Slow Viscous Flows

Qualitative Features And Quantitative Analysis Using Complex Eigenfunction Expansions (With Cd-rom)

By: P N Shankar

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Paperback | 26 September 2007

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600 Pages

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This unique book provides a unified and systematic account of internal, external and unsteady slow viscous flows, including the latest advances of the last decade, some of which are due to the author. The book shows how the method of eigenfunctions, in conjunction with least squares, can be used to solve problems of low Reynolds number flows, including three-dimensional internal and unsteady flows, which until recently were considered intractable. Although the methods used are quantitative, much stress is laid on understanding the qualitative nature of these intriguing flows. A secondary purpose of the book is to explain how the complex eigenfunction method can be used to solve problems in science and engineering.Although primarily aimed at graduate students, academics and research engineers in the areas of fluid mechanics and applied mathematics, care has been taken, through the use of numerous diagrams and much discussion, to explain to the non-specialist the qualitative features of these complex flows.

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It should be a valuable addition to the literature, presenting an in-depth treatment of a subject not covered elsewhere in similar detail. -- Mathematical Reviews "Mathematical Reviews"

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Non-Fiction Science Physics Classical Mathematics Fluid Mechanics Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Engineering & Technology Mechanical Engineering & Materials Materials Science Mechanics of Fluids

Flow Applied Mathematics Science in General Booktopia Publisher Services World Scientific Publishing

Foreword	p. v
Preface	p. vii
Acknowledgments	p. xi
Introduction	p. xxiii
Physical and Mathematical Background	p. 1
Physical Aspects of the Motion of a Viscous Fluid	p. 3
The Continuity, Navier-Stokes and Energy Equations	p. 3
The Boundary Conditions	p. 9
Non-Dimensionalization: The Reynolds and Other Dimensionless Numbers	p. 11
Slow Viscous or Low Reynolds Number Flows: The Stokes Equation	p. 13
The Pressure and the Vorticity	p. 15
Streamlines and the Stream Function	p. 17
The Method of Eigenfunction Expansions	p. 21
Fourier Series	p. 22
A Boundary Value Problem for Laplace's Equation	p. 25
Convergence of Fourier Series and Completeness of the Eigenfunctions	p. 28
The Relationship of the Eigenfunction Expansions of Classical Physics to the Ones That will Arise in This Book	p. 34
The Method of Least Squares	p. 37
Curve fitting by least squares	p. 37
Least squares and the expansion of an arbitrary function in terms of an infinite, complete set of functions	p. 40
Least squares and Fourier series	p. 42
Extensions to more difficult, non-classical situations: complex eigenfunction expansions	p. 46
Eigenfunction expansions involving a single, infinite sequence of complex scalars	p. 48
Eigenfunction expansions involving an infinite sequence of real scalars and an infinite sequence of complex scalars	p. 50
Systematic Procedures for Finding the Real and Complex Roots of Transcendental Equations	p. 53
Steady Planar Flows	p. 59
Two-Dimensional Flow in a Rectangular Container Generated by the Motion of the Walls	p. 61
The Field Equations, the Stream Function and Boundary Conditions	p. 63
Symmetric and Antisymmetric Eigenfunctions for the Rectangle	p. 65
Symmetric-Antisymmetric Decomposition in z	p. 69
The Solution of the General Boundary Value Problem in the Rectangle	p. 70
A rectangular container of infinite depth	p. 71
A rectangular container of finite depth h	p. 73
Corner Data, Some Computational Details and Accuracy	p. 76
The Flow Field and Eddy Structure in the Lid Driven Container	p. 87
The primary eddy structure	p. 87
The shallow container	p. 89
Corner eddies and the evolution of the primary eddies with increasing h	p. 91
The pressure and vorticity fields in the container	p. 93
The superiority of using velocity boundary conditions	p. 97
Some General Conclusions on Rectangular Flow Fields	p. 100
Other Methods of Solution	p. 102
Biorthogonality and methods based on its use	p. 102
Boundary integral and singularity methods	p. 106
The superposition method or the method of real Fourier series	p. 108
Other approaches	p. 109
Similarity Solutions, Streamline Patterns and Eddies in Planar Flows	p. 111
Separable Solutions of the Stokes Equations in Plane Polar Coordinates	p. 112
Similarity Solutions of the Paint-Scraper Problem and One with a Free Surface	p. 113
Flow Near a Sharp Corner: The Corner or Moffatt Eddies	p. 115
Flow Near a Sharp Corner: Large Corner Angles	p. 121
Corner Flows with a Source at the Vertex	p. 123
Isolated Stagnation Points in Planar Flows	p. 124
Viscous Flow Fields Near Points on the Boundary Where the Skin Friction Vanishes	p. 126
Blocked Flows	p. 129
Some Interesting Examples	p. 131
Limitations	p. 135
The Method of Embedding for Complex Geometries	p. 138
The Embedding Method	p. 139
The Method of Embedding for Planar Stokes Flows	p. 141
A complete set of Stokes flow eigenfunctions in the rectangle	p. 141
A complete set of Stokes flow eigenfunctions in the circle	p. 144
Planar Stokes flow in a container of arbitrary shape	p. 146
Stokes flow in an annular sector: comparison with the results of Krasnopolskaya et al.	p. 150
Applications to Non-Simple Geometries	p. 154
The effect of container height on the eddy structure in a round-bottomed container	p. 154
A container with a trench-like bottom	p. 155
Eddy structure in a container with a concave bottom	p. 158
Fields in containers which are not symmetric about x = 1/2	p. 161
The Strengths and Limitations of the Embedding Method	p. 163
On Mathematical Issues and Some Special Two-Dimensional Flows	p. 171
Mathematical Issues	p. 172
Canonical and non-canonical problems in the semi-infinite strip	p. 172
Eigenfunction expansions and completeness	p. 173
Convergence of the expansions	p. 176
The non-uniform nature of the Stokes equations	p. 178
The physical and mathematical bases of the singular LDC problem	p. 178
Non-Homogeneous and Discontinuous Corner Data in the Eigenfunction Method	p. 181
The Singular LDC Problem in the Rectangular Geometry	p. 183
Stokes Flow in Other Special Geometries	p. 187
The double lid driven container	p. 190
Flow in an annulus and in an annular sector	p. 194
Flow in an infinite wedge driven by belts on the sidewalks	p. 196
Meniscus roll coating and stress free sidewalls: real eigenvalues	p. 200
Cusped interfaces in low Reynolds number flows	p. 203
Planar Convective Flows	p. 207
The Field Equations, Non-Dimensionalization and the Boussinesq Approximation	p. 210
Planar Natural and Mixed Convection in Rectangular Containers	p. 213
The case where the lid is heated and the bottom is cooled: analysis	p. 213
The case where the lid is heated and the bottom is cooled: some results	p. 216
Convection fields with other wall heating protocols	p. 220
Thermal Convection Involving a Free Surface	p. 222
Thermocapillary or Marangoni Flows	p. 230
Steady Three-Dimensional Flows	p. 237
Some General Features of Three-Dimensional Flows	p. 239
Three-Dimensional Flows	p. 239
A Useful General Solution for the Stokes System	p. 240
Other General Solutions of the Stokes Equations	p. 242
An Atypically Simple Three-Dimensional Field: Eddies in an Infinite Channel	p. 244
Critical Points, Limit Cycles and Vortices in Three-Dimensional Flows	p. 250
Three-Dimensional Stokes Flow in a Cylinder of Circular Section	p. 258
The Governing Equations	p. 259
The Derivation of the Vector Eigenfunctions	p. 260
Expansion of the Field in Terms of the Vector Eigenfunctions	p. 264
The Flow Field in the Cylinder	p. 267
The flow field in a deep cylinder	p. 267
The primary eddy structure in the cylinder and the corner eddies	p. 269
Corner eddy merger and the evolution of the primary eddies with h	p. 273
Concluding Remarks	p. 279
Three-Dimensional Flow in a Cylindrical Liquid Bridge	p. 280
Analysis of the Three-Dimensional Field	p. 281
The Field Structure When the Bottom Wall Alone Moves	p. 285
Flow Fields Generated When Both Boundaries Move	p. 292
Summary, Implications and Extensions	p. 298
Three-Dimensional Corner Eddies	p. 299
Moffatt Eddies in the Cone	p. 302
Similarity solutions and the inner eigenfunctions	p. 303
Moffatt eddies	p. 309
The flow in the lid driven cone	p. 316
The outer eigenfunctions and the flow field in a conical container	p. 318
Flow fields in the cone and in the conical container	p. 319
Corner Eddies in the Wedge	p. 323
The work of Sano and Hasimoto	p. 325
The studies of Moffatt and collaborators	p. 329
The field near the corner of a semi-wedge	p. 336
Summary of Results	p. 344
Stokes Flow in a Rectangular Container	p. 345
The Nature of the Difficulties in This Geometry	p. 346
The Analysis of the Field in the Container	p. 348
The z-symmetric/antisymmetric decomposition	p. 348
A new set of z-antisymmetric three-dimensional eigenfunctions in Cartesian coordinates	p. 349
The solution of the boundary value problem for the z-antisymmetric field	p. 356
The boundary value problem for the z-symmetric field	p. 359
The calculation of the field in the container	p. 360
The Nature of the Recirculating Flow Field in the Container	p. 360
Some comparisons with other work	p. 362
Fields in containers of spanwise aspect ratio A = 1	p. 363
Containers of spanwise aspect ratio A = 3	p. 368
A very large aspect ratio container, A = 6	p. 371
A case where the lid speed vanishes for y > yo, A = 6	p. 373
Three-Dimensional Convection in a Cylindrical Container	p. 377
Introduction	p. 377
Analysis	p. 378
The field equations and the solution strategy	p. 378
The temperature field	p. 381
A solution of the inhomogeneous Stokes equations	p. 381
The radial eigenfunctions: Stokes flow eigenfunctions satisfying homogeneous boundary conditions on the cylindrical sidewall	p. 382
Axial eigenfunctions: Stokes flow eigenfunctions satisfying homogeneous boundary conditions on the flat endwalls	p. 384
Some useful Stokes flow fields	p. 386
The solution of the natural convection problem	p. 387
Natural Convective Flow Fields	p. 388
Some Other Applications of the Above Extensions	p. 394
Unsteady Flows	p. 397
Slow Unsteady Viscous Flows	p. 399
The Oscillating Lid Driven Container	p. 404
Formulation of the Unsteady Problem	p. 404
The Eigenvalue Problem and the Distribution of the Eigenvalues	p. 406
The Eigenfunction Expansion for the Velocity Field	p. 410
Oscillatory Flow Fields	p. 412
The periodic field in an infinitely deep container	p. 413
The general eddy structure in containers of finite depth	p. 415
The mechanism for the reversal of the general circulation	p. 419
The formation and decay of the second primary eddy, h = 1.5	p. 422
Corner eddies	p. 423
Moderate Reynolds numbers	p. 424
Applications	p. 425
Mixing by Advection in Lid Driven Flows	p. 428
Computation of Velocity Fields	p. 430
Particle Trajectories and Their Computation	p. 431
Mixing Estimates	p. 432
Mixing in a DLDC	p. 433
Experiment	p. 435
Computation	p. 436
Mixing calculations	p. 437
Case C[subscript 1]	p. 437
Case C[subscript 2]	p. 438
Mixing Calculations with Inertial Effects	p. 443
Summary	p. 445
Viscous Wave Attenuation	p. 446
Fluid Interfaces, Free Surfaces and Contact Lines	p. 446
Formulation of the Problem	p. 447
The Solution	p. 450
The case of infinite depth	p. 450
An eigenvalue problem for [omega]	p. 451
The case of finite depth	p. 454
Results and Discussion	p. 455
The case of infinite depth	p. 455
The case of finite depth	p. 459
External Flows	p. 471
External Flows Past Bodies	p. 473
The Oseen Equations for External Flows	p. 474
Different View Points Regarding the Use of the Oseen Equations	p. 476
The Embedding Method for External Flows	p. 478
Planar Bluff Body Flows at Low Reynolds Numbers	p. 480
The Flow Past a Circular Cylinder	p. 481
The Filon eigenfunctions	p. 481
The flow past a stationary circular cylinder	p. 486
The lift and drag coefficients	p. 488
The rotating cylinder in uniform flow	p. 490
Flows Past Cylindrical Bluff Bodies of Essentially Arbitrary Shape	p. 495
The Flow Past an Ellipse	p. 498
Flows Past More Complex Shapes	p. 500
Low Reynolds Number Flows Past Streamlined Bodies	p. 505
The Flow Past an Elliptic Cylinder at an Angle of Attack	p. 506
Elliptic coordinates	p. 506
Formulation in elliptic coordinates	p. 508
Separable solutions in elliptic coordinates and solutions aperiodic in [eta]	p. 509
An expansion for the field in terms of the Hasimoto eigenfunctions	p. 512
Viscous flow past elliptic sections	p. 515
Aerofoils in Viscous Flow	p. 521
A symmetrical section: the NACA 0012 aerofoil	p. 521
A cambered section: the NACA 4412 aerofoil	p. 528
Further Directions	p. 532
Spherical Harmonics	p. 534
Mathieu Functions	p. 536
Aerofoil Sections	p. 543
Bibliography and Author Index	p. 545
Subject Index	p. 557
Table of Contents provided by Ingram. All Rights Reserved.

Slow Viscous Flows

Qualitative Features And Quantitative Analysis Using Complex Eigenfunction Expansions (With Cd-rom)

At a Glance

Paperback

Industry Reviews

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