Slow Viscous Flows

The following sections are included:

• Foreword

• Preface

• Acknowledgments

• Contents

• Introduction

The following sections are included:

• The Continuity, Navier–Stokes and Energy Equations

• The Boundary Conditions

• Non-Dimensionalization: The Reynolds and Other Dimensionless Numbers

• Slow Viscous or Low Reynolds Number Flows: The Stokes Equation

• The Pressure and the Vorticity

• Streamlines and the Stream Function

The following sections are included:

• Fourier Series

• A Boundary Value Problem for Laplace's Equation

• Convergence of Fourier Series and Completeness of the Eigenfunctions

• The Relationship of the Eigenfunction Expansions of Classical Physics to the Ones That Will Arise in This Book

• The Method of Least Squares
  • Curve fitting by least squares
  • Least squares and the expansion of an arbitrary function in terms of an infinite, complete set of functions
  • Least squares and Fourier series
  • Extensions to more difficult, non-classical situations: complex eigenfunction expansions
  • Eigenfunction expansions involving a single, infinite sequence of complex scalars
  • Eigenfunction expansions involving an infinite sequence of real scalars and an infinite sequence of complex scalars

• Systematic Procedures for Finding the Real and Complex Roots of Transcendental Equations

The following sections are included:

• The Field Equations, the Stream Function and Boundary Conditions

• Symmetric and Antisymmetric Eigenfunctions for the Rectangle

• Symmetric–Antisymmetric Decomposition in z

• The Solution of the General Boundary Value Problem in the Rectangle
  • A rectangular container of infinite depth
  • A rectangular container of finite depth h

• Corner Data, Some Computational Details and Accuracy

• The Flow Field and Eddy Structure in the Lid Driven Container
  • The primary eddy structure
  • The shallow container
  • Corner eddies and the evolution of the primary eddies with increasing h
  • The pressure and vorticity fields in the container
  • The superiority of using velocity boundary conditions

• Some General Conclusions on Rectangular Flow Fields

• Other Methods of Solution
  • Biorthogonality and methods based on its use
  • Boundary integral and singularity methods
  • The superposition method or the method of real Fourier series
  • Other approaches

The following sections are included:

• Separable Solutions of the Stokes Equations in Plane Polar Coordinates

• Similarity Solutions of the Paint-Scraper Problem and One with a Free Surface

• Flow Near a Sharp Corner: The Corner or Moffatt Eddies

• Flow Near a Sharp Corner: Large Corner Angles

• Corner Flows with a Source at the Vertex

• Isolated Stagnation Points in Planar Flows

• Viscous Flow Fields Near Points on the Boundary Where the Skin Friction Vanishes

• Blocked Flows

• Some Interesting Examples

• Limitations

The following sections are included:

• The Embedding Method

• The Method of Embedding for Planar Stokes Flows
  • A complete set of Stokes flow eigenfunctions in the rectangle
  • A complete set of Stokes flow eigenfunctions in the circle
  • Planar Stokes flow in a container of arbitrary shape
  • Stokes flow in an annular sector: comparison with the results of Krasnopolskaya et al.

• Applications to Non-Simple Geometries
  • The effect of container height on the eddy structure in a round-bottomed container
  • A container with a trench-like bottom
  • Eddy structure in a container with a concave bottom
  • Fields in containers which are not symmetric about x = 1/2

• The Strengths and Limitations of the Embedding Method

The following sections are included:

• Mathematical Issues
  • Canonical and non-canonical problems in the semi-infinite strip
  • Eigenfunction expansions and completeness
  • Convergence of the expansions
  • The non-uniform nature of the Stokes equations
  • The physical and mathematical bases of the singular LDC problem

• Non-Homogeneous and Discontinuous Corner Data in the Eigenfunction Method

• The Singular LDC Problem in the Rectangular Geometry

• Stokes Flow in Other Special Geometries
  • The double lid driven container
  • Flow in an annulus and in an annular sector
  • Flow in an infinite wedge driven by belts on the sidewalls
  • Meniscus roll coating and stress free sidewalls: real eigenvalues
  • Cusped interfaces in low Reynolds number flows

The following sections are included:

• The Field Equations, Non-Dimensionalization and the Boussinesq Approximation

• Planar Natural and Mixed Convection in Rectangular Containers
  • The case where the lid is heated and the bottom is cooled: analysis
  • The case where the lid is heated and the bottom is cooled: some results
  • Convection fields with other wall heating protocols

• Thermal Convection Involving a Free Surface

• Thermocapillary or Marangoni Flows

The following sections are included:

• Three-Dimensional Flows

• A Useful General Solution for the Stokes System

• Other General Solutions of the Stokes Equations

• An Atypically Simple Three-Dimensional Field: Eddies in an Infinite Channel

• Critical Points, Limit Cycles and Vortices in Three-Dimensional Flows

The following sections are included:

• The Governing Equations

• The Derivation of the Vector Eigenfunctions

• Expansion of the Field in Terms of the Vector Eigenfunctions

• The Flow Field in the Cylinder
  • The flow field in a deep cylinder
  • The primary eddy structure in the cylinder and the corner eddies
  • Corner eddy merger and the evolution of the primary eddies with h

• Concluding Remarks

The following sections are included:

• Analysis of the Three-Dimensional Field

• The Field Structure When the Bottom Wall Alone Moves

• Flow Fields Generated When Both Boundaries Move

• Summary, Implications and Extensions

The following sections are included:

• Moffatt Eddies in the Cone
  • Similarity solutions and the inner eigenfunctions
  • Moffatt eddies
  • The flow in the lid driven cone
  • The outer eigenfunctions and the flow field in a conical container
  • Flow fields in the cone and in the conical container

• Corner Eddies in the Wedge
  • The work of Sano and Hasimoto
  • The studies of Moffatt and collaborators
  • The field near the corner of a semi-wedge

• Summary of Results

The following sections are included:

• The Nature of the Difficulties in This Geometry

• The Analysis of the Field in the Container
  • The z-symmetric/antisymmetric decomposition
  • A new set of z-antisymmetric three-dimensional eigenfunctions in Cartesian coordinates
  • The solution of the boundary value problem for the z-antisymmetric field
  • The boundary value problem for the z-symmetric field
  • The calculation of the field in the container

• The Nature of the Recirculating Flow Field in the Container
  • Some comparisons with other work
  • Fields in containers of spanwise aspect ratio
  • Containers of spanwise aspect ratio
  • A very large aspect ratio container,
  • A case where the lid speed vanishes for y > y₀,

The following sections are included:

• Introduction

• Analysis
  • The field equations and the solution strategy
  • The temperature field
  • A solution of the inhomogeneous Stokes equations
  • The radial eigenfunctions: Stokes flow eigenfunctions satisfying homogeneous boundary conditions on the cylindrical sidewall
  • Axial eigenfunctions: Stokes flow eigenfunctions satisfying homogeneous boundary conditions on the flat endwalls
  • Some useful Stokes flow fields
  • The solution of the natural convection problem

• Natural Convective Flow Fields

• Some Other Applications of the Above Extensions

All the flow fields that have been discussed in detail so far have been steady fields, i.e. ones that do not change with time. Needless to say, in reality these flow fields must have been unsteady during the time taken for them to have been established from some initial quiescent state. There are also situations where the mechanisms generating the motion are themselves unsteady; in such cases, the fields also will be unsteady. These aspects which have so far been ignored will now be addressed…

The following sections are included:

• Formulation of the Unsteady Problem

• The Eigenvalue Problem and the Distribution of the Eigenvalues

• The Eigenfunction Expansion for the Velocity Field

• Oscillatory Flow Fields
  • The periodic field in an infinitely deep container
  • The general eddy structure in containers of finite depth
  • The mechanism for the reversal of the general circulation
  • The formation and decay of the second primary eddy, h = 1.5
  • Corner eddies
  • Moderate Reynolds numbers

• Applications

The following sections are included:

• Computation of Velocity Fields

• Particle Trajectories and Their Computation

• Mixing Estimates

• Mixing in a DLDC
  • Experiment
  • Computation
  • Mixing calculations
    • Case C₁
    • Case C₂

• Mixing Calculations with Inertial Effects

• Summary

The following sections are included:

• Fluid Interfaces, Free Surfaces and Contact Lines

• Formulation of the Problem

• The Solution
  • The case of infinite depth
    • An eigenvalue problem for Ω
  • The case of finite depth

• Results and Discussion
  • The case of infinite depth
  • The case of finite depth

The following sections are included:

• The Oseen Equations for External Flows

• Different View Points Regarding the Use of the Oseen Equations

• The Embedding Method for External Flows

The following sections are included:

• The Flow Past a Circular Cylinder
  • The Filon eigenfunctions
  • The flow past a stationary circular cylinder
  • The lift and drag coefficients
  • The rotating cylinder in uniform flow

• Flows Past Cylindrical Bluff Bodies of Essentially Arbitrary Shape

• The Flow Past an Ellipse

• Flows Past More Complex Shapes

The following sections are included:

• The Flow Past an Elliptic Cylinder at an Angle of Attack
  • Elliptic coordinates
  • Formulation in elliptic coordinates
  • Separable solutions in elliptic coordinates and solutions aperiodic in η
  • An expansion for the field in terms of the Hasimoto eigenfunctions
  • Viscous flow past elliptic sections

• Aerofoils in Viscous Flow
  • A symmetrical section: the NACA 0012 aerofoil
  • A cambered section: the NACA 4412 aerofoil

• Further Directions

The following sections are included:

• Appendix 1 Spherical Harmonics

• Appendix 2 Mathieu Functions

• Appendix 3 Aerofoil Sections

• Bibliography & Author Index

• Subject Index