The aim of this book is to bring together classical and recent developments in the particular field of Newtonian flow at low Reynolds numbers. The methods are developed from first principles, alternative formulations are compared, a variety of configurations are addressed, the proper mathematical framework is discussed in the context of functional analysis and integral-equation-theory, and procedures of numerical solution in the context of the boundary element method are introduced. The text contains a fair amount of original material pertaining, in particular, to the properties and explicit form of the Green's functions, and the theory of the integral equations that arise from boundary integral representations.
Author(s): C. Pozrikidis
Series: Cambridge Texts in Applied Mathematics
Publisher: CUP
Year: 1992
Language: English
Pages: 271
Contents......Page 7
Preface......Page 11
1.1 Linearization of the equations of fluid flow......Page 13
1.2 The equations of Stokes flow......Page 15
1.3 Reversibility of Stokes flow......Page 19
1.4 The reciprocal identity......Page 21
1.5 Uniqueness of solution, significance of homogeneous boundary conditions, and further properties of Stokes flow......Page 26
1.6 Unsteady Stokes flow......Page 28
2.1 Green's functions of Stokes flow......Page 31
2.2 The free-space Green's function......Page 34
2.3 The boundary integral equation......Page 36
Infinite flow......Page 41
Simplification by the use of proper Green's functions......Page 42
Eliminating the single-layer or double-layer potential......Page 43
The boundary integral equation as a Fredholm integral equation......Page 47
2.4 Flow in an axisymmetric domain......Page 50
The far flow due to an immersed particle......Page 57
The viscosity of a suspension of force-free particles......Page 60
Generalized Faxen relations......Page 63
Boundary effects on the motion of a particle......Page 67
2.6 Two-dimensional Stokes flow......Page 70
The two-dimensional Stokeslet......Page 72
The boundary integral equation......Page 73
Semi-direct and indirect boundary integral methods......Page 74
The unsteady Stokeslet......Page 78
The boundary integral equation......Page 80
2.8 Swirling flow......Page 83
3.1 Properties of Green's functions......Page 88
3.2 Properties of the pressure and stress......Page 92
3.3 Computation of Green's functions......Page 94
Flow bounded by an infinite plane wall......Page 96
Flow bounded internally by a solid sphere......Page 99
3.4 Axisymmetric flow......Page 100
A ring of point forces in a cylindrical tube......Page 101
An array of rings of point forces in a cylindrical tube......Page 103
A point force above an infinite plane wall......Page 105
An array of point forces in an infinite fluid......Page 106
An array of point forces above a plane wall......Page 107
A point force between two parallel plane walls......Page 108
A point force in the exterior of a circular cylinder......Page 111
3.6 Unsteady flow......Page 112
4 Generalized boundary integral methods......Page 115
4.1 The single-layer potential......Page 116
4.2 Representation of a flow in terms of a single-layer potential......Page 119
4.3 The double-layer potential......Page 121
4.4 Representation of a flow in terms of a double-layer potential......Page 125
4.5 The eigenvalues of the double-layer potential......Page 126
Internal flow......Page 128
External flow......Page 129
4.6 Regularizing the double-layer potential: removing the marginal eigenvalues......Page 132
Removing the jJ = 1 eigenvalue......Page 133
Removing both eigenvalues......Page 136
4.7 A compound double-layer representation for external flow......Page 139
4.8 The resistance problem......Page 142
4.9 The mobility problem......Page 145
5.1 Introduction......Page 151
5.2 The boundary integral formulation......Page 153
5.3 The single-layer formulation......Page 155
5.4 Investigation of the integral equations......Page 157
5.5 The discontinuity in the interfacial surface force......Page 159
Interfaces with isotropic tension......Page 160
Interfaces with elastic behaviour......Page 163
Incompressible interfaces......Page 167
Evolving interfaces......Page 168
Stationary interfaces......Page 169
6.1 General procedures......Page 171
6.2 Boundary element representations......Page 174
Planar boundaries......Page 175
Three-dimensional boundaries......Page 179
Three-dimensional boundary elements......Page 183
6.3 Adaptive representation of evolving planar boundaries......Page 188
6.4 Numerical computation of the boundary integrals......Page 189
6.5 Accuracy of boundary element methods......Page 192
6.6 Computer implementations......Page 194
7.1 Introduction......Page 202
Free-space singularities......Page 204
Singularities of bounded flow......Page 208
Singularities of internal flow......Page 210
Two-dimensional flow......Page 211
7.3 Singularity representations......Page 213
A translating solid sphere......Page 214
A sphere in linear flow......Page 215
Flow due to a rotating sphere......Page 217
A translating liquid drop......Page 218
Thin and slender bodies......Page 220
7.4 Numerical methods......Page 222
7.5 Unsteady flow......Page 224
Answers and keys......Page 227
References......Page 261
Index......Page 269
