Introductory Fluid Mechanics for Physicists and Mathematicians

490 Pages·2013·6.377 MB·English

by  Pert, Geoffrey

INTRODUCTORY FLUID MECHANICS
for
Physicists and Mathematicians
Frontispiece The first observation of shock waves generated by the passage of a bullet
throughairbyMachandSalcher(1887).Behindtheprimaryshockcanbeseenthefollowing
rarefaction. Note also the turbulet wake leaving the rear of the bullet. Since the bullet was
blunt the shock is slightly detached, although this cannot be clearly seen. The two vertical
lines are for timing. The photograph was taken using a shadowgraph technique.
INTRODUCTORY FLUID MECHANICS
for
Physicists and Mathematicians
Geoffrey J. Pert
Department of Physics, University of York, UK
A John Wiley & Sons, Ltd., Publication
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This book is dedicated
with grateful thanks and appreciation to
my doctoral supervisor
John Pain
who introduced me to the science of
shock waves and fluid dynamics
and who provided help and support
in times of stress
(even when we attacked the Tirpitz)

Contents
Preface xvii
1 Introduction 1
1.1 Fluids as a State of Matter . . . . . . . . . . . . . . . . . . . . 1
1.2 The Fundamental Equations for Flow of a Dissipationless Fluid 3
1.3 Lagrangian Frame . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . 6
1.3.2 Conservation of Momentum–Euler’s Equation . . . . . 7
1.3.3 Conservation of Angular Momentum . . . . . . . . . . 8
1.3.4 Conservation of Energy . . . . . . . . . . . . . . . . . . 8
1.3.5 Conservation of Entropy . . . . . . . . . . . . . . . . . 8
1.4 Eulerian Frame . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Conservation of Mass–Equation of Continuity . . . . . 8
1.4.2 Conservation of Momentum . . . . . . . . . . . . . . . 9
1.4.3 Conservation of Angular Momentum . . . . . . . . . . 10
1.4.4 Conservation of Energy . . . . . . . . . . . . . . . . . . 11
1.4.5 Conservation of Entropy . . . . . . . . . . . . . . . . . 11
1.5 Hydrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5.1 Isothermal Fluid–Thermal and Mechanical Equilibrium 12
1.5.2 Adiabatic Fluid–Lapse Rate . . . . . . . . . . . . . . . 12
1.5.3 Stability of an Equilibrium Configuration . . . . . . . . 15
1.6 Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Bernoulli’s Equation: Weak Form . . . . . . . . . . . . . . . . 16
1.8 Polytropic Gases . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8.1 Applications of Bernoulli’s Theorem . . . . . . . . . . 19
1.8.1.1 Vena Contracta . . . . . . . . . . . . . . . . . 19
1.8.1.2 Flow of gas along a pipe of varying cross-section 20
Casestudy1.I Munroe Effect–Shaped Charge Explosive . . 22
2 Flow of Ideal Fluids 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 26
vii
viii Contents
2.2.1 Vorticity and Helmholtz’s Theorems . . . . . . . . . . 27
2.2.1.1 Simple or rectilinear vortex . . . . . . . . . . 29
2.2.1.2 Vortex sheet . . . . . . . . . . . . . . . . . . . 29
2.3 Irrotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1 Crocco’s Equation . . . . . . . . . . . . . . . . . . . . . 31
2.4 Irrotational Flow–Velocity Potential and the Strong Form of
Bernoulli’s Equation . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Incompressible Flow–Streamfunction . . . . . . . . . . . . . . 33
2.5.1 Planar Systems . . . . . . . . . . . . . . . . . . . . . . 33
2.5.2 Axisymmetric Flow–Stokes Streamfunction . . . . . . 34
2.6 Irrotational Incompressible Flow . . . . . . . . . . . . . . . . . 35
2.6.1 Simply and Multiply Connected Spaces . . . . . . . . . 37
2.7 Induced Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7.1 Streamlined Flow around a Body Treated
as a Vortex Sheet . . . . . . . . . . . . . . . . . . . . . 41
2.8 Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.8.1 Doublet Sources . . . . . . . . . . . . . . . . . . . . . . 43
2.8.1.1 Doublet sheets . . . . . . . . . . . . . . . . . 45
2.8.2 Flow Around a Body Treated as a Source Sheet . . . . 45
2.8.3 Irrotational Incompressible Flow Around a Sphere . . . 48
Casestudy2.I Rankine Ovals . . . . . . . . . . . . . . . . . 49
2.9 Two-Dimensional Flow . . . . . . . . . . . . . . . . . . . . . . 51
2.9.1 Irrotational Incompressible Flow. . . . . . . . . . . . . 51
2.10 Applications of Analytic Functions in Fluid Mechanics . . . . 52
2.10.1 Flow from a Simple Source and a Simple Vortex . . . . 52
2.10.1.1 Free vortex . . . . . . . . . . . . . . . . . . . 53
2.10.1.2 Two-dimensional doublets and vortex loops . 55
2.10.2 Flow Around a Body Treated as a Sheet of Complex
Sources and Doublets . . . . . . . . . . . . . . . . . . . 55
Casestudy2.II Application of Complex Function Analysis to
the Flow around a Thin Wing . . . . . . . . . . . . . . 59
2.10.3 Flow Around a Cylinder with Zero Circulation . . . . . 62
2.10.4 Flow Around a Cylinder with Circulation . . . . . . . 64
2.10.5 The Flow Around a Corner . . . . . . . . . . . . . . . 66
2.11 Force on a Body in Steady Two-Dimensional Incompressible
Ideal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.12 Conformal Transforms . . . . . . . . . . . . . . . . . . . . . . 69
Appendix2.A Drag in Ideal Flow . . . . . . . . . . . . . . . . . . 70
2.A.1 Helmholtz’s Flow and Separation . . . . . . . . . . . . 71
2.A.2 Lines of Vortices . . . . . . . . . . . . . . . . . . . . . 72
2.A.2.1 Single infinite row of vortices . . . . . . . . . 72