in Magnetohydrodynamics
    Stationary Boundary Layer: von Mises Variables
    Continuation of Two-Dimensional Boundary Layer
    Asymptotic Behavior of the Velocity Component along the Boundary Layer
    Conditions for Boundary Layer Separation
    Self-Similar Solutions of the Boundary Layer Equations
    Solving the Continuation Problem by the Line Method
    On Three-Dimensional Boundary Layer Equations
    Comments
    Stationary Boundary Layer: Crocco Variables
    Axially Symmetric Stationary Boundary Layer
    Symmetric Boundary Layer
    The Problem of Continuation of the Boundary Layer
    Weak Solutions of the Boundary Layer System
    Nonstationary Boundary Layer
    Axially Symmetric Boundary Layer
    The Continuation Problem for a Nonstationary Axially Symmetric Boundary Layer
    Continuation of the Boundary Layer: Successive Approximations
    On t-Global Solutions of the Prandtl System for Axially Symmetric Flows
    Stability of Solutions of the Prandtl System
    Time-Periodic Solutions of the Nonstationary Boundary Layer System
    Solving the Nonstationary Prandtl System by the Line Method in the Time Variable
    Formation of the Boundary Layer
    Solutions and Asymptotic Expansions for the Problem of Boundary Layer formation: The Case of Gradual Acceleration
    Formation of the Boundary Layer about a Body that Suddenly Starts to Move
    Comments
    Finite-Difference Method
    Solving the Boundary Layer Continuation Problem by the Finite Difference Method
    Solving the Prandtl System for Axially Symmetric Flows by the Finite Difference Method
    Comments
    Diffraction Problems for the Prandtl System
    Boundary Layer with Unknown Border between Two media
    Mixing of Two Fluids with Distinct Properties at the Interface between Two Flows
    Comments
    Boundary Layer in Non-Newtonian Flows
    Symmetric Boundary Layer in Pseudo-Plastic Fluids
    Weak Solutions of the Boundary Layer Continuation Problem for Pseudo-Plastic Fluids
    Nonstationary Boundary Layer for Pseudo-Plastic Fluids
    Continuation of the Boundary Layer in Dilatable Media
    Symmetric Boundary Layer in Dilatable Media
    Comments
    Boundary Layer in Magnetic Hydrodynamics
    Continuation of the MHD Boundary Layer in Ordinary Fluids
    Solving the Equations of the MHD Boundary Layer in Pseudo-Plastic Fluids
    Self-Similar Solutions of the MHD Boundary Layer System for a Dilatable Fluid
    Solving the Equations of Boundary Layer for Dilatable Conducting Fluids in a Transversal Magnetic Field
    Comments
    Homogenization of Boundary Layer Equations
    Homogenization of the Prandtl System with Rapidly Oscillating Injection and Suction
    Homogenization of the Equations of the MHD Boundary Layer in a Rapidly Oscillating Magnetic Field
    Comments
    Some Open Problems
    References
    Index

    Biography

    Samokhin, V.N.