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# Low Reynolds Number Hydrodynamics: with special applications to particulate media

## byJ. Happel, H. Brenner

### Paperback | September 30, 1983

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One studying the motion of fluids relative to particulate systems is soon impressed by the dichotomy which exists between books covering theoretical and practical aspects. Classical hydrodynamics is largely concerned with perfect fluids which unfortunately exert no forces on the particles past which they move. Practical approaches to subjects like fluidization, sedimentation, and flow through porous media abound in much useful but uncorrelated empirical information. The present book represents an attempt to bridge this gap by providing at least the beginnings of a rational approach to fluid particle dynamics, based on first principles. From the pedagogic viewpoint it seems worthwhile to show that the Navier-Stokes equations, which form the basis of all systematic texts, can be employed for useful practical applications beyond the elementary problems of laminar flow in pipes and Stokes law for the motion of a single particle. Although a suspension may often be viewed as a continuum for practical purposes, it really consists of a discrete collection of particles immersed in an essentially continuous fluid. Consideration of the actual detailed boundary value problems posed by this viewpoint may serve to call attention to the limitation of idealizations which apply to the overall transport properties of a mixture of fluid and solid particles.

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Title:Low Reynolds Number Hydrodynamics: with special applications to particulate mediaFormat:PaperbackDimensions:565 pagesPublished:September 30, 1983Publisher:Springer Netherlands

The following ISBNs are associated with this title:

ISBN - 10:9024728770

ISBN - 13:9789024728770

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Table of Contents

1. Introduction.- 1-1 Definition and purpose, 1. 1-2 Historical review, 8. 1-3 Application in science and technology, 13..- 2. The Behavior of Fluids in Slow Motion.- 2-1 The equations of change for a viscous fluid, 23. 2-2 Mechanical energy dissipation in a viscous fluid, 29. 2-3 Force and couple acting on a body moving in a viscous fluid, 30. 2-4 Exact solutions of the equations of motion for a viscous fluid, 31. 2-5 Laminar flow in ducts, 33. 2-6 Simplifications of the Navier-Stokes equations, especially for slow motion, 40. 2-7 Paradoxes in the solution of the creeping motion equations, 47. 2-8 Molecular effects in fluid dynamics, 49. 2-9 Non-newtonian flow, 51. 2-10 Unsteady creeping flows, 52..- 3. Some General Solutions and Theorems Pertaining to the Creeping Motion Equations.- 3-1 Introduction, 58. 3-2 Spherical coordinates, 62. 3-3 Cylindrical coordinates, 71. 3-4 Integral representations, 79. 3-5 Generalized reciprocal theorem, 85. 3-6 Energy dissipation, 88..- 4. Axisymmetrical Flow.- 4-1 Introduction, 96. 4-2 Stream function, 96. 4-3 Relation between stream function and local velocity, 98. 4-4 Stream function in various coordinate systems, 99. 4-5 Intrinsic coordinates, 100. 4-6 Properties of the stream function, 102. 4-7 Dynamic equation satisfied by the stream function, 103. 4-8 Uniform flow, 106. 4-9 Point source or sink, 106. 4-10 Source and sink of equal strength, 107. 4-11 Finite line source, 108. 4-12 Point force, 110. 4-13 Boundary conditions satisfied by the stream function, 111. 4-14 Drag on a body, 113. 4-15 Pressure, 116. 4-16 Separable coordinate systems, 117. 4-17 Translation of a sphere, 119. 4-18 Flow past a sphere, 123. 4-19 Terminal settling velocity, 124. 4-20 Slip at the surface of a sphere, 125. 4-21 Fluid sphere, 127. 4-22 Concentric spheres, 130. 4-23 General solution in spherical coordinates, 133. 4-24 Flow through a conical diffuser, 138. 4-25 Flow past an approximate sphere, 141. 4-26 Oblate spheroid, 145. 4-27 Circular disk, 149. 4-28 Flow in a venturi tube, 150. 4-29 Flow through a circular aperture, 153. 4-30 Prolate spheroid, 154. 4-31 Elongated rod, 156. 4-32 Axisymmetric flow past a spherical cap, 157..- 5. The Motion of a Rigid Particle of Arbitrary Shape in an Unbounded Fluid.- 5-1. Introduction, 159. 5-2 Translational motions, 163. 5-3 Rotational motions, 169. 5-4 Combined translation and rotation, 173. 5-5 Symmetrical particles, 183. 5-6 Nonskew bodies, 192. 5-7 Terminal settling velocity of an arbitrary particle, 197. 5-8 Average resistance to translation, 205. 5-9 The resistance of a slightly deformed sphere, 207. 5-10 The settling of spherically isotropic bodies, 219. 5-11 The settling of orthotopic bodies, 220..- 6. Interaction between Two or More Particles.- 6-1 Introduction, 235. 6-2 Two widely spaced spherically isotropic particles, 240: 6-3 Two spheres by the method of reflections and similar techniques, 249. 6-4 Exact solution for two spheres falling along their line of centers, 270. 6-5 Comparison of theories with experimental data for two spheres, 273. 6-6 More than two spheres, 276. 6-7 Two spheroids in a viscous liquid, 278. 6-8 Limitations of creeping motion equations, 281..- 7. Wall Effects on the Motion of a Single Particle.- 7-1 Introduction, 286. 7-2 The translation of a particle in proximity to container walls, 288. 7-3 Sphere moving in an axial direction in a circular cylindrical tube, 298. 7-4 Sphere moving relative to plane walls, 322. 7-5 Spheroid moving relative to cylindrical and plane walls, 331. 7-6 k-coefficients for typical boundaries, 340. 7-7 One- and two-dimensional problems, 341. 7-8 Solid of revolution rotating symmetrically in a bounded fluid, 346. 7-9 Unsteady motion of a sphere in the presence of a plane wall, 354..- 8. Flow Relative to Assemblages of Particles.- 8-1 Introduction, 358. 8-2 Dilute systems-no interaction effects, 360. 8-3 Dilute systems-first-order interaction effects, 371. 8-4 Concentrated systems, 387. 8-5 Systems with complex geometry, 400. 8-6 Particulate suspensions, 410. 8-7 Packed beds, 417. 8-8 Fluidization, 422..- 9. The Viscosity of Particulate Systems.- 9-1 Introduction, 431. 9-2 Dilute systems of spheres-no interaction effects, 438. 9-3 Dilute systems-first-order interaction effects, 443. 9-4 Concentrated systems, 448. 9-5 Nonspherical and nonrigid particles, 456. 9-6 Comparison with data, 462. 9-7 Non-newtonian behavior, 469..- Appendix A. Orthogonal Curvilinear Coordinate Systems.- A-l Curvilinear coordinates, 474. A-2 Orthogonal curvilinear coordinates, 477. A-3 Geometrical properties, 480. A-4 Differentiation of unit vectors, 481. A-5 Vector differential invariants, 483. A-6 Relations between cartesian and orthogonal curvilinear coordinates, 486. A-7 Dyadics in orthogonal curvilinear coordinates, 488. A-8 Cylindrical coordinate systems, 490. A-9 Circular cylindrical coordinates, 490. A-10 Conjugate cylindrical coordinate systems, 494. A-ll Elliptic cylinder coordinates, 495. A-12 Bipolar cylinder coordinates, 497. A-l3 Parabolic cylinder coordinates, 500. A-14 Coordinate systems of revolution, 501. A-l5 Spherical Coordinates, 504. A-l6 Conjugate coordinate systems of revolution, 508. A-17 Prolate spheroidal coordinates, 509. A-18 Oblate spheroidal coordinates, 512. A-19 Bipolar coordinates, 516. A-20 Toroidal coordinates, 519. A-21 Paraboloidal Coordinates, 521..- Appendix B. Summary of Notation and Brief Review of Polyadic Algebra.- Name Index.