April 1, 1999 9:15 CB150/Slattery FM Advanced Transport ...assets.cambridge.org/97805216/32034/sample/9780521632034...Slattery, John Charles, 1932– Advanced transport phenomena

April 1, 1999 9:15 CB150/Slattery FM

Advanced TransportPhenomena

John C. Slattery

v


P U B L I S H E D B Y T H E P R E S S S Y N D I C A T E O F T H E U N I V E R S I T Y O F C A M B R I D G E

The Pitt Building, Trumpington Street, Cambridge, United Kingdom

C A M B R I D G E U N I V E R S I T Y P R E S S

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c© Cambridge University Press 1999

This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 1999

Printed in the United States of America

Typeset in Gill Sans and Times Roman LATEX 2ε [TB]

A catalog record for this book is available from the British Library

Library of Congress Cataloging-in-Publication Data

Slattery, John Charles, 1932–Advanced transport phenomena / John C. Slattery.

p. cm. – (Cambridge series in chemical engineering)ISBN 0-521-63203-X (hb). – ISBN 0-521-63565-9 (pb)1. Transport theory. 2. Chemical engineering. I. Title.

II. Series.TP156.T7S57 1999660′.2842 – dc21 98-44872

CIPISBN 0 521 63203X hardbackISBN 0 521 635659 paperback

vi

April 1, 1999 9:15 CB150/SlatteryFM

Contents

Preface pagexvList of Notation xix

1 Kinematics 11.1 Motion 11.2 Frame 9

1.2.1 Changes of Frame 91.2.2 Equivalent Motions 15

1.3 Mass 181.3.1 Conservation of Mass 181.3.2 Transport Theorem 181.3.3 Differential Mass Balance 211.3.4 Phase Interface 231.3.5 Transport Theorem for a Region Containing a Dividing Surface 231.3.6 Jump Mass Balance for Phase Interface 251.3.7 Stream Functions 26

2 Foundations for Momentum Transfer 282.1 Force 282.2 Additional Postulates 30

2.2.1 Momentum and Moment of Momentum Balances 302.2.2 Stress Tensor 322.2.3 Differential and Jump Momentum Balances 332.2.4 Symmetry of Stress Tensor 35

2.3 Behavior of Materials 372.3.1 Some General Principles 372.3.2 Simple Constitutive Equation for Stress 392.3.3 Generalized Newtonian Fluid 432.3.4 Noll Simple Fluid 48

2.4 Summary 502.4.1 Differential Mass and Momentum Balances 502.4.2 Stream Function and the Navier–Stokes Equation 552.4.3 Interfacial Tension and the Jump Mass and Momentum Balances 58


viii Contents

3 Differential Balances in Momentum Transfer 653.1 Philosophy 663.2 Complete Solutions 67

3.2.1 Flow in a Tube 673.2.2 Flow of a Generalized Newtonian Fluid in a Tube 733.2.3 Tangential Annular Flow 783.2.4 Wall That Is Suddenly Set in Motion 843.2.5 Rotating Meniscus 90

3.3 Creeping Flows 993.3.1 Flow in a Cone–Plate Viscometer 1013.3.2 Flow Past a Sphere 1113.3.3 Thin Draining Films 1163.3.4 Melt Spinning 121

3.4 Nonviscous Flows 1313.4.1 Bernoulli Equation 1333.4.2 Potential Flow Past a Sphere 1363.4.3 Plane Potential Flow in a Corner 140

3.5 Boundary-Layer Theory 1433.5.1 Plane Flow Past a Flat Plate 1433.5.2 More on Plane Flow Past a Flat Plate 1483.5.3 Plane Flow Past a Curved Wall 1533.5.4 Solutions Found by Combination of Variables 1583.5.5 Flow in a Convergent Channel 1623.5.6 Flow Past a Body of Revolution 165

3.6 Stability 1723.6.1 Stability of a Liquid Thread 174

4 Integral Averaging in Momentum Transfer 1824.1 Time Averaging 182

4.1.1 Time Averages 1834.1.2 The Time Average of a Time-Averaged Variable 1854.1.3 Empirical Correlations forT(t) 1864.1.4 Wall Turbulence in Flow Through a Tube 190

4.2 Area Averaging 1944.2.1 Flow from Rest in a Circular Tube 195

4.3 Local Volume Averaging 1974.3.1 The Concept 1984.3.2 Theorem for the Local Volume Average of a Gradient 2004.3.3 The Local Volume Average of the Differential Mass Balance 2034.3.4 The Local Volume Average of the Differential

Momentum Balance 2034.3.5 Empirical Correlations forg 2054.3.6 Summary of Results for an Incompressible Newtonian Fluid 2094.3.7 Averages of Volume-Averaged Variables 2104.3.8 Flow through a Packed Tube 2114.3.9 Neglecting the Divergence of the Local Volume-Averaged

Extra Stress 215


Contents ix

4.4 Integral Balances 2174.4.1 The Integral Mass Balance 2184.4.2 The Integral Mass Balance for Turbulent Flows 2204.4.3 The Integral Momentum Balance 2224.4.4 Empirical Correlations forF andG 2254.4.5 The Mechanical Energy Balance 2294.4.6 Empirical Correlations forE 2344.4.7 Integral Moment-of-Momentum Balance 2364.4.8 Examples 239

5 Foundations for Energy Transfer 2505.1 Energy 250

5.1.1 Energy Balance 2505.1.2 Radiant and Contact Energy Transmission 2515.1.3 Differential and Jump Energy Balances 253

5.2 Entropy 2565.2.1 Entropy Inequality 2565.2.2 Radiant and Contact Entropy Transmission 2575.2.3 The Differential and Jump Entropy Inequalities 259

5.3 Behavior of Materials 2615.3.1 Implications of the Differential Entropy Inequality 2615.3.2 Restrictions on Caloric Equation of State 2705.3.3 Energy and Thermal Energy Flux Vectors 2735.3.4 Stress Tensor 275

5.4 Summary of Useful Equations 277

6 Differential Balances in Energy Transfer 2836.1 Philosophy 2836.2 Conduction 284

6.2.1 Cooling of a Semi-Infinite Slab: Constant Surface Temperature 2856.2.2 Cooling a Semi-Infinite Slab: Newton’s “Law” of Cooling 2876.2.3 Cooling a Flat Sheet: Constant Surface Temperature 290

6.3 More Complete Solutions 2946.3.1 Couette Flow of a Compressible Newtonian Fluid 2946.3.2 Couette Flow of a Compressible Newtonian Fluid with Variable

Viscosity and Thermal Conductivity 3026.3.3 Rate at Which a Fluid Freezes (or a Solid Melts) 309

6.4 When Viscous Dissipation Is Neglected 3146.4.1 Natural Convection 316

6.5 No Convection 3246.6 No Conduction 326

6.6.1 Speed of Propagation of Sound Waves 3276.7 Boundary-Layer Theory 331

6.7.1 Plane Flow Past a Flat Plate 3316.7.2 More on Plane Flow Past a Flat Plate 3356.7.3 Plane Flow Past a Curved Wall 3426.7.4 Flow Past a Wedge 345


x Contents

6.7.5 Flow Past a Body of Revolution 3496.7.6 Energy Transfer in the Entrance of a Heated Section of a Tube 3516.7.7 More on Melt Spinning 359

6.8 More About Energy Transfer in a Heated Section of a Tube 3616.8.1 Constant Temperature at Wall 3626.8.2 Constant Energy Flux at Wall 366

7 Integral Averaging in Energy Transfer 3747.1 Time Averaging 374

7.1.1 The Time-Averaged Differential Energy Balance 3747.1.2 Empirical Correlation for the Turbulent Energy Flux 3757.1.3 Turbulent Energy Transfer in a Heated Section of a Tube 378

7.2 Area Averaging 3827.2.1 A Straight Cooling Fin of Rectangular Profile 383

7.3 Local Volume Averaging 3867.3.1 The Local Volume Average of the Differential Energy Balance 3877.3.2 Empirical Correlations forh 3897.3.3 Summary of Results for a Nonoriented,

Uniform-Porosity Structure 3927.3.4 Transpiration Cooling 394

7.4 More on Integral Balances 3997.4.1 The Integral Energy Balance 3997.4.2 Empirical Correlations forQ 4077.4.3 More About the Mechanical Energy Balance 4107.4.4 The Integral Entropy Inequality 4177.4.5 Integral Entropy Inequality for Turbulent Flows 4187.4.6 Example 418

8 Foundations for Mass Transfer 4238.1 Viewpoint 423

8.1.1 Body, Motion, and Material Coordinates 4248.2 Species Mass Balance 426

8.2.1 Differential and Jump Balances 4268.2.2 Concentration, Velocities, and Mass Fluxes 428

8.3 Revised Postulates 4308.3.1 Conservation of Mass 4328.3.2 Momentum Balance 4338.3.3 Moment of Momentum Balance 4348.3.4 Energy Balance 4358.3.5 Entropy Inequality 435

8.4 Behavior 4368.4.1 Implications of the Differential Entropy Inequality 4378.4.2 Restrictions on Caloric Equation of State 4428.4.3 Energy Flux Vectors 4498.4.4 Mass Flux Vector 4518.4.5 Mass Flux Vector in Binary Solutions 4598.4.6 Mass Flux Vector: Limiting Cases in Ideal Solutions 462


Contents xi

8.4.7 Constitutive Equations for the Stress Tensor 4648.4.8 Rates of Reactions 464

8.5 Summary 4728.5.1 Summary of Useful Equations for Mass Transfer 4728.5.2 Summary of Useful Equations for Energy Transfer 477

9 Differential Balances in Mass Transfer 4829.1 Philosophy of Solving Mass-Transfer Problems 4829.2 Energy and Mass Transfer Analogy 483

9.2.1 Film Theory 4859.3 Complete Solutions for Binary Systems 488

9.3.1 Unsteady-State Evaporation 4899.3.2 Rate of Isothermal Crystallization 5019.3.3 Rate of Nonisothermal Crystallization 5079.3.4 Silicon Oxidation 5139.3.5 Pressure Diffusion in a Natural Gas Well 5159.3.6 Forced Diffusion in Electrochemical Systems 5209.3.7 Film Theory: Condensation of Mixed Vapors 5259.3.8 Two- and Three-Dimensional Problems 532

9.4 Complete Solutions for Multicomponent Systems 5339.4.1 Film Theory: Steady-State Evaporation 5339.4.2 More on Unsteady-State Evaporation 5379.4.3 Oxidation of Iron 545

9.5 Boundary-Layer Theory 5579.5.1 Plane Flow Past a Flat Plate 5589.5.2 Flow Past Curved Walls and Bodies of Revolution 561

9.6 Forced Convection in Dilute Solutions 5619.6.1 Unsteady-State Diffusion with a First-Order

Homogeneous Reaction 5629.6.2 Gas Absorption in a Falling Film with Chemical Reaction 566

10 Integral Averaging in Mass Transfer 57310.1 Time Averaging 573

10.1.1 The Time-Averaged Differential Mass Balance for SpeciesA 57310.1.2 Empirical Correlations for the Turbulent Mass Flux 57510.1.3 Turbulent Diffusion from a Point Source in a Moving Stream 576

10.2 Area Averaging 58010.2.1 Longitudinal Dispersion 580

10.3 Local Volume Averaging 58510.3.1 Local Volume Average of the Differential Mass Balance

for SpeciesA 58510.3.2 When Fick’s First Law Applies 58610.3.3 Empirical Correlations for Tortuosity Vectors 58810.3.4 Summary of Results for a Liquid or Dense Gas in a

Nonoriented, Uniform-Porosity Structure 59210.3.5 When Fick’s First Law Does Not Apply 59310.3.6 Knudsen Diffusion 595


xii Contents

10.3.7 The Local Volume Average of the Overall DifferentialMass Balance 597

10.3.8 The Effectiveness Factor for Spherical Catalyst Particles 59710.4 Still More on Integral Balances 603

10.4.1 The Integral Mass Balance for SpeciesA 60410.4.2 Empirical Correlations forJ(A) 60610.4.3 The Integral Overall Balances 60910.4.4 Example 614

A Tensor Analysis 619A.1 Spatial Vectors 619

A.1.1 Definition** 619A.1.2 Position Vectors 621A.1.3 Spatial Vector Fields 621A.1.4 Basis 622A.1.5 Basis for the Spatial Vector Fields 624A.1.6 Basis for the Spatial Vectors 625A.1.7 Summation Convention 626

A.2 Determinant 627A.2.1 Definitions 627

A.3 Gradient of Scalar Field 630A.3.1 Definition** 630

A.4 Curvilinear Coordinates 632A.4.1 Natural Basis** 632A.4.2 Dual Basis* 638A.4.3 Covariant and Contravariant Components* 641

A.5 Second-Order Tensors 644A.5.1 Components of Second-Order Tensor Fields 645A.5.2 Transpose of a Second-Order Tensor Field** 650A.5.3 Inverse of a Second-Order Tensor Field 654A.5.4 Trace of a Second-Order Tensor Field** 655

A.6 Gradient of Vector Field 657A.6.1 Definition** 657A.6.2 Covariant Differentiation* 658

A.7 Third-Order Tensors 664A.7.1 Definition 664A.7.2 Components of Third-Order Tensor Fields 665A.7.3 Another View of Third-Order Tensor Fields 668

A.8 Gradient of Second-Order Tensor Field 669A.8.1 Definition 669A.8.2 More on Covariant Differentiation* 670

A.9 Vector Product and Curl 673A.9.1 Definitions** 673A.9.2 More on the Vector Product and Curl 674

A.10 Determinant of Second-Order Tensor 677A.10.1 Definition 677


Contents xiii

A.11 Integration 679A.11.1 Spatial Vector Fields** 679A.11.2 Green’s Transformation 680A.11.3 Change of Variable in Volume Integrations** 683

B More on the Transport Theorem 685

References 687Author/Editor Index 699Index 703

December 18, 1998 17:7 CB150/Slattery CB150-DRV

1

Kinematics

THIS ENTIRE CHAPTER IS INTRODUCTORYin much the same way as Ap-pendix A is. In Appendix A, I introduce the mathematical language that I shall be using

in describing physical problems. In this chapter, I indicate some of the details involved in rep-resenting from the continuum point of view the motions and deformations of real materials.This chapter is important not only for the definitions introduced, but also for the viewpointtaken in some of the developments. For example, the various forms of the transport theoremwill be used repeatedly throughout the text in developing differential equations and integralbalances from our basic postulates.

Perhaps the most difficult point for a beginner is to properly distinguish between thecontinuum model for real materials and the particulate or molecular model. We can all agreethat the most factually detailed picture of real materials requires that they be represented interms of atoms and molecules. In this picture, mass is distributed discontinuously throughoutspace; mass is associated with the protons, neutrons, electrons,. . . , which are separated byrelatively large voids. In the continuum model for materials, mass is distributed continu-ously through space, with the exception of surfaces of discontinuity, which represent phaseinterfaces or shock waves.

The continuum model is less realistic than the particulate model but far simpler. For manypurposes, the detailed accuracy of the particulate model is unnecessary. To our sight andtouch, mass appears to be continuously distributed throughout the water that we drink and theair that we breathe. The problem is analogous to a study of traffic patterns on an expressway.The speed and spacings of the automobiles are important, but we probably should not worryabout whether the automobiles have four, six, or eight cylinders.

This is not to say that the particulate theories are of no importance. Information is lostin a continuum picture. It is only through the use of statistical mechanics that a complete apriori prediction about the behavior of the material can be made. I will say more about thisin the next chapter.

1.1 Motion

My goal in this book is to lay the foundation for understanding a wide variety of operationsemployed in the chemical and petroleum industries. To be specific, consider the extrusion of


2 1. Kinematics

Figure 1.1.0-1. A rubber ball in threeconfigurations as it strikes a wall and re-bounds. The particle that was in the po-sition zκ in the reference configurationis in the positionz at timet .

a molten polymer to produce a fiber, catalytic cracking in a fluidized reaction, the productionof oil and gas from a sandstone reservoir, or the flow of a coal slurry through a pipeline. Oneimportant feature that these operations have in common is that at least some of the materialsconcerned are undergoing deformation and flow.

How might we describe a body of material as it deforms? Figure 1.1.0-1 shows a rubberball in three configurations as it strikes a wall and rebounds. How should we describe thedeformation of this rubber body from its original configuration as a sphere? How shouldvelocity be defined in order to take into account that it must surely vary as a function ofposition within the ball as well as time as the ball reaches the wall and begins to deform?We need a mathematical description for a body that allows us to describe where its variouscomponents go as functions of time.

Let’s begin by rather formally defining a body to be set, any elementζ of which is calleda particle or amaterial particle. A material particle is aprimitive, in the sense that it is notdefined but its properties are described. I will give an experimentally oriented descriptionof a material particle a little later in this section. Meanwhile, be careful not to confuse amaterial particle with a molecule. Molecules play no role in continuum mechanics; they areintroduced in the context of the other model for real materials – statistical mechanics.

A one-to-one continuous mapping of this set of material particles onto a region of thespaceE studied in elementary geometry exists and is called aconfigurationof the body:

z= X(ζ ) (1.1.0-1)

ζ = X−1(z) (1.1.0-2)

The pointz= X(ζ ) of E is called the place occupied by the particleζ , andζ = X−1(z), theparticle whose place inE is z.

It is completely equivalent to describe the configuration of a body in terms of the positionvectorz of the pointz with respect to the originO (Section A.1.2):

z= χ(ζ ) (1.1.0-3)

ζ = χ−1(z) (1.1.0-4)


1.1. Motion 3

Hereχ−1 indicates the inverse mapping ofχ. With an originO having been defined, it isunambiguous to refer toz = χ(ζ ) as the place occupied by the particleζ andζ = χ−1(z)as the particle whose place isz.

In what follows, I choose to refer to points inE by their position vectors relative to apreviously defined originO.

A motionof a body is a one-parameter family of configurations; the real parametert istime. We write

z= χ(ζ, t) (1.1.0-5)

and

ζ = χ−1(z, t) (1.1.0-6)

I have introduced the material particle as a primitive concept, without definition but witha description of its attributes. A set of material particles is defined to be a body; there is aone-to-one continuous mapping of these particles onto a region of the spaceE in which wevisualize the world about us. But clearly we need a link with what we can directly observe.

Whereas the bodyB should not be confused with any of its spatial configurations, it isavailable to us for observation and study only in these configurations. We will describe amaterial particle by its position in areference configurationκ of the body. This referenceconfiguration may be, but need not be, one actually occupied by the body in the course ofits motion. The place of a particle inκ will be denoted by

zκ = κ(ζ ) (1.1.0-7)

The particle at the placezκ in the configurationκ may be expressed as

ζ = κ−1(zκ ) (1.1.0-8)

The motion of a body is described by

z = χ(ζ, t)

= χκ (zκ , t)≡ χ(κ−1(zκ ), t) (1.1.0-9)

Referring to Figure 1.1.0-1, we find that the particle that was in the positionzκ in the referenceconfiguration is at timet in the positionz. This expression defines a family ofdeformationsfrom the reference configuration. The subscript. . .κ is to remind you that the form ofχκdepends upon the choice of reference configurationκ.

The position vectorzκ with respect to the originO may be written in terms of its rectan-gular Cartesian coordinates:

zκ = zκ i ei (1.1.0-10)

The zκ i (i = 1, 2, 3) are referred to as thematerial coordinatesof the material particleζ .They locate the position ofζ relative to the originO, when the body is in the referenceconfigurationκ. In terms of these material coordinates, we may express (1.1.0-9) as

z = χκ (zκ , t)

= χ̂κ (zκ1, zκ2, zκ3, t) (1.1.0-11)


4 1. Kinematics

Let A be any quantity: scalar, vector, or tensor. We shall have occasion to talk about thetime derivative ofA following the motion of a particle. We define

d(m) A

dt≡(∂A

∂t

)ζ

≡(∂A

∂t

)zκ

≡(∂A

∂t

)zκ1,zκ2,zκ3

(1.1.0-12)

We refer to the operationd(m)/dt as thematerial derivative[or substantial derivative (Birdet al. 1960, p. 73)]. For example, thevelocity vectorv represents the time rate of change ofposition of a material particle:

v ≡ d(m)zdt

≡[∂χ (ζ, t)

∂t

]ζ

≡[∂χκ (zκ , t)

∂t

]zκ

≡[∂χ̂κ (zκ1, zκ2, zκ3, t)

∂t

]zκ1,zκ2,zκ3

(1.1.0-13)

We are involved with several derivative operations in the chapters that follow. Bird et al.(1960, p. 73) have suggested some examples that serve to illustrate the differences.

The partial time derivative ∂c/∂t Suppose we are in a boat that is anchored securely in a river,some distance from the shore. If we look over the side of our boat and note the concentrationof fish as a function of time, we observe how the fish concentration changes with time at afixed position in space:

∂c

∂t≡(∂c

∂t

)z

≡(∂c

∂t

)z1,z2,z3

The material derivative d(m)c/dt Suppose we pull up our anchor and let our boat drift along withthe river current. As we look over the side of our boat, we report how the concentration offish changes as a function of time while following the water (the material):

d(m)c

dt= ∂c

∂t+∇c · v (1.1.0-14)

The total derivative dc/dt We now switch on our outboard motor and race about the river,sometimes upstream, sometimes downstream, or across the current. As we peer over the


1.1. Motion 5

side of our boat, we measure fish concentration as a function of time while following anarbitrary path across the water:

dc

dt= ∂c

∂t+∇c · v(b) (1.1.0-15)

Herev(b) denotes the velocity of the boat.

Exercise 1.1.0-1 Let A be any real scalar field, spatial vector field, or second-order tensor field.Show that1

d(m) A

dt= ∂A

∂t+∇A · v

Exercise 1.1.0-2 Let a= a(z, t) be some vector field that is a function of position and time.

i) Show that

d(m)adt=(∂an

∂t+ an,i v

i

)gn

ii) Show that

d(m)adt=(∂an

∂t+ an,i v

i

)gn

Exercise 1.1.0-3 Consider the second-order tensor fieldT = T(z, t).

i) Show that

d(m)Tdt=(∂Ti j

∂t+ Ti j ,k v

k

)gi g j

ii) Show that

d(m)Tdt=(∂Ti

j

∂t+ Ti

j ,k vk

)gi g j

Exercise 1.1.0-4 Show that

d(m)(a · b)

dt= d(m)

dt(ai bi )

= d(m)ai

dtbi + ai d(m)bi

dt

1 Where I write (∇A) · v, some authors say insteadv · (∇A). WhenA is a scalar, there is no difference.When A is either a vector or second-order tensor, the change in notation is the result of a differentdefinition for the gradient operation. See Sections A.6.1 and A.8.1.


6 1. Kinematics

Exercise 1.1.0-5

i) Starting with the definition for the velocity vector, prove that

v = d(m)xi

dtgi

ii) Determine that, with respect to the cylindrical coordinate system defined in ExerciseA.4.1-4,

v = d(m)r

dtgr + r

d(m)θ

dtgθ + d(m)z

dtgz

iii) Determine that, with respect to the spherical coordinate system defined in ExerciseA.4.1-5,

v = d(m)r

dtgr + r

d(m)θ

dtgθ + r sinθ

d(m)ϕ

dtgϕ

Exercise 1.1.0-6 Path lines The curve in space along which the material particleζ travels is referredto as thepath line for the material particleζ . The path line may be determined from themotion of the material as described in Section 1.1:

z= χ(zκ , t)

Herezκ represents the position of the material particleζ in the reference configurationκ;time t is a parameter along the path line that corresponds to any given positionzκ .

The path lines may be determined conveniently from the velocity distribution, sincevelocity is the derivative of position with respect to time following a material particle. Theparametric equations of a particle path are the solutions of the differential system

dzdt= v

or

dzi

dt= vi for i = 1, 2, 3

The required boundary conditions may be obtained by choosing the reference configurationto be a configuration that the material assumed at some timet0.

As an example, let the rectangular Cartesian components ofv be

v1 = z1

1+ t, v2 = z2

1+ 2t, v3 = 0

and let the reference configuration be that which the material assumed at timet = 0. Provethat, in the planez3 = zκ3, the particle paths or the path lines have the form

z2

zκ2=(

2z1

zκ1− 1

)1/2


1.1. Motion 7

Exercise 1.1.0-7 Streamlines The streamlines for timet form that family of curves to which thevelocity field is everywhere tangent at a fixed timet . The parametric equations for thestreamlines are solutions of the differential equations

dzi

dα= vi for i = 1, 2, 3

Hereα is a parameter with the units of time, anddz/dα is tangent to the streamline [see(A.4.1-1)]. Alternatively, we may think of the streamlines as solutions of the differentialsystem

dzdα∧ v = 0

or

ei jkdzj

dαvk = 0 for i = 1, 2, 3

As an example, show that, for the velocity distribution introduced in Exercise 1.1.0-6, thestreamlines take the form

z2 = z2(0)

(z1

z1(0)

)(1+t)/(1+2t)

for different reference points (z1(0), z2(0)).Experimentalists sometimes sprinkle particles over a gas–liquid phase interface and take

a photograph in which the motion of the particles is not quite stopped (see Figures 3.5.1-1and 3.5.1-3). The traces left by the particles are proportional to the velocity of the fluidat the surface (so long as we assume that very small particles move with the fluid). For asteady-state flow, such a photograph may be used to construct the particle paths. For anunsteady-state flow, it depicts the streamlines, the family of curves to which the velocityvector field is everywhere tangent.

In two-dimensional flows, the streamlines have a special significance. They are curvesalong which the stream function (Sections 1.3.7) is a constant. See Exercise 1.3.7-2.

Exercise 1.1.0-8 For the limiting case of steady-state, plane potential flow past a stationary cylinderof radiusa with no circulation, the physical components of velocity in cylindrical coordinatesare (see Exercise 3.4.2-2)

vr = V

(1− a2

r 2

)cosθ

vθ = −V

(1+ a2

r 2

)sinθ

and

vz = 0

Show that the family of streamlines is described by(1− a2

r 2

)r sinθ = C

Plot representative members of this family as in Figure 1.1.0-2.


8 1. Kinematics

-3 -2 -1 1 2 3

-1.5

-1

-0.5

0.5

1

1.5

z2

z1

Figure 1.1.0-2. Streamlines for the limiting case of steady-state, plane potential flow past a stationary cylinder with nocirculation corresponding toC = 0.2, 0.4, 0.6, 0.8, 1.

Exercise 1.1.0-9 Streak lines The streak line through the pointz(0) at timet represents the positionsat timet of the material particles that at any timeτ ≤ t have occupied the placez(0).

Experimentally we might visualize that smoke, dust, or dye are continuously injected intoa fluid at a positionz(0) and that the resulting trails are photographed as functions of time.Each photograph shows a streak line corresponding to the positionz(0) and the time at whichthe photograph was taken.

We saw in Section 1.1 that the motionχ describes the positionz at timet of the materialparticle that occupied the positionzκ in the reference configuration:

z= χ(zκ , t)

In constructing a streak line, we focus our attention on those material particles that were inthe placez(0) at any timeτ ≤ t :

zκ = χ−1(z(0), τ

)The parametric equations of the streak line through the pointz(0) at timet are obtained byeliminatingzκ between these equations:

z= χ (χ−1(z(0), τ

), t)

Time τ ≤ t is the parameter along the streak line.As an example, show that, for the velocity distribution of Exercise 1.1.0-6, the streak line

throughz(0) at timet is specified by

z1 = z1(0)

(1+ t

1+ τ)

z2 = z2(0)

(1+ 2t

1+ 2τ

)1/2

z3 = z3(0)

A streak line corresponding tot = 4 is shown in Figure 1.1.0-3. This figure also presentstwo of the path lines from Exercise 1.1.0-6 corresponding toτ = 0 and 0.5 that contribute


1.2. Frame 9

2 3 4 5

1.5

2

2.5

3

0.25= 0

t = 4

z1

z2

τ

Figure 1.1.0-3. Starting from the top, we see two path linesfrom Exercise 1.1.0-6) corresponding toτ = 0 and 0.25. Thebottom curve is the streak line corresponding tot = 4.

to this streak line. The path line corresponding toτ = 0 extends to the right tip of the streakline. It represents the first particle contributing to the streak line. A path line correspondingto τ = 4 would merely be a point at the left tip of the streak line, the origin of the streakline (0, 0). It would represent the particle currently leaving the origin.

Exercise 1.1.0-10 Show that, for a velocity distribution that is independent of time, the path lines,streamlines, and streak lines coincide.

Hint: In considering the path line, take as the boundary condition

at t = τ : z= z(0)

This suggests the introduction of a new variableα ≡ t − τ , which denotes time measuredsince the particle passed through the positionz(0).

1.2 Frame

1.2.1 Changes of Frame

The Chief of the United States Weather Bureau in Milwaukee announces that a tornado wassighted in Chicago at 3P.M. (Central Standard Time). In Chicago, Harry reports that he sawa black funnel cloud about two hours ago at approximately 800 North and 2400 West. Bothmen described the same event with respect to their own particular frame of reference.

The time of some occurrence may be specified only with respect to the time of someother event, theframe of reference for time. This might be the time at which a stopwatch wasstarted or an electric circuit was closed. The Chief reported the time at which the tornadowas sighted relative to the mean time at which the sun appeared overhead on the Greenwichmeridian. Harry gave the time relative to his conversation.


10 1. Kinematics

b(1)

b(2)

Figure 1.2.1-1. Pencil points away from the di-rection ofb(1) and toward the direction ofb(2).

A frame of reference for positionmight be the walls of a laboratory, the fixed stars, or theshell of a space capsule that is following an arbitrary trajectory. When the Chief specifiedChicago, he meant the city at 41◦ north and 87◦ west measured relative to the equator andthe Greenwich meridian. Harry thought in terms of eight blocks north of Madison Avenueand 24 blocks west of State Street. More generally, a frame of reference for position is a setof objects whose mutual distances remain unchanged during the period of observation andwhich do not all lie in the same plane.

To help you get a better physical feel for these ideas, let us consider two more examples.Extend your right arm and take as your frame of reference for position the direction of

your right arm, the direction of your eyes, and the direction of your spine. Stand out at thestreet with your eyes fixed straight ahead. A car passes in the direction of your right arm.If you were standing facing the street on the opposite side, the automobile would appear topass in the opposite direction from your right arm.

Lay a pencil on your desk as shown in Figure 1.2.1-1 and take the edges of the desk thatmeet in the left-hand front corner as your frame of reference for position. The pencil pointsaway fromb(1) and towardb(2). Without moving the pencil, walk around to the left-handside and take as your new frame of reference for position the edges of the desk that meet atthe left-hand rear corner. The pencil now appears to point toward the intersection ofb∗(1) andb∗(2) in Figure 1.2.1-2.

Since all of the objects defining a frame of reference do not lie in the same plane, we mayvisualize replacing them by three mutually orthogonal unit vectors. Let us view a typicalpoint z in this space with respect to two such frames of reference: theb(i ) (i = 1, 2, 3) inFigure 1.2.1-3 and theb∗( j ) ( j = 1, 2, 3) in Figure 1.2.1-4.

An orthogonal transformation preserves both lengths and angles (Section A.5.2). LetQbe the orthogonal transformation that describes the rotation and (possibly) reflection thattakes theb(i ) in Figure 1.2.1-3 into the vectorsQ · b(i ), which are seen in Figure 1.2.1-4 withrespect to the starred frame of reference for position. A reflection allows for the possibilitythat an observer in the new frame looks at the old frame through a mirror. Alternatively,a reflection allows for the possibility that two observers orient themselves oppositely, onechoosing to work in terms of a right-handed frame of reference for position and the other interms of a left-handed one. [For more on this point, I suggest that you read Truesdell (1966a,p. 22) as well as Truesdell and Noll (1965, pp. 24 and 47).]

The vector(z− z(0)

)in Figure 1.2.1-3 becomesQ ·

(z− z(0)

)when viewed in the starred

frame shown in Figure 1.2.1-4. From Figure 1.2.1-4, it follows as well that

z∗ − z∗(0) = Q ·(z− z(0)

)(1.2.1-1)


1.2. Frame 11

b(1)

b(2)*

*

Figure 1.2.1-2. Pencil pointstoward the direction ofb∗(1)andb∗(2).

b(1)

b(2)

b(3)

z

z(0)

z(0)

z

z-z(0)

Figure 1.2.1-3. The pointsz and z(0) are locatedby the position vectorsz andz(0) with respect to theframe of reference for position

(b(1), b(2), b(3)

).

Similarly,(z∗ − z∗(0)

)in Figure 1.2.1-4 is seen asQT ·

(z∗ − z∗(0)

)when observed with

respect to the unstarred frame in Figure 1.2.1-5. Figure 1.2.1-5 also makes it clear that

z− z(0) = QT ·(z∗ − z∗(0)

)(1.2.1-2)

Let z andt denote a position and time in the old frame;z∗ andt∗ are the correspondingposition and time in the new frame. We can extend the discussion above to conclude that the


12 1. Kinematics

b(1)b(2)

b(3)

z(0)

z(0)

zQ z-z(0)

z

Q z

Q z(0)

Q b(1)

Q b(2)

Q b(3)

* *

*

*

*

. ( ).

.

..

.

Figure 1.2.1-4. The pointsz and z(0) are located by theposition vectorsz∗ andz∗(0) with respect to the starred frame

of reference for position(b∗(1), b∗(2), b∗(3)

). With respect to

the starred frame of reference, the unstarred frame is seen as(Q · b(1), Q · b(2), Q · b(3)

).

QT b(1)

QT b(2)

QT b(3)

QT z

QT z(0) QT z -z(0)

z(0)

z

z

z(0)

b(1)

b(2)

b(3)

.

** *

*

*

*

*

.

.

. . ( )

.

Figure 1.2.1-5. With respect to the unstarred frameof reference, the starred frame is seen as(QT · b∗(1), QT · b∗(2), QT · b∗(3)

).


1.2. Frame 13

most general change of frame is of the form

z∗ = z∗(0)(t)+Q(t) ·(z− z(0)

)(1.2.1-3)

t∗ = t − a (1.2.1-4)

where we allow the two frames discussed in Figures 1.2.1-3 and 1.2.1-4 to rotate and translatewith respect to one another as functions of time. The quantitya is a real number. Equivalently,we could also write

z= z(0)(t)+QT ·(z∗ − z∗(0)

)(1.2.1-5)

t = t∗ + a (1.2.1-6)

It is important to carefully distinguish between a frame of reference for position and acoordinate system. Any coordinate system whatsoever can be used to locate points in spacewith respect to three vectors defining a frame of reference for position and their intersection,although I recommend that admissible coordinate systems be restricted to those whose axeshave a time-invariant orientation with respect to the frame. Let(z1, z2, z3) be a rectangularCartesian coordinate system associated with the frame of reference

(b(1), b(2), b(3)

); simi-

larly, let(z∗1, z∗2, z∗3

)be a rectangular Cartesian coordinate system associated with another

frame of reference(b∗(1), b∗(2), b∗(3)

). We will say that these two coordinate systems are the

sameif the orientation of the basis fieldsei with respect to the vectorsb( j ) is identical to theorientation of the basis fieldse∗i with respect to the vectorsb∗( j ):

ei · b( j ) = e∗i · b∗( j ) for all i, j = 1, 2, 3 (1.2.1-7)

We will generally find it convenient to use the same coordinate system in discussing twodifferent frames of reference.

Let us use thesamerectangular Cartesian coordinate system to discuss the change offrame illustrated in Figures 1.2.1-4 and 1.2.1-5. The orthogonal tensor

Q = Qi j e∗i ej (1.2.1-8)

describes the rotation (and possibly reflection) that transforms the basis vectorsej ( j =1, 2, 3) into the vectors

Q · ej = Qi j e∗i (1.2.1-9)

which are vectors expressed in terms of the starred frame of reference for position. Therectangular Cartesian components ofQ are defined by the angles between thee∗i and theQ · ej :

Qi j = e∗i ·(Q · ej

)(1.2.1-10)

The vector(z− z(0)

)in Figure 1.2.1-3 becomes

Q ·(z− z(0)

) = Qi j

(zj − z(0) j

)e∗i (1.2.1-11)

when viewed in the starred frame shown in Figure 1.2.1-4. From Figure 1.2.1-4, it followsas well that

z∗i e∗i = z∗(0)i e∗i + Qi j

(zj − z(0) j

)e∗i (1.2.1-12)


14 1. Kinematics

We speak of a quantity as beingframe indifferentif it remains unchanged or invariantunder all changes of frame. Aframe-indifferent scalar bdoes not change its value:

b∗ = b (1.2.1-13)

A frame-indifferent spatial vectorremains the same directed line element under a change offrame in the sense that if

u = z1− z2

then

u∗ = z∗1 − z∗2

From (1.2.1-3),

u∗ = Q · (z1− z2)

= Q · u (1.2.1-14)

A frame-indifferent second-order tensoris one that transforms frame-indifferent spatialvectors into frame-indifferent spatial vectors. If

u = T · w (1.2.1-15)

the requirement thatT be a frame-indifferent second-order tensor is

u∗ = T∗ · w∗ (1.2.1-16)

where

u∗ = Q · u(1.2.1-17)

w∗ = Q · w

This means that

Q · u = T∗ · Q · w= Q · T · w (1.2.1-18)

which implies

T = QT · T∗ · Q (1.2.1-19)

or

T∗ = Q · T · QT (1.2.1-20)

The importance of changes of frame will become apparent in Section 2.3.1, where theprinciple of frame indifference is introduced. This principle will be used repeatedly indiscussing representations for material behavior and in preparing empirical data correlations.

The material in this section is drawn from Truesdell and Toupin (1960, p. 437), Truesdelland Noll (1965, p. 41), and Truesdell (1966a, p. 22).


1.2. Frame 15

Exercise 1.2.1-1 Let T be a frame-indifferent scalar field. Starting with the definition of thegradient of a scalar field in Section A.3.1, show that the gradient ofT is frame indifferent:

∇T∗ ≡ (∇T)∗ = Q · ∇T

Exercise 1.2.1-2 In order thatε (defined in Exercise A.7.2-11) be a frame-indifferent third-ordertensor field, prove that

ε∗ = (detQ)ei jk ei ej ek

1.2.2 Equivalent Motions

In Section 1.1, I described the motion of a material with respect to some frame of referenceby

z= χ(zκ , t) (1.2.2-1)

where we understand that the form of this relation depends upon the choice of referenceconfigurationκ. According to our discussion in Section 1.2.1, the same motion with respectto some new frame of reference is represented by

z∗ = χ∗(z∗κ , t∗)= z∗0(t)+Q(t) · [χ(zκ , t)− z0] (1.2.2-2)

We will say that any two motionsχ andχ∗ related by an equation of the form of (1.2.2-2)areequivalent motions.

Let us write (1.2.2-2) in an abbreviated form:

z∗ = z∗0 +Q · (z− z0) (1.2.2-3)

The material derivative of this equation gives

v∗ = dz∗0dt+ dQ

dt· (z− z0)+Q · v (1.2.2-4)

or

v∗ −Q · v = dz∗0dt+ dQ

dt· (z− z0) (1.2.2-5)

In view of (1.2.2-3), we may write

z− z0 = QT · Q · (z− z0)

= QT ·(z∗ − z∗0

)(1.2.2-6)

This allows us to express (1.2.2-5) as

v∗ −Q · v = dz∗0dt+(

dQdt· QT

)·(z∗ − z∗0

)= dz∗0

dt+ A ·

(z∗ − z∗0

)(1.2.2-7)


16 1. Kinematics

where

A ≡ dQdt· QT (1.2.2-8)

We refer to the second-order tensorA as theangular velocity tensor of the starred framewith respect to the unstarred frame(Truesdell 1966a, p. 24).

SinceQ is an orthogonal tensor,

Q · QT = I ∗ (1.2.2-9)

Taking the material derivative of this equation, we have

A = dQdt· QT = −Q · dQT

dt

= −Q ·(

dQdt

)T

= −AT(1.2.2-10)

In this way we see that the angular velocity tensor is skew symmetric.Theangular velocity vector of the unstarred frame with respect to the starred frameω is

defined as

ω ≡ 1

2tr(ε∗ · A) (1.2.2-11)

The third-order tensorε is introduced in Exercises A.7.2-11 and A.7.2-12 (see also Exercise1.2.1-2), where tr denotes the trace operation defined in Section A.7.3. Let us consider thefollowing spatial vector in rectangular Cartesian coordinates:

ω ∧(z∗ − z∗0

) = tr(ε∗ ·

[(z∗ − z∗0

)ω])

= tr

(ε∗ ·

{[z∗ − z∗0

] [1

2tr(ε∗ · A)

]})

= ei jk

(z∗k − z∗0k

) (1

2ejmnAnm

)e∗i

= 1

2

(z∗k − z∗0k

)(Ai j − Aki )e∗i

= (z∗k − z∗0k

)Aike∗i

= A ·(z∗ − z∗0

)(1.2.2-12)

We may consequently write (1.2.2-7) in terms of the angular velocity of the unstarred framewith respect to the starred frame (Truesdell and Toupin 1960, p. 437):

v∗ = dz∗0dt+ ω ∧ [Q · (z− z0)] +Q · v (1.2.2-13)


1.2. Frame 17

The material in this section is drawn from Truesdell and Noll (1965, p. 42) and Truesdell(1966a, p. 22).

Exercise 1.2.2-1

i) Show that velocity is not frame indifferent.ii) Show that at any position in euclidean point space a difference in velocities with respect

to the same frame is frame indifferent.

Exercise 1.2.2-2 Acceleration

i) Determine that (Truesdell 1966a, p. 24)

d(m)v∗

dt= d2z∗0

dt2+ 2A ·

(v∗ − dz∗0

dt

)+(

dAdt− A · A

)·(z∗ − z∗0

)+Q · d(m)vdt

ii) Prove that (Truesdell and Toupin 1960, p. 440)

d(m)v∗

dt= d2z∗0

dt2+(

d(m)Adt+ A · A

)· Q · (z− z0)

+ 2A · Q · v+Q · d(m)vdt

iii) Prove that

ω ∧[ω ∧

(z∗ − z∗0

)] = A · A ·(z∗ − z∗0

)dω

dt∧ ∧

(z∗ − z∗0

) = dAdt·(z∗ − z∗0

)and

ω ∧ (Q · v) = A · Q · v

iv) Conclude that (Truesdell and Toupin 1960, p. 438)

d(m)v∗

dt= d2z∗0

dt2+ dω

dt∧ · [Q · (z− z0)]

+ ω∧ {ω ∧ [Q · (z− z0)]}

+ 2ω ∧ · (Q · v)+Q · d(m)vdt

Exercise 1.2.2-3 Give an example of a scalar that is not frame indifferent.

Hint: What vector is not frame indifferent?


18 1. Kinematics

Exercise 1.2.2-4 Motion of a rigid body Determine that the velocity distribution in a rigid body maybe expressed as

v∗ = dz∗0dt+ ω ∧ (

z∗ − z∗0)

What is the relation of the unstarred frame to the body in this case?

1.3 Mass

1.3.1 Conservation of Mass

This discussion of mechanics is based upon several postulates. The first is

Conservation of mass The mass of a body is independent of time.

Physically, this means that, if we follow a portion of a material body through any numberof translations, rotations, and deformations, the mass associated with it will not vary as afunction of time. Ifρ is themass densityof the body, the mass may be represented as∫

R(m)

ρ dV

HeredV denotes that a volume integration is to be performed over the regionR(m) of spaceoccupied by the body in its current configuration; in generalR(m), or the limits on thisintegration, is a function of time. The postulate of conservation of mass says that

d

dt

∫R(m)

ρ dV = 0 (1.3.1-1)

Notice that, like the material particle introduced in Section 1.1,massis a primitive concept.Rather than defining mass, we describe its properties. We have just examined its mostimportant property: It is conserved. In addition, I will require that

ρ > 0 (1.3.1-2)

and that the mass density be a frame-indifferent scalar,

ρ∗ = ρ (1.3.1-3)

Our next objective will be to determine a relationship that expresses the idea of conserva-tion of mass at each point in a material. To do this, we will find it necessary to interchangethe operations of differentiation and integration in (1.3.1-1). Yet the limits on this integraldescribe the boundaries of the body in its current configuration and generally are functionsof time. The next section explores this problem in more detail.

1.3.2 Transport Theorem

Let us consider the operation

d

dt

∫R(m)

9 dV

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