Physical Hydrodynamics

by Guyon, Etienne; Hulin, Jean-Pierre; Petit, Luc; Mitescu, Catalin D.

• ISBN13: 9780198517467
• ISBN10: 0198517467
• Format: Hardcover
• Copyright: 2001-06-28
• Publisher: Oxford University Press

Summary

In the course of the last twenty years, teaching and research in fluid mechanics has expanded considerably into the physics and chemistry communities, who in their turn developed new approaches to the teaching of this topic. These approaches are mainly oriented towards the comprehension offluids of different hierarchies, the development of various experimental tools, and explanations in terms of elementary physical mechanics. Physical Hydrodynamics presents this original approach for the first time. The elementary microscopic basics of the statistical theory of fluids is discussed,as are the classical aspects of deformation and pressure and the laws of conservation. The problem of Low-Reynolds-Number Flows will be addressed, its applications to suspensions and porous media explained. A discussion of the aspects of boundary layers, high-velocity flows and instabilitiesconclude this presentation of incompressible fluid hydrodynamics. The present book provides a thorough introduction into the topic from a primarily physical point of view and will be a useful textbook and reference work for graduate students, lecturers and researchers.

Table of Contents

The physics of fluids	p. 1
The liquid state	p. 1
The different states of matter: model systems and real media	p. 2
The solid--liquid transition: a sometimes nebulous process	p. 7
Macroscopic transport coefficients	p. 8
Thermal conductivity	p. 9
Mass diffusion	p. 18
Microscopic models for transport coefficients	p. 21
A different approach to mass diffusion: the random walk	p. 21
Transport coefficients for an ideal gas	p. 24
Diffusive transport phenomena in liquids	p. 28
Surface and surface tension effects	p. 31
Surface tension	p. 31
The pressure difference between the two sides of a curved interface: Laplace's law	p. 32
Variations in the surface tension due to a surfactant	p. 35
The Rayleigh--Taylor instability	p. 37
The spectroscopy of liquids	p. 40
Some common techniques for probing the microscopic structure of liquids	p. 40
The form factor and elastic X-ray diffraction: an example of the use of scattering on an atomic scale	p. 42
Elastic and quasi-elastic scattering of light: a tool for the study of the structure and diffusive transport in liquids	p. 46
Inelastic scattering of light in liquids	p. 52
Typical orders of magnitude for a number of physical parameters characteristic of the interfacial properties of ordinary liquids	p. 55
The diffusion of momentum under various flow conditions	p. 57
Diffusive and convective momentum transport in flowing fluids	p. 57
Diffusion and convection of momentum: two illustrative experiments	p. 57
Momentum transport in shear flow: an introduction to the concept of viscosity	p. 59
Microscopic models of viscosity	p. 64
The viscosity of gases	p. 64
The viscosity of liquids	p. 67
Numerical simulation of the particle trajectories in a flowing fluid	p. 69
A comparison of diffusion and convection mechanisms	p. 71
The Reynolds number	p. 71
Convective and diffusive mass and heat transport	p. 73
The description of different flow regimes	p. 76
Different flow regimes in the wake of a cylinder	p. 77
Transitions in the shedding of vortices behind a cylinder: the Landau model	p. 79
The kinematics of fluids	p. 89
The description of motion of a fluid	p. 89
Characteristic linear scales and the hypothesis of continuity	p. 89
The Eulerian and Lagrangian descriptions of fluid motion	p. 90
Acceleration of a particle of fluid	p. 91
Streamlines and stream-tubes, pathlines, and streaklines	p. 93
Visualization of flows	p. 95
Deformations in flows	p. 99
The local components of the velocity gradient field	p. 100
Analysis of the symmetric component: pure strain (deformation)	p. 100
Analysis of the antisymmetric component: pure rotation	p. 104
Small and large deformations	p. 106
The conservation of mass in fluid flow	p. 110
The equation of continuity	p. 110
The incompressibility of a fluid	p. 112
Analogies with electromagnetic theory	p. 114
The stream function	p. 115
The introduction and significance of the stream function	p. 115
Examples of two-dimensional flows and of their stream functions	p. 117
Axially symmetric flows	p. 121
Some measurements of velocity and of velocity gradients in fluid flows	p. 122
Measurement of the local velocity of a fluid: laser Doppler anemometry	p. 122
Determination of the local velocity gradients	p. 125
The dynamics of fluids: local equations	p. 128
Surface forces	p. 128
The general expression for the surface forces	p. 128
The characteristics of the viscous shear stress tensor	p. 132
The viscous shear stress for a Newtonian fluid	p. 134
Non-Newtonian fluids	p. 136
The equation of motion for a fluid	p. 140
The general equation for the dynamics of a fluid	p. 140
The Navier-Stokes equation of motion for a Newtonian fluid	p. 142
Euler's equation of motion for an ideal fluid	p. 143
The dimensionless form of the Navier-Stokes equation	p. 143
Boundary conditions for fluid flow	p. 144
The boundary condition at a solid wall	p. 144
Boundary conditions at the interface between two fluids: surface tension effects	p. 145
A few specific solutions of the Navier-Stokes equations	p. 147
The Navier-Stokes equation for one-dimensional flow	p. 147
Simple shear flow (plane Couette flow)	p. 148
Poiseuille flow (a viscous fluid flowing in a stationary conduit)	p. 149
Oscillating flows in a viscous fluid	p. 155
Flow driven by a gradient in the surface tension (the Marangoni effect)	p. 160
Cylindrical Couette flow	p. 163
Representation of the stress tensor, the equation of continuity, and the Navier-Stokes equations, for Newtonian fluids, in the most commonly used co-ordinate systems	p. 167
Cartesian co-ordinates (x, y, z)	p. 167
Cylindrical co-ordinates ([rho], [open phi], z)	p. 167
Spherical polar co-ordinates (r, [theta], [open phi])	p. 168
The conservation laws	p. 170
Conservation of mass	p. 170
Conservation of momentum	p. 171
The local equation	p. 171
The integral expression of the law of conservation of momentum	p. 172
The conservation of kinetic energy: Bernoulli's equation	p. 176
The conservation of energy for a flowing incompressible fluid with or without viscosity	p. 177
Bernoulli's equation: applications	p. 180
Applications of the laws of conservation of energy and momentum	p. 189
A jet incident on to a plane	p. 189
The exit jet from an opening in a reservoir	p. 192
The force on the walls of an axially symmetric conduit with variable cross-section	p. 194
The hydraulic jump	p. 197
Another application: a discharge sluice gate in a channel	p. 205
Potential flow	p. 208
Introduction	p. 208
Definitions, properties, and examples of potential flow	p. 210
Characteristics and examples of the velocity potential	p. 210
The uniqueness of the velocity potential	p. 210
Velocity potentials for simple flows and combinations of potential functions	p. 214
Examples of simple potential flows	p. 221
Forces acting on an obstacle in potential flow	p. 230
Two-dimensional flow	p. 230
The case of an obstacle in three dimensions	p. 236
Linear surface waves on an ideal fluid	p. 240
Swells, cat's paws, and breaking waves	p. 241
Trajectories of fluid particles during the passing of a wave	p. 245
Solitons	p. 246
An electrical analogue for two-dimensional potential flows	p. 248
Direct analogue	p. 249
Inverse analogue	p. 249
The complex velocity potential	p. 252
The definition of a complex potential	p. 252
Complex potentials for several types of flow	p. 253
Conformal mapping	p. 256
Velocity potentials and stream functions for two-dimensional flows	p. 266
Appendix A2	p. 267
Derivation of the velocity components from the stream function	p. 267
Derivation of the velocity components from the velocity potential	p. 267
Vorticity: dynamics of vortices	p. 268
Vorticity and its electromagnetic analogue	p. 268
The vorticity vector	p. 268
The electromagnetic analogue	p. 269
Straight vortex tubes: the analogy with the magnetic field due to a current-carrying wire	p. 271
The application of the electromagnetic analogy in dealing with arbitrary distributions of vorticity	p. 277
The dynamics of circulation	p. 279
Kelvin's theorem: the conservation of circulation	p. 280
Sources of circulation in the flow of viscous or compressible fluids, or in the presence of non-conservative forces	p. 284
The dynamics of vorticity	p. 289
The transport equation for vorticity, and its consequences	p. 289
Equilibrium between elongation and diffusion in the dynamics of vorticity	p. 295
A few examples of distributions of vorticity concentrated along singularities: systems of vortex lines	p. 298
A few cases with vorticity concentrated in vortex filaments	p. 298
The dynamics of a system of parallel line vortices	p. 300
Vortex rings	p. 305
Flow at low Reynolds numbers	p. 311
Examples of low-Reynolds-number flows	p. 311
The equation of motion at low Reynolds number	p. 313
The Stokes equation	p. 313
Further equivalent representations of the Stokes equation	p. 314
Properties of solutions of the Stokes equation	p. 315
Dimensional-analysis predictions for flows at low Reynolds number	p. 323
The forces and torques acting on a moving solid body	p. 324
Linear proportionality between the velocity of the solid body and the external forces	p. 325
General symmetry properties of the tensors A[subscript ij], B[subscript ij], C[subscript ij], and D[subscript ij]	p. 326
The effect of the symmetry of solid bodies on the applied forces and torques	p. 327
Uniform-velocity motion of a sphere in a viscous fluid	p. 333
The velocity field around a moving sphere	p. 333
The force acting on a moving sphere in a fluid of infinite extent: the drag coefficient	p. 338
The generalization of the solution of the Stokes equation to other experiments	p. 340
Limitations on the Stokes treatment of flow at low Reynolds numbers: the Oseen equation	p. 343
Quasi-parallel flows at low Reynolds numbers: lubrication	p. 347
Dynamics of suspensions	p. 351
The rheology of suspensions	p. 352
Sedimentation of particles in a suspension	p. 357
Flow in porous media	p. 361
A few characteristic examples of the different types of flows	p. 361
Parameters characterising a porous medium	p. 362
Flow in porous media: Darcy's law	p. 366
Permeability models for media with cylindrical pores	p. 370
The permeability of porous media containing channels of variable cross-section	p. 373
The flow of immiscible fluids in a porous medium	p. 377
Laminar boundary layers	p. 383
Introduction	p. 383
A qualitative physical discussion of the structure of the boundary layer near a flat plate in uniform flow	p. 385
The equations of motion within the boundary layer: Prandtl theory	p. 388
The equations of motion near a flat plate	p. 388
Transport of vorticity in the boundary layer	p. 390
Self-similarity of the velocity profiles in the boundary layer for the case of uniform, constant, external velocity	p. 390
Velocity profiles within boundary layers	p. 393
The Blasius equation for uniform external flow along a flat plate	p. 393
An approximate solution of the Blasius equation	p. 394
The frictional force on a flat plate in a uniform flow	p. 397
The thickness of boundary layers	p. 397
The hydrodynamic stability of a laminar boundary layer: turbulent boundary layers	p. 399
The laminar boundary layer in the presence of an external pressure gradient: boundary layer separation	p. 400
A simplified physical treatment of the problem	p. 400
Self-similar velocity profiles: flows such that U(x) = Cx[superscript m]	p. 401
Boundary layers with constant thickness	p. 406
Flows lacking self-similarity: boundary layer separation	p. 407
The practical consequences of boundary layer separation	p. 409
Separation of turbulent boundary layers: the decrease of the drag force	p. 409
A few applications of boundary layer separation problems in aerodynamics	p. 412
The aerodynamics of airplane wings	p. 412
Controlling boundary layer separation by suction	p. 417
The control of boundary layer separation by adjustment of the profile of the solid object	p. 417
Thermal and mass boundary layers	p. 420
Thermal boundary layers	p. 421
Concentration boundary layers and polarography	p. 428
The laminar wake	p. 432
A qualitative approach to the problem	p. 432
The solution of the equation of motion in the wake far from the object	p. 433
The drag force on a body: the relationship with the velocity profile in the wake	p. 435
Hydrodynamic instabilities	p. 439
Thermal convection	p. 439
Convective transport equations for heat	p. 439
Thermal convection resulting from a horizontal temperature gradient	p. 440
The Rayleigh-Benard instability	p. 443
A description of the Rayleigh-Benard instability	p. 444
The mechanism of the Rayleigh-Benard instability, and orders of magnitude	p. 445
The two-dimensional solution of the Rayleigh-Benard problem	p. 448
Other examples of threshold instabilities	p. 455
The Taylor-Couette instability	p. 455
The Benard-Marangoni instability	p. 459
Other classes of instability	p. 462
The Kelvin-Helmholtz instability	p. 463
Poiseuille flow in a tube, and between parallel plates	p. 469
The role of the shape of the velocity and vorticity profiles	p. 470
Transition to chaos	p. 471
Experiments in fully developed turbulence	p. 476
Two-dimensional flows	p. 477
Three-dimensional flows	p. 479
Superfluid helium: an (almost) ideal fluid	p. 482
Important properties of Helium II at finite temperatures	p. 482
The two-fluid model for Helium II	p. 482
Quantization of the circulation of the superfluid velocity v[subscript s]	p. 483
Experimental evidence for the existence of a superfluid component flowing with no energy dissipation	p. 484
Vortices in superfluid helium	p. 485
The existence of vortex filaments in superfluid helium	p. 485
Setting a volume of superfluid helium in rotation	p. 485
Experimental evidence for the quantisation of circulation in superfluid helium: the Hall and Vinen experiment	p. 486
Dynamics of vortex rings in superfluid helium	p. 488
Bibliography	p. 489
Index	p. 496
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