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FLUID MECHANICSFoundations and Applications of Mechanics

Volume II, Third edition

C. S. Jog

Cambridge University Press978-1-107-09129-0 - Fluid Mechanics: Foundations and Applications of Mechanics: Volume II,: Third EditionC. S. JogFrontmatterMore information

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Library of Congress Cataloging in Publication DataJog, C. S.Fluid mechanics / Chandrashekhar S. Jog. – 3rd edition.pages cm– (Foundations and applications of mechanics ; volume II)Includes bibliographical references and index.Summary: “Discusses the fundamental theories related to kinematics and governingequations, hydrostatics, surface waves and ideal fluid flow, followed by theirapplications”– Provided by publisher.ISBN 978-1-107-09129-0 (hardback)1. Fluid mechanics. I. Title.QC145.2.J64 2015620.1’06–dc232014048627

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Contents

List of Figures viiiList of Tables xvPreface xviiNotation xix

1 Kinematics and Governing Equations 1

1.1 Tensor Identities 11.2 Kinematics 2

1.2.1 Lagrangian and Eulerian descriptions 21.2.2 Flow lines 51.2.3 Analysis of deformation 51.2.4 Vortex lines and circulation 8

1.3 Governing Equations 9

1.3.1 Conservation of mass 111.3.2 Balance of linear momentum 121.3.3 Balance of angular momentum 141.3.4 Principle of material frame-indifference 151.3.5 Definition of a fluid 211.3.6 The first law of thermodynamics 231.3.7 The second law of thermodynamics 261.3.8 The Navier–Stokes and energy equations 361.3.9 Summary of the governing equations for a Newtonian fluid 37

1.4 Nature of the Governing Equations 381.5 Boundary Conditions on Velocity and Temperature 39

1.5.1 Kinematic boundary conditions on the velocity field 401.5.2 Dynamical boundary conditions at a fluid-rigid solid interface 411.5.3 Dynamical boundary conditions at a fluid–fluid interface 421.5.4 Boundary conditions on temperature 52

1.6 Dimensionless Parameters 531.7 Special Forms of the Governing Equations 55

1.7.1 First integral of the momentum equation 561.7.2 Bernoulli’s equation 651.7.3 Kelvin’s circulation theorem 701.7.4 Helmholtz’s vorticity equation 72






iv Contents

1.8 Example Applications 73

1.8.1 Flow past a cylinder 731.8.2 von Karman momentum integral 761.8.3 Flow of inviscid fluid over a bump 781.8.4 Jet impacting on a moving plate 831.8.5 Rocket propulsion 871.8.6 Impulse turbine 881.8.7 Slip-stream analysis of a wind turbine 921.8.8 Flow through a pipe network 95

2 Hydrostatics 103

2.1 Force on a Plane Surface 1042.2 Forces on a Curved Surface 1062.3 Example Applications 110

3 Ideal Fluid Flow 119

3.1 Simplification of the Euler Equations for Potential Flow 1193.2 Two-Dimensional Potential Flow 1203.3 Circulation and Volume Flow Rate 1253.4 Elementary Complex Potentials 1263.5 Rankine Oval 1333.6 Uniform Flow Past a Cylinder 1353.7 Uniform Flow Past an Elliptic Cylinder/Motion

of an Elliptic Cylinder in a Stationary Fluid 1383.8 Method of Images 1393.9 Blasius Force Theorems 1423.10 Flow Past a Cylinder with Point Vortex at the Origin 1483.11 Circle Theorem 1513.12 Conformal Mapping 1543.13 Free-Streamline Theory 1623.14 Flow Inside/Outside a Rotating Prismatic Tube 1663.15 Two-Dimensional Flow with Vorticity 1713.16 Three-Dimensional Potential Flow 1733.17 Elementary Potentials 1783.18 Force on a Rigid Body in Potential Flow 1813.19 Flow Around a Sphere 1823.20 Flow Around a Prolate Spheroid 1843.21 Kinetic Energy of a Moving Fluid 1853.22 Motion of a Sphere Through a Stationary Fluid 1873.23 Motion of a Bubble through a Stationary Fluid 189






Contents v

4 Surface Waves 201

4.1 Governing Equations for Surface Waves 2014.2 Small Amplitude Two-Dimensional Waves 2034.3 Small-Amplitude Traveling Waves 205

4.3.1 Particle paths for traveling waves 2074.3.2 Stream function for traveling waves 208

4.4 The Dispersion Relation 2094.5 Standing Waves 2104.6 Traveling Waves at the Interface of Two Liquids 2114.7 Group Velocity 2144.8 Shallow-Water Finite Waves 217

4.8.1 The wave equation 2194.8.2 Finite-amplitude waves: The method of characteristics 223

4.9 The Dam Breaking Problem 2274.9.1 The dam break problem without tailwater 2274.9.2 Shock waves 2304.9.3 Dam break problem with tailwater 234

4.10 Lagrangian formulation 236

5 Exact Solutions to Flow Problems of an Incompressible Viscous Fluid 242

5.1 Governing Equations 2435.2 Plane Steady Unidirectional Flows 2465.3 Poiseuille Flow 2505.4 Stagnation-Point Flow 2675.5 Flow in a Converging or Diverging Channel 2705.6 Flow between Rotating Cylinders 2735.7 Fluid in a Rotating Body 2765.8 Flow over an Infinite Rotating Disc 2775.9 Unsteady Flow Problems 278

5.9.1 Radial growth or collapse of a spherical bubble 2785.9.2 Stokes’ problems 2795.9.3 Generalized Couette (or generalized Stokes) flow in a channel 2845.9.4 Unidirectional flow through a circular pipe 2905.9.5 Transient flow inside or outside a rotating cylinder 2995.9.6 Transient flow in an annular duct 3025.9.7 Transient flow in a duct of rectangular cross section 3055.9.8 Squeeze-film flows 306

5.10 Wind Driven Ocean Currents: The Ekman Layer 313

6 Laminar Boundary Layer Theory 326

6.1 Governing Equations 327






vi Contents

6.2 Blasius Solution 3306.3 Falkner–Skan Solution 3346.4 Displacement and Momentum Thickness 3386.5 Approximate Techniques: The von Karman Momentum Integral 3396.6 Thermal Boundary Layer 343

7 Low-Reynolds Number Hydrodynamics 352

7.1 Lubrication Theory 3527.2 Low-Inertia External Flows 3557.3 Three-Dimensional Axisymmetric Flows 357

7.3.1 Solution using separation of variables 3597.3.2 Uniform flow around a sphere/spherical cap 3637.3.3 Motion of a spherical bubble in a fluid 3657.3.4 Flow over a prolate spheroid 3697.3.5 Flow through a conical tube 371

7.4 Flow Between Rotating Axially Symmetric Bodies 3717.5 General Formulation for Two-Dimensional Problems 377

7.5.1 Uniform flow around a circular cylinder 3787.5.2 Elliptical cylinder rotating in an unbounded viscous fluid 379

8 Compressible Fluid Flow 381

8.1 One-Dimensional Flow Equations 3828.2 Acoustic Theory: The Wave Equation 384

8.2.1 Straight duct with specified acceleration at the left end andp∆ 0 or Bp∆Bx 0 at the other 391

8.2.2 Straight duct with specified acceleration at the left endand spring-mass system at the other 399

8.2.3 Pulsating sphere 4008.2.4 Accelerated rigid sphere 4118.2.5 Generalization of pulsating and oscillating sphere examples 4188.2.6 Vibrating circular piston in an infinite rigid baffle 4208.2.7 Pulsating circular cylinder 4238.2.8 Accelerated circular cylinder in a rigid cylinder 430

8.3 Isentropic Flow Relations for a One-Dimensional Steady Flow Field 4348.4 Solution of the One-Dimensional Flow Equations 4368.5 Normal Shock 442

8.5.1 Stationary normal shock 4428.5.2 Moving shock wave 4448.5.3 Structure of a normal shock 447

8.6 Oblique Shock and Expansion Waves 4518.6.1 Mach waves 4528.6.2 Oblique shock relations 453






Contents vii

8.6.3 Prandtl–Meyer expansion waves 4628.7 Shock-Expansion Theory 4658.8 Isentropic Flow through Variable-Area Ducts 4708.9 Area-Velocity Relation 4728.10 Flow Through a Convergent-Divergent Nozzle 4748.11 Unsteady One-Dimensional Flow Problems: Finite Waves 4828.12 The Shock Tube Problem 4848.13 Compressible Couette Flow 487

Appendices 492

A Governing Equations in Cylindrical Coordinates 492

B Governing Equations in Spherical Coordinates 494

C Governing Equations in Elliptic Cylindrical Coordinates 496

D Governing Equations in Bipolar Cylindrical Coordinates 498

E A General Solution to the Axisymmetric Laplace andBiharmonic Equations in Spherical Coordinates 500

F The Laplace Transform Method 507

G Fourier-series Expansion of some Functions 509

Bibliography 543Answers and Hints to Selected Exercises 550Index 565






List of Figures

1.1 Reference and current configurations. 2

1.2 Section of a vortex tube. 9

1.3 Traction vectors and the corresponding stress components on an element; inthe figure tpiq stands for tpx, t, eiq. Tractions and stresses on a hidden face areequal and opposite to those on the corresponding visible face having anopposite normal. 14

1.4 Frames of reference. 151.5 Two observers in rotatory relative motion. 161.6 Computation of body force in the xyz frame. 201.7 Volume element for proving the normal velocity boundary condition. 401.8 Motion of a fluid–fluid interface. 411.9 Surface tension at a fluid–fluid interface. 421.10 Liquid drop and soap bubble. 451.11 Contact angle at a liquid–solid interface. 461.12 Liquid–vertical boundary interface. 471.13 The elongated shape assumed by an initially spherical drop in the spinning

drop method. 481.14 (a) Sessile and (b) pendant drops for the case ρ1 ¡ ρ2;

(c) choice of coordinate system. 521.15 Flow through an orifice in a tank. 571.16 Numerical solution of Eqn. (1.195) for H 5 m, and (i) m 5 (ii) m 100. 601.17 Oscillatory motion of fluid in a U-tube. 611.18 Water sprinkler with (a) straight horizontal pipe;

(b) horizontal pipe with bent ends. 631.19 Pitot tube. 681.20 Flow past a cylinder. 731.21 Control volume for flow past a cylinder problem. 741.22 Control volume analysis for flow past a flat plate. 761.23 Channel flow. 781.24 Control volume for channel flow example. 791.25 Graph of specific energy versus height. 821.26 Water jet impinging on an inclined plate in motion. 84






List of Figures ix

1.27 Vertically moving rocket. 871.28 Water jet impinging on a turbine. 891.29 Control volume for the turbine problem; the arrows indicate the tractions

acting on the fluid due to the blades. 891.30 Pelton turbine: (a) Schematic of the turbine (b) View of bucket. 911.31 Slipstream definitions and control volume for a turbine. 931.32 Control volume for the turbine problem. 931.33 Flow through a pipe network. 951.34 Problem 12 991.35 Problem 13 991.36 Problem 14 1001.37 Problem 15 1001.38 Problem 16 1011.39 Problem 17 1011.40 Problem 18 1021.41 Problem 19 1022.1 Simple manometer. 1042.2 Submerged plane surface in a liquid. 1052.3 Pressure diagram. 1062.4 Pressure acting on area dS on a curved surface. 1072.5 Buoyant force on a body immersed in a fluid. 1082.6 Flat plate subjected to static fluid loading. 1102.7 Pressure diagram. 1112.8 One-eighth part of a sphere immersed in a fluid. 1122.9 Projected area of the body on the vertical plane. 1122.10 Container with fluid traveling with uniform acceleration. 1132.11 Balloon in an accelerating container. 1142.12 Free-body diagram for the balloon example. 1142.13 Cylinder rotating at constant speed. 1152.14 Problem 1 1162.15 Problem 2 1172.16 Problem 3 1172.17 Problem 4 1182.18 Problem 5 1183.1 Volume flux across a curve. 1223.2 Uniform flow at an angle α. 1263.3 Flow source. 1273.4 Point vortex. 128






x List of Figures

3.5 (a) Superposition of a source and sink; (b) Streamline pattern for doublet. 1303.6 Flow in a sector. 1313.7 Flow around a wedge. 1323.8 Flow around an edge. 1333.9 Flow past a Rankine oval. 1333.10 Variation of the pressure coefficient on the periphery of a cylinder immersed

in a uniform flow assuming ideal fluid flow conditions. 1373.11 Flow over an elliptical cylinder. 1383.12 Source near a wall; (a) Actual problem (b) Modified problem using method

of images. 1403.13 Contour used to find the flow rate in the method of images. 1413.14 Closed contour surrounding a solid body. 1433.15 Free-fall of a cylinder in an ideal fluid. 1453.16 Location of stagnation points for varying magnitude of circulation. 1503.17 Conformal mapping. 1543.18 Joukowski transformation. 1563.19 Flow around an ellipse. 1563.20 Flow around a flat airfoil. 1573.21 Joukowski transformation for the airfoil problem. 1573.22 Stagnation points (a) without using Kutta condition; (b) using Kutta condition. 158

3.23 The mapping z cξβπ . 1593.24 Schwartz–Cristoffel transformation. 1603.25 Flow between parallel planes. 1603.26 Flow due to a source located in a corner in a rectangular channel. 1613.27 Jet flow through an orifice. 1623.28 Mapping of the Ω and the W planes into the λ plane. 1643.29 Flow inside a rotating tube whose cross section is an ellipse. 1663.30 Cylinder immersed in a rotational flow. 1713.31 (a) Three-dimensional potential flow; (b) Spherical coordinates. 1743.32 Superposition of a source and sink to produce a doublet. 1803.33 Force on a body in potential flow. 1813.34 Axisymmetric potential flow around a prolate spheroid. 1853.35 Body moving through a stationary fluid. 1863.36 Bubble moving through a stationary fluid with constant speed U. 1903.37 Discretization of the bubble shape. 1923.38 Problem 1 1943.39 Pair of vortices. 1953.40 Problem 2 195






List of Figures xi

3.41 Problem 3 1963.42 Problem 4 1963.43 Problem 8 1973.44 Problem 9 1983.45 Problem 10: (a) Configuration used to generate a Rankine solid;

(b) Flow pattern around the Rankine solid. 1983.46 Problem 11 1993.47 Problem 12 1993.48 Problem 14 2004.1 Surface waves. 2024.2 A sinusoidal wave. 2054.3 Variation of particle trajectory with depth. 2084.4 Plot of phase velocity versus wavelength. 2094.5 Traveling waves at an interface. 2114.6 Phase and group velocity. 2154.7 Large amplitude shallow water waves. 2174.8 Characteristics for the wave equation. 2204.9 Boundary data for the wave problem. 2214.10 Domain of dependence and range of influence. 2254.11 Characteristics in a (a) Uniform region (b) Simple region. 2254.12 The dam break problem. 2284.13 The dam break problem without tailwater: (a) Gradual acceleration

(b) Instantaneous acceleration. 2294.14 Profile of the wave at various times in the dam breaking problem. 2304.15 Dam moving into the reservoir (a) gradually (b) instantaneously. 2314.16 Hydraulic jump. 2324.17 Dam break problem with tailwater. 2354.18 Particle path in a Gerstner wave. 2384.19 Problem 1 2404.20 Problem 2 2415.1 Couette flow. 2465.2 Control volume for verifying the mechanical energy balance in Couette flow. 2475.3 Plane Poiseuille flow. 2485.4 Velocity profile for general Couette flow. 2495.5 Flow down an incline. 2505.6 Control volume for computing the head loss for Poiseuille flow. 2515.7 Poiseuille flow through a circular pipe. 2545.8 Fully developed flow through a concentric annulus. 257






xii List of Figures

5.9 Flow through an eccentric annulus. 2585.10 Poiseuille flow through a duct whose cross-section is bounded by

confocal ellipses. 2615.11 Poiseuille flow through a duct whose cross section is an isosceles

right-angled triangle. 2635.12 Poiseuille flow through a duct whose cross section is an annular sector. 2645.13 Flow through a rectangular duct. 2665.14 Stagnation-point flow. 2675.15 Plot of F and F1 as a function of η. 2705.16 Flow in converging and diverging channels. 2715.17 Velocity profile for flow in a (a) convergent channel

(b) divergent channel. (Re1 ¡ Re2 ¡ Re3) 2725.18 Flow between rotating cylinders. 2735.19 Cylinder rotating in an infinite expanse of fluid. 2765.20 Fluid in a rotating body. 2765.21 Stokes problems. 2805.22 Velocity profile for Stokes’ first problem. 2815.23 Velocity profile for Stokes’ second problem. 2845.24 Generalized Couette flow in a channel. 2855.25 Squeezed flow between a vertically moving and a stationary plate;

(a) Planar flow; (b) Axisymmetric flow. 3075.26 Plot of gpηq and g1pηq when Re=10 and H 10. 3105.27 Ocean currents. 3145.28 The Ekman spiral. 3165.29 Problem 3 3175.30 Problem 4 3185.31 Problem 5 3185.32 Problem 8 3195.33 Problem 10 3195.34 Problem 11 3205.35 Problem 12 3215.36 Problem 13 3225.37 Problem 14 3226.1 Boundary layer around a solid body immersed in a flowing fluid. 3266.2 Curvilinear coordinates on the surface of a solid body. 3286.3 Boundary layer on a plane surface. 3306.4 Velocity distribution in a boundary layer over a flat plate. 3336.5 Separation of boundary layer. 337






List of Figures xiii

6.6 Displacement thickness. 3386.7 Thermal boundary layer. 3446.8 Problem 1 3476.9 Problem 3: (a) Schematic of flow; (b) Physical picture. 3486.10 Problem 4 3496.11 Problem 5 3507.1 Sliding bearing. 3537.2 (a) Typical flow problem; (b) Spherical coordinates. 3577.3 Motion of a spherical bubble in another fluid. 3667.4 Streamline pattern inside and outside the bubble. 3687.5 Flow between rotating spheres. 3727.6 Extensional flow. 3808.1 Control volume analysis for one-dimensional flow. 3828.2 A ‘fluid-structure’ interaction problem. 3998.3 Fluid-structure interaction problem. 4008.4 Accelerated sphere problem. 4118.5 Schematic of a normal shock. 4428.6 Pitot tube immersed in a supersonic flow. 4448.7 (a) Moving shock wave (b) Stationary shock wave. 4458.8 Reflection of a normal shock. 4478.9 Structure of a normal shock. 4488.10 (a) Oblique shock wave (b) Expansion fan. 4528.11 Wave patterns set up by a particle in (a) subsonic (b) supersonic flow. 4538.12 (a) Schematic of an oblique shock wave (b) Control volume. 4548.13 (a) Attached shock wave (b) Detached shock wave. 4608.14 Centered expansion fan. 4638.15 Coordinate system used to find the velocity field in the centered expansion fan. 4638.16 Flat plate airfoil in a supersonic flow. 4678.17 Diamond-shaped wedge in supersonic flow. 4698.18 Control volume for quasi-one-dimensional flow. 4718.19 (a) Subsonic flow, (b) Supersonic flow, through converging and diverging

nozzles. 4738.20 (a) Minimum-area flow; (b) Maximum-area flow. 4748.21 Convergent-Divergent nozzle. 4748.22 Pressure distribution caused by various back pressures. 4758.23 Calculation of properties at section 4 knowing properties at section 1. 4788.24 Shock tube: Configurations before and after the diaphragm is broken. 4858.25 Characteristics for the shock tube problem. 486






xiv List of Figures

8.26 Couette flow in a compressible fluid. 4888.27 Problem 10 491G.1 Stagnation points for Problem 3. 555G.2 Flow in a channel. 556






List of Tables

8.1 Mach number and the Prandtl–Meyer function in degrees. 466

8.2 Table for determining pcr1 and pcr3. (γ 1.4) 479






Preface

The purpose of this volume is to show how most of the approximate theories that areused to solve real-life problems in fluid mechanics follow from the more exact continuumtheories that have been presented in detail in Vol. I. The outline of this volume is shown inFig. 1. We begin by presenting a summary of the governing equations for a Newtonianfluid. As shown in Fig. 1, we proceed from the general to the particular, so that it is clearat each stage what assumptions have been made to obtain a particular approximation,and also where that particular approximation fits into the general framework. Theseapproximate theories are in turn used for the solution of special problems.

Fig. 1 Outline of Volume II.

Keeping in mind that many readers may be interested not so much in the detailedderivations of the governing equations as in how these equations are applied to solveflow problems, we have tried to keep this volume as self-contained as possible. In keepingwith this goal, a detailed summary of all the governing equations has been presented atthe outset. Also in keeping with this goal, we have tried to follow closely the notationused in the continuum mechanics literature in Vol. I, and the notation used in the fluidmechanics literature in Vol. II. This has necessitated a slight change in notation in goingfrom Vol. I to Vol. II. For example, u is usually used to denote the displacement vector inbooks on continuum mechanics, while it is usually used to denote the velocity vector in






xviii Preface

the fluid mechanics literature. Since a table of notation has been included at the beginningof each volume, and since all the relevant equations are restated using the new notation inVol. II, we hope that readers interested in learning only about the applied aspects of fluidmechanics will be able to do so simply by reading Vol. II.

Even those readers who are interested in the detailed derivations of the governingequations for a fluid need not read Vol. I in its entirety; the relevant sections and chaptersfrom Vol. I are Sections 1.1–1.6, 1.9.5, 1.11, 2.1–2.4, Chapter 3 excluding Sections 3.8 and3.9, and Section 7.3.

Chapters 2–4 can be used in a course on ‘Ideal Fluid Flow’, while Chapters 5–8 can beused in a course on ‘Viscous Fluid Flow’, with appropriately selected parts of Chapter 1serving as background material for the two courses.

Although we feel that this book treats many topics in a new way, the topic where thetreatment is significantly different is that of compressible flow. We have tried to presentclosed-form solutions wherever possible, some of which we believe to be new, e.g., theinversion of the θ–β–M relation, transient solutions to the acoustic equations etc. We havealso emphasized obtaining the solutions directly by numerical techniques, rather than bythe classical approach of using ‘gas tables’. This approach is in keeping with thewidespread use of computers for obtaining approximate solutions.

The topic of turbulence is not treated in this book. However, direct numericalsimulations (DNS) seem to indicate that the Navier–Stokes equations remain valid forturbulent flows of Newtonian fluids. Thus, no new governing equations seem to beinvolved. Among the numerical methods for solving turbulent flow problems, theabove-mentioned DNS technique, although computationally intensive, seems to be themost reliable.

The author is indebted to Professor R. E. Johnson from whose lectures he first learnt thesubject, and who also contributed many of the exercise problems. The responsibility forthe correctness of the answers is solely the author’s, though. He would also like toacknowledge the many helpful discussions he had with Professor Anoop Das andProfessor Jaywant Arakeri.

Suggestions and comments for improving this book are welcome.






Notation

a speed of sound

a acceleration vector

α surface tension coefficient

b body force vector per unit mass

c phase velocity

cg group velocity

cp specific heat at constant pressure

cv specific heat at constant volume

χ mapping characterizing the deformation

DDt material derivative

D rate-of-deformation (rate-of-strain) tensor

δ boundary-layer thickness

e specific internal energy

ei Cartesian basis vectors

F deformation gradient

φ velocity potential for irrotational flows (u ∇φ)

Φ potential for conservative body force (b ∇Φ)

g temperature gradient

γ ratio of specific heats (cpcv)

Γ circulation

h specific enthalpy

hL head loss

J determinant of F

k thermal conductivity, wavenumber

κ bulk viscosity

L velocity gradient tensor

λ dilatational viscosity, wavelength

λ unit tangent to a contour

9m mass flow rate






xx Notation

µ shear or dynamic viscosity

n unit normal to a surface

ν kinematic viscosity

p thermodynamic pressure

p mean pressure

P modified pressure

q heat flux vector

Qh heat generated per unit mass per unit time

R universal gas constant

ρ density

s specific entropy

S control surface

Sptq surface of material volume

σ viscous stress tensor

t time

t traction vector

T absolute temperature

τ stress tensor

u velocity vector

v specific volume

V control volume

Vptq material volume

Wv viscous work

Ws shaft work

Wpzq complex potential

W vorticity tensor

W the skew tensor 9QQT

ω vorticity vector (axial vector of 2W)

w angular velocity (axial vector of W)

Ω axial vector of QTWQ (equal to QTw)

x spatial coordinates

X material coordinates

ψ free energy, stream function






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