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2
Foundations for Momentum Transfer
I
N WHAT FOLLOWS, the principal tools for studying fluid mechanics are developed.
We begin by introducing the concept of force. Notice that force is not defined; it is a
primitive concept in the same sense as is mass and the material particle in Chapter 1. This
forms the basis for introducing our second and third postulates: the momentum balance and
the moment of momentum balance. The stress tensor is introduced in order to derive the
equations that describe at each point in a material the local balances for momentum and
moment of momentum. The differential mass and momentum balances together with the
symmetry of the stress tensor form the foundation for fluid mechanics.
We conclude our discussion with an outline of what must be said about real material
behavior if we are to analyze any practical problems. It is especially at this point that
statistical mechanics, based upon the molecular viewpoint of real materials, can be used to
supplement the concepts developed in continuum mechanics. In continuum mechanics, we
can indicate a number of rules that constitutive equations for the stress tensor must satisfy
(the principle of determinism, the principle of local action, the principle of material frame
indifference,...), but from first principles we cannot derive an explicit relationship between
stress and deformation. If we work strictly within the bounds of continuum mechanics, we
can derive such a relationship only by making some sort of assumption about its form. Our
feelings are that the most interesting advances in describing material behavior result from
predictions based upon simple molecular models that are generalized through the use of the
statements about material behavior that have been postulated in continuum mechanics. For
an excellent brief summary of what can be said about material behavior from a molecular
point of view, see Bird et al. (1960, Chap. 1).
Although our direct concerns here are for momentum transfer, practically all of the ideas
developed will be applied again in examining energy and mass transfer. We firmly believe
that the best foundation for energy and mass-transfer studies is a clear understanding of fluid
mechanics.
2.1
Force
Like the material particle, force is a primitive concept. It is not defined. Instead we describe its attributes in a series of five axioms.
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2.1. Force
29
Corresponding to each body B, there is a distinct set of bodies Be such that the mass of
the union of these bodies is the mass of the universe. We refer to Be as the exterior or the
surroundings of the body B.
a) A system of forces is a vector-valued function f(B, C) of pairs of bodies. The value of
f(#, C) is called the force exerted on the body B by the body C.
b) For a specified body B, f(C, B e ) is an additive function defined over the subbodiesC of B.
c) Conversely, for a specified body B, f(#, C) is an additive function defined over the subbodies C of Be.
d) In any particular problem, we regard the forces exerted upon a body as being given a priori
to all observers; all observers would assume the same set of forces in a given problem.
In prescribing these forces, we specify a particular dynamic problem. We consequently
assume that all forces are independent of the observer or are frame indifferent (Truesdell
1966a, p. 27) (see Section 1.2.1):
f* = Q . f
(2.1.0-1)
There are three types of forces with which we may be concerned:
External forces These arise at least in part from outside the body and act upon the material particles
of which the body is composed. One example is the uniform force of gravity. Another example
would be the electrostatic force between two charged bodies. Let P indicate a portion of a
body B as illustrated in Figure 2.1.0-1. Taking fe to be external force per unit mass that the
surroundings Be exert on the body B, we write the total external force acting on P in terms
of a volume integral over the region occupied by P:
f
RP
In general, the external force per unit mass is a function of position, and fe should be regarded
as a spatial vector field.
Mutual forces These arise within a body and act upon pairs of material particles. The long-range
intermolecular forces acting between a thin film of liquid and the solid upon which it rests
are mutual forces. We can imagine a body in which there is a distribution of electrostatic
charge; we would speak of the electrostatic force between one portion of the body with a net
positive charge and some other element of the body with a net negative charge as being a
Figure 2.1.0-1. The body B of which P is a
portion.
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30
2. Foundations for Momentum Transfer
mutual force. Let fm be the mutual force per unit mass that B — Pl exerts upon P; the total
mutual force acting upon P may be represented as an integral over the volume of P?
f
pfmdV
JRP
We should expect the mutual force per unit mass generally to be a function of position within
the material; fm should be viewed as a spatial vector field.
Contact forces These forces are not assignable as functions of position but are to be imagined as
acting upon the bounding surface of a portion of material in such a way as to be equivalent
to the force exerted by one portion of the material upon another beyond that accounted for
through mutual forces. In typing you exert a contact force upon the keys of the typewriter. If
we deform some putty in our hands, during the deformation any one portion of the putty exerts
a contact force upon the remainder at their common boundary. Let t = t(z, P) represent the
stress vector or force per unit area that B — P exerts upon the boundary of P at the position
z. This force per unit area t is usually referred to as stress. The total contact force that B — P
exerts upon P may be written as an integral over the bounding surface of P:
f
tdA
JSp
The fifth axiom, the stress principle, specifies the nature of the contact load.
e) Stress principle There is a vector-valued function t(z, n) defined for all points z in a
body B and for all unit vectors n such that the stress that B — P exerts upon any portion
P of B is given by
t(z, P) = t(z, n)
(2.1.0-2)
Here n is the unit normal that is outwardly directed with respect to the closed bounding
surface of P. The spatial vector t = t(z, n) is referred to as the stress vector at the position
z acting upon the oriented surface element with normal n; n points into the material that
exerts the stress t upon the surface element.
The material in this section is drawn from Truesdell (1966b, p. 97), Truesdell and Toupin
(1960, pp. 531 and 536) and Truesdell and Noll (1965, p. 39).
2.2
Additional Postulates
2.2.1
Momentum and Moment of Momentum Balance
In Section 1.3.1, we introduced our first postulate, conservation of mass. Our second postulate
is (Truesdell 1966b, p. 97; Truesdell and Toupin 1960, pp. 531 and 537; Truesdell and Noll
1965, p. 39)3
1
2
3
We define B - P to be such that B = (B - P) U P and (B - P) n P = 0.
We recognize here that the sum of the mutual forces exerted by any two parts of P upon each other is
zero (Truesdell and Toupin 1960, p. 533).
Truesdell and Toupin (1960, pp. 531 and 534) point out that "the laws of Newton . . . are neither
unequivocally stated nor sufficiently general to serve as a foundation for continuum mechanics."
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2.2. Additional Postulates
3I
Momentum balance The time rate of change of the momentum of a body relative to an inertial
frame of reference is equal to the sum of the forces acting on the body.
Let the volume and closed bounding surface of a body or any portion of a body be denoted,
respectively, as Rim) and S(m). Referring to our discussion of forces in the introduction to
Section 2.1, in an inertial frame of reference we may express the momentum balance as
d
It
f
p\dV = f
tdA+
f
pfdV
(2.2.1-1)
Here f is the field of external and mutual forces per unit mass:
f=f,+fw
(2.2.1-2)
In most cases, the effect of mutual forces can be neglected with respect to external forces,
one of the primary exceptions being thin films, where long-range, intermolecular forces
exerted by the adjoining phases must be taken into account. Hereafter we will assume that
mutual forces have been dismissed, and we will refer to f as the field of external forces per
unit mass.
Our understanding is that (2.2.1-1) is written with respect to an inertialframe of reference.
In reality, we define an inertial frame of reference to be one in which (2.2.1-1) and (2.2.1-3)
are valid. Somewhat more casually we describe an inertial frame of reference to be one that
is stationary with respect to the fixed stars.
Our third postulate is (Truesdell 1966b, p. 97; Truesdell and Toupin 1960, pp. 531 and
537; Truesdell and Noll 1965, p. 39).
Moment of momentum balance The time rate of change of the moment of momentum of a body
relative to an inertial frame of reference is equal to the sum of the moments of all the forces
acting on the body.
In an inertial frame of reference, the moment of momentum balance assumes the form
— /
p(pA\)dV = I
pAtdA+
f
p(pAf)dV
(2.2.1-3)
In writing the moment of momentum balance in this manner we confine our attention to
the so-called nonpolar4 case [i.e., we assume that all torques acting on the body are the
result of forces acting on the body (Truesdell and Toupin 1960, pp. 538 and 546; Curtiss
1956; Livingston and Curtiss 1959; Dahler and Scriven 1961)]. For example, it is possible
to induce a local source of moment of momentum by use of a suitable rotating electric field
(Lertes 1921a,b,c; Grossetti 1958,1959). In such a case it might also be necessary to account
for the flux of moment of momentum at the bounding surface of the body. Effects of this
type have not been investigated thoroughly, but they are thought to be negligibly small for
all but unusual situations. Consequently, they are neglected here.
4
When molecules are referred to as nonpolar, it indicates that their dipole moment is zero. This is an
entirely different use of the word than that intended here, where nonpolar means that all torques acting
on the material are the result of forces.
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32
2. Foundations for Momentum Transfer
Figure 2.2.2-1. A material body in the form of a tetrahedron.
Exercise 2.2.1 -1 Cauchy's lemma Consider two neighboring portions of a continuous body. Apply
the momentum balance to each portion and to their union. Deduce that on their common
boundary
t(z, n) = -t(z, - n )
or
Cauchy's lemma The stress vectors acting upon opposite sides of the same surface at a given point
are equal in magnitude and opposite in direction.
2.2.2
Stress Tensor
We ask here how t(z, n) varies as the position z is held fixed and n changes.
At any point in a body, consider the tetrahedron shown in Figure 2.2.2-1. Three sides
are mutually orthogonal and coincide with a set of rectangular Cartesian coordinate planes
intersecting at z; the fourth side has an outwardly directed normal n. Let the altitude of the
tetrahedron be h; let the area of the inclined face be A. In terms of the rectangular Cartesian
basis fields, we may write n = «,e,.
Let us apply the momentum balance (2.2.1-1) to the material in the tetrahedron at time t
to obtain
—f
pvdV = f
tdA+ f
pfdV
(2.2.2-1)
Applying the form of the transport theorem introduced in Exercise 1.3.3-1 to the term on
the left and using the theorem of mean value to evaluate the surface integral, we have
L(
d(m)\
dV=Alt*
+ —
(2.2.2-2)
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2.2. Additional Postulates
33
Here the asterisk denotes a mean value of the function, t* denotes the contact stress that
the surroundings exert upon the inclined face, and t* denotes the contact stress that the
surroundings exert upon the face whose inwardly directed unit normal is e?. Applying the
theorem of mean value to the term on the left, we write
^
= A(t* + « 1 tt + «2^ + n3t*)
(2.2.2-3)
Since the cosines of the angles between the Cartesian coordinate planes and the inclined
plane are «i, ri2, and « 3 , respectively, the areas of the faces of the tetrahedron lying in the
coordinate planes are n\A, /22A, and «3A. Consider a sequence of geometrically similar
tetrahedrons; in the limit as h approaches zero, we obtain
t = -(t n i +t/!2 + t*3)
(2.2.2-4)
where all stress vectors are evaluated at the point z. From their definitions, the quantities
ti, t2, and t3 do not depend upon n.
With the convention that 7im is the kth component of the stress vector acting upon the
positivt side of the plane zm = constant (such that the unit normal is em), by Cauchy's lemma
(Exercise 2.2.1-1) we write
-ti = Tei
- t = 7-e /
(2.2.2-5)
-t =
This permits us to express (2.2.2-4) as
t = njTijei
(2.2.2-6)
The matrix [7}y] defines a second-order stress tensor T:
T = Tije&j
(22.2-1)
and (2.2.2-6) becomes
t =T•n
(2.2.2-8)
Remember in using (2.2.2-8) that n is the unit normal directed into the material that is
exerting the force per unit area t(z, n) at the position z on the surface.
Exercise 2.2.2-1
Show that the stress tensor is frame indifferent.
Exercise 2.2.2-2
In going from (2.2.2-2) to (2.2.2-3), prove that
m = ^-,
A
i = 1,2,3
2.2.3 Differential and Jump Momentum Balances
A modification of the transport theorem that takes into account the postulate that mass is
conserved (see Exercise 1.3.6-3) may be used to express the momentum balance (2.2.1-1)
as
p^-dV
dt
+ f [pv(v - u) • (\dA = f tdA+ f pfdV
JE
JS(m)
JR{m)
(2.2.3-1)
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34
2. Foundations for Momentum Transfer
If we express the stress vector t in terms of the stress tensor T as suggested in Section 2.2.2,
the first term on the right of this equation may be rearranged by an application of Green's
transformation (Section A.I 1.2):
f tdA = f T •ndA
'sim}
Jslm)
= ff divlW + / [T•£]dA
(2.2.3-2)
Substituting (2.2.3-2) into (2.2.3-1), we have
f
dV + [ [pv(v-u) '£-T.£\dA
=0
(2.2.3-3)
Remember that (2.2.3-1) is written for an arbitrary portion P of a body. Whether P is
very large or arbitrarily small or whether P consists of one phase or many phases,
(2.2.3-3) remains valid. This implies that the differential momentum balance (Cauchy' s first
"law"),
p
(2.2.3-4)
dt
must be satisfied at each point within each phase and that the jump momentum balance,
[pv(v-u) - £ - T . £ ] = 0
(2.2.3-5)
must be obeyed at each point on each phase interface. Here u • £ is the speed of displacement
of the phase interface; the boldface bracket notation is defined in Section 1.3.5.
Exercise 2.2.3-1 Archimedes principles Prove the following two theorems of Archimedes (Rouse
andlnce l957, p. 17):
If a solid lighter than a fluid be forcibly immersed in it, the solid will be driven upwards
by a force equal to the difference between its weight and the weight of the fluid displaced.
A solid heavier than a fluid will, if placed in it, descend to the bottom of the fluid, and
the solid will, when weighed in the fluid, be lighter than its true weight by the weight of
the fluid displaced.
Hint: Apply the momentum balance and extend the definition of fluid pressure into the
region of space occupied by the solid, recognizing that pressure in the fluid is a constant in
any horizontal plane.
Exercise 2.2.3-2 Another theorem of Archimedes Prove (Rouse and Ince 1957, p. 17):
Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the
weight of the solid will be equal to the weight of the fluid displaced.
Hint: Proceed as in Exercise 2.2.3-1, making use of the jump momentum balance
(2.2.3-5).
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2.2. Additional Postulates
35
Exercise 2.2.3-3 Another frame of reference
i) Use the result of Exercise 1.2.2-2 to write the differential momentum balance for an
arbitrary frame of reference:
- 2a; A (y* - -*^\
1 = divT* + pf*
ii) Determine that the differential momentum balance assumes the form of (2.2.3-4) in
every frame of reference that moves at a constant velocity (without rotation) relative to
the inertial frame of reference.
2.2.4
Symmetry of Stress Tensor
We again restrict ourselves to a two-phase body in which all quantities are smooth functions
of position and time as described at the beginning of Section 2.2.3.
After an application of the transport theorem (see Exercise 1.3.6-3) and the differential
mass balance, the moment of momentum balance (2.2.1-3) may be written as
f
{
JR
-n)dA+
pA(T-n)dA + f
**S{m)
p(pAf)dV
(2.2.4-1)
JR(m)
In writing this equation we have also expressed the stress vector in terms of the stress tensor
as described in Section 2.2.2. Let us consider these expressions individually.
From the left-hand term, we have
^
) t P A V)
dt
^
dt
Av +
P
A ^
(2.2.4-2)
dt
But since we have defined (Section 1.1)
v^ ^
(2.2.4-3)
dt
we are left with
H p
Av) = p A ^
(2.2.4-4)
By an application of Green's transformation (Section A.I 1.2), the first term on the right
of (2.2.4-1) becomes
f p A (T • n)dA = f
~ (eijkZjTkm)
Jsm
JR(m) dZm
dV e, + f [ p A (T . £)\dA
JZi
(2.2.4-5)
We find that
T
j T k m )
i
= eijk—- L T
ozm
km
"lien,
+ eiJkZj——-
(2.2.4-6)
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36
2. Foundations for Momentum Transfer
But
= Sim
(2.2.4-7)
Equations (2.2.4-6) and (2.2.4-7) allow us to conclude that
— ,„_,_„>;
= eijkTkjti + p A (divT)
(2.2.4-8)
dzm
Equations (2.2.4-4), (2.2.4-5), and (2.2.4-8) enable us to express (2.2.4-1) in the form
/ [P A (P^df ~ dlVT ~ Pf ) " ei*Tk'A dV
+ I [p A v(v - u) • £ - T • £]dA
= 0
(2.2.4-9)
In view of the differential momentum balance and the jump momentum balance (Section
2.2.3), this becomes
y e
/
d V
=0
(2.2.4-10)
But (2.2.4-1) was written for a portion of a body with the understanding that the portion
considered might be arbitrarily large or small. This implies from (2.2.4-10) that
e ij kTkjei=0
(2.2.4-11)
Since the rectangular Cartesian basis fields are linearly independent, we have
eijkTkj = 0
(2.2.4-12)
We may also write (see Exercise A.2.1-2)
eimnerkTk- — 0
Tnm -Tmn=0
(2.2.4-13)
Equation (2.2.4-13) expresses
Symmetry of stress tensor A necessary and sufficient condition for the moment of momentum
balance to be satisfied at every point within a phase is that the stress tensor be symmetric
(Cauchy's second "law"):
T = Tr
(2.2.4-14)
as long as the differential momentum balance is satisfied and the body conforms to the
nonpolar case (Section 2.2.1).
Hereafter, we need not worry about satisfying the moment of momentum balance, as long
as we take the components of the stress tensor to be symmetric.
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2.3. Behavior of Materials
37
Exercise 2.2.4-1 Orientation of suspended body An irregular solid body is suspended in a fluid from
a swivel that permits it to assume an arbitrary orientation. The density of the solid may be
either less than or greater than the density of the fluid. If iSWiVei is the position vector locating
the point at which the swivel is attached to the body and if
zc = /
(p (s) - p(f>) zdV \ f
(p(s) - p(f)) d
prove that the orientation of the body will be such that zc — zswivei is parallel to gravity.
Exercise 2.2.4-2 Orientation of floating body An irregular solid body is floating at an interface
between two fluids, A and B. Let R^ be the region of the solid surrounded by fluid A and
the A-B interface; let R(B) be the region of the solid surrounded by fluid B and the A-B
interface. If
and
prove that the orientation of the floating solid will be such that zcA — ZCB is parallel to gravity.
2.3
Behavior of Materials
2.3.1
Some General Principles
It should occur to you that we have said nothing as yet about the behavior of materials. Mass
conservation, the momentum balance, and the moment of momentum balance are stated
for all materials. Yet our experience tells us that under similar circumstances air and steel
respond to forces in drastically different manners. Somewhere in our theoretical structure
we must incorporate this information.
To confirm this intuitive feeling, let us consider the mathematical structure we have
developed. For simplicity, assume that the material is incompressible so that mass density
p is a known constant. Let us also assume that the description of the external force field f is
given; for example, we are commonly concerned with physical situations in an essentially
uniform gravitational field. As unknowns in some arbitrary coordinate system we are left
with the three components of the velocity vector v and the six components of the symmetric
stress tensor T (Section 2.2.4). As equations, we have the differential mass balance (Section 1.3.3) and the three components of the differential momentum balance (Section 2.2.3).
This means we have four equations in nine unknowns. This reinforces our intuitive feeling
that further information is required.
While we have assumed the nature of the external force is known, we have said nothing
about the character of the force that one portion of a body exerts on its neighboring portion.
We indicated in Section 2.2.1 that we would neglect mutual forces. This leaves only the
contact force. We must describe how the contact forces in a body depend upon the motion
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38
2. Foundations for Momentum Transfer
and deformation of the body. More specifically, we must say how the stress tensor T varies
with the motion and deformation of the body.
Before plunging ahead to relate the stress tensor to motion, let us see if our everyday
experience in observing materials in deformations and motions will help us to lay down
some rules governing such a relation. For example, it seems obvious that what happens to
a body in the future is going to have no influence on the present stress tensor field. This
suggests stating (Truesdell 1966b, p. 6; Truesdell and Noll 1965, p. 56)
The principle of determinism The stress in a body is determined by the history of the motion that
the body has undergone.
Our experience is that motion in one portion of a body does not necessarily have any
effect on the state of stress in another portion of the body. For example, if we lay down
a bead of caulking compound, we may shape one portion of the bead with a putty knife
without disturbing the rest. From a somewhat different point of view, the physical idea of a
contact force suggests that the circumstances in the immediate neighborhood of the point in
question determine it. We may state this as (Truesdell 1966b, p. 6; Truesdell and Noll 1965,
p. 56)
The principle of local action The motion of the material outside an arbitrarily small neighborhood
of the material point f may be ignored in determining the stress at this material point.
Let us consider an experiment in which a series of weights are successively added to
one end of a spring, the other end having been attached to the ceiling of a laboratory. Two
experimentalists observe this experiment, one standing on the floor of the laboratory near the
spring and the other standing across the room on a turntable that rotates with some angular
velocity. The frame of reference for the first observer might be his backbone, his shoulders,
and his nose or perhaps the walls of the laboratory. The frame of reference for the second
observer consists of the turntable's axis and a series of lines painted upon the turntable; to
this observer, the spring and weights appear to be revolving in a circle. Yet we expect both
observers to come to the same conclusions regarding the behavior of the spring under stress.
Referring to Sections 1.2.1 and 1.2.2 and to Exercise 2.2.2-1, we can summarize our feeling
here with (Truesdell and Noll 1965, p. 44)5
The principle of frame indifference Descriptions of material behavior must be invariant under
changes of frame of reference.
If a description of stress-deformation behavior is satisfied for a process in which the stress
tensor and motion are given by
T = T(z K ,t)
5
(2.3.1-1)
In writing this text, I have made an effort to refer the reader to the more lucid reference rather than the
historical "first" when a choice has been necessary. A choice was necessary here, because the essential
idea of the principle of material frame indifference had been stated by several authors prior to Noll
(1958). Oldroyd (1950) in particular attracted considerable interest with his viewpoint. Truesdell and
Noll (1965, p. 45) have gone back to the seventeenth century to trace the development of this idea
through the literature.
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2.3. Behavior of Materials
39
and
Z = X,(W)
(2.3.1-2)
then it must also be satisfied for any equivalent process described with respect to another
frame of reference. In particular, the description of material behavior must be satisfied for a
process in which the stress tensor and motion are given by
(2.3.1-3)
+ Q(0 • [X&K, 0 - z0]
(2.3.1-4)
and
t* = t-a
(2.3.1-5)
It is possible to make further statements in much the same manner as above6 (Truesdell
and Toupin 1960, p. 700; Truesdell and Noll 1965, p. 101). These principles may be used to
help in the construction of particular constitutive equations for the stress tensor. The type of
argument involved is illustrated in the following section.
2.3.2 Simple Constitutive Equation for Stress
In the preceding section we discussed three principles that every description of stressdeformation behavior should satisfy. Let us propose a simple stress-deformation relationship
that is consistent with these principles.
We can satisfy the principle of determinism by requiring the stress to depend only upon a
description of the present state of motion in the material. Both the principle of determinism
and the principle of local action are satisfied if we assume that the stress at a point is a
function of the velocity and velocity gradient at that point:
T = H(v, Vv)
(2.3.2-1)
It is understood that stress may also depend upon local thermodynamic state variables, but
since this dependence is not of primary concern as yet, it is not shown explicitly.
Every second-order tensor can be written as the sum of a symmetric tensor and a skewsymmetric tensor. For example, the velocity gradient may be expressed as
1
1
T
Vv = -[Vv + (Vv)r] + -[Vv - D+ W
6
r
(Vvf]
(2.3.2-2)
It is traditional for discussions rooted in the thermodynamics of irreversible processes to apply Curie's
"law" and the Onsager-Casimir reciprocal relations in developing linear descriptions of material behavior (Bird 1993). Usually these relations are employed in place of the principle of frame indifference
and the entropy inequality (the second law of thermodynamics, discussed in Sections 5.2 and 5.3) (Bird
1993). Truesdell (1969, p. 134) follows the historical evolution of Curie's law and the Onsager-Casimir
reciprocal relations, and he finds them to be without merit.
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40
2. Foundations for Momentum Transfer
where
D=
[Vv + (Vv)]
(2.3.2-3)
is the rate of deformation tensor and
W = ~[Vv - (Vv) r ]
(2.3.2-4)
is the vorticity tensor. This allows us to rewrite (2.3.2-1) as
T = H(v, D + W)
(2.3.2-5)
The principle of frame indifference discussed in Section 2.3.1 requires that
T* = Q • T • Q r = H(v*, D* + W*)
(2.3.2-6)
From Section 1.2.2, Exercise 2.3.2-1, and Equation (2.3.2-5), we find that the function H
must be such that
Q . H ( v , D + W) - Q
r
=
( ^
^
+ ^ ^ r Q)
dt
Let us choose a particular change of frame such that
WQ
dt
r
dt
(2.3.2-7)
(2.3.2-8)
and
^ = - Q. W
dt
With this change of frame, we have from (2.3.2-6) and (2.3.2-7)
(2.3.2-9)
T* = G(D*)
= H ( 0 , Q • D • Q r +0)
(2.3.2-10)
Applying the principle of frame indifference, we conclude from (2.3.2-10) that
T = G(D)
(2.3.2-11)
Equation (2.3.2-7) requires that the function G must satisfy
Q•G(D)•Qr = G(Q•D•Qr)
(2.3.2-12)
Let us try to give a physical interpretation to our use of the principle of frame indifference
in the preceding paragraphs. We begin with an observer at the position z in an arbitrary
frame of reference who assumes that the stress tensor depends upon velocity, the rate of
deformation tensor, and the vorticity tensor; (2.3.2-5). Equations (2.3.1-4), (2.3.2-8), and
(2.3.2-9) describe another observer at the position z* in a new frame of reference. The
observer at z* rotates and translates with the material in such a way that for him the velocity
v* and the vorticity tensor W* are zero. He can see no dependence of the stress tensor T*
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2.3. Behavior of Materials
41
upon v* and W*; he sees a dependence of T* on D* alone in (2.3.2-10). But the principle
of frame indifference requires that all observers come to the same conclusions about the
behavior of materials. We consequently conclude that (2.3.2-5) reduces to (2.3.2-11).
The most general form that (2.3.2-11) can take in view of (2.3.2-12) is (Truesdell and
Noll 1965, p. 32)
D
(2.3.2-13)
where
Kk
= Kk(IDJ ID,IIID)
(2.3.2-14)
Here /#, IID, and HID are the three principal invariants of the rate of deformation tensor
(i.e., the coefficients in the equation for the principal values of D):
det(D - ml) = -m3 + hm2
- IIDm + IIID
= 0
(2.3.2-15)
where
/D = t r D
= divv
[(/ O ) -HD]
(2.3.2-16)
(2.3.2-17)
/7D s i r (D-D)
(2.3.2-18)
I I I = detD
(2.3.2-19)
Equation (2.3.2-13) was first obtained by Reiner (1945) and Prager (1945) for functions
G(D) in the form of a tensor power series (Truesdell and Noll 1965, p. 33).
Notice that this constitutive equation for stress automatically satisfies the symmetry of
the stress tensor, since the rate of deformation tensor is symmetric.
It follows immediately from (2.3.2-13) that the most general linear relation between the
stress tensor and the rate of deformation tensor that is consistent with the principle of material
frame indifference is
T = (a + k div v)I + 2 / I B
(2.3.2-20)
In Section 5.3.4 (Truesdell and Noll 1965, p. 357), we find that this reduces to the
Newtonian fluid,
(2.3.2-21)
where P is the thermodynamic pressure (Section 5.3.1),
fi > 0
(2.3.2-22)
is the shear viscosity, and
2
X > —-/ x
(2.3.2-23)
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42
2. Foundations for Momentum Transfer
It is often stated that the bulk viscosity
2
K = A + -//,
(2.3.2-24)
= 0
and that this has been substantiated for low-density monatomic gases. In fact, this result is
implicitly assumed in that theory (Truesdell 1952, Sec. 61 A). To our knowledge, experimental measurements indicate that X is positive and that for many fluids it is orders of magnitude
greater than fi (Truesdell 1952, Sec. 61A; Karim and Rosenhead 1952).
Another special case of (2.3.2-20) is the incompressible Newtonian fluid:
T = -pi + 2/xD
(2.3.2-25)
Equation (2.3.2-25) is sometimes described as a special case of (2.3.2-21). (Note that the
thermodynamie pressure P is not defined for an incompressible fluid.) The quantity p is
known as the mean pressure. From (2.3.2-25), we see that we may take as its definition
p = --trT
(2.3.2-26)
When discussing stress-deformation behavior, it is common to speak in terms of the
viscous portion of the stress tensor,
S = T + PI
(2.3.2-27)
In this way, the strictly thermodynamie quantity P is separated from those effects arising
from deformation. For incompressible fluids, we define the viscous portion of the stress
tensor as
S = T + pi
(2.3.2-28)
where p is the mean pressure (2.3.2-26).
Although the Newtonian fluid (2.3.2-21) has been found to be useful in describing the
behavior of gases and low-molecular-weight liquids, it has not been established that any
fluid requires a nonzero value for /C2 or a dependence of K\ upon IIID in (2.3.2-13). However,
a number of empirical relations based upon limiting forms of (2.3.2-13) have been found to
be of some engineering value. A few of these are discussed in the next section.
Exercise 2.3.2-!
(Truesdell 1966a, p. 25)
i) Let us define the deformation gradient as
F=
where %K is
the
deformation of the body (see Section 1.1),
and grad denotes the gradient operation in the reference configuration of the material.
Note that we have used the same rectangular Cartesian coordinate system (or set of basis
vectors) both in the current configuration of the body and in the reference configuration
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2.3. Behavior of Materials
43
of the body. Use the definition of the material derivative (1.1.0-12), the definition of the
velocity vector (1.1.0-13), and the chain rule to show that
A, JS
= (Vv) • F
dt
and that
V
v
dt
F
ii) Let Q be a time-dependent orthogonal transformation associated with a change of frame
(Section 1.2.1) and let the motions x a n d x* be referred to the same reference configuration. Starting with (2.3.1-4), show that
F* = Q • F
iii) Take the material derivative of this equation and show that
(Vv)* • F* = Q • (Vv) • Q r • F* + — • Q r • F*
dt
Here the asterisk indicates an association with the new frame.
iv) The decomposition of a second-order tensor into skew-symmetric and symmetric portions is unique. Make use of this fact to show that
D* = Q • D . Q r
and
W* = Q • W • Q r + A
We conclude that the rate of deformation tensor D is frame indifferent, whereas the
vorticity tensor W is not. Note that the angular velocity tensor A defined by (1.2.2-8) is
shown to be skew symmetric in (1.2.2-10).
Exercise 2.3.2-2
Starting with tr D, show that
trD = divv
Exercise 2.3.2-3 The existence of a hydrostatic pressure Consider a fluid described by (2.3.2-11)
and (2.3.2-12). We see from (2.3.2-13) that, when there is no flow,
Prove this result directly from (2.3.2-11) and (2.3.2-12) by means of Exercise A.5.2-4.
2.3.3
Generalized Newtonian Fluid
At the present time it has not been established that any of the many fluids for which the
Newtonian fluid is inadequate can be described by (2.3.2-13). But empirical models based
upon (2.3.2-13) may have some utility, in that they predict some aspects of real fluid behavior
in a restricted class of flows known as viscometric flows.
In a viscometric flow, a material particle is subjected to a constant deformation history,
so that memory effects are wiped out. Examples of viscometric flows are flow through a
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44
2. Foundations for Momentum Transfer
tube, Couette flow, and flow in a cone-plate viscometer under conditions such that inertial
effects can be neglected. Unsteady-state flows, flow in a periodically constricted tube, flow
through a porous rock, and flow through a pump are examples of nonviscometric flows.
The most common class of empirical models for incompressible fluids based upon
(2.3.2-13) is the generalized Newtonian fluid, which can be written in two forms.
Primary Form
The primary form of the generalized Newtonian fluid can be expressed as
S = T + pi
= 2t](y)D
(2.3.3-1)
where
Y =
= <v/2tr(D.D)
(2.3.3-2)
We refer to S as the viscous portion of the stress tensor; it is common to call rj(y) the apparent
viscosity by analogy with (2.3.2-25). From the differential entropy inequality, we can show
that (Exercise 5.3.4-1)
t](y) > 0
(2.3.3-3)
It is well to keep in mind that rj(y) should include a dependence upon all of the parameters
required to describe a fluid's behavior. Truesdell (1964; Truesdell and Noll 1965, p. 65)
argues that, no matter how many parameters describe a fluid's behavior, only two of them
have dimensions, a characteristic viscosity /xo and a characteristic (relaxation) time 50:
1 = rj(y, /x0, so)
(2.3.3-4)
With this assumption, the Buckingham-Pi theorem (Brand 1957) requires that
— = if (soY)
Mo
(2.3.3-5)
A number of suggestions have been made for the definitions of /XO and SO (Bird et al. 1977).
For example, /XQ could be identified as the viscosity of the fluid in the limit y < yo (all fluids
are Newtonian in the limit y —• 0).
One of the most common two-parameter generalized Newtonian models is the Ostwald—
de Waele model or power-law fluid (Reiner 1960, p. 243):
= Hom* (soy)""1
= m{yf~l
(2.3.3-6)
Here
m = /xom V 1 " 1
(2.3.3-7)
and n are parameters that must be determined empirically. (Since s0 and /xom* appear only in
combination, we don't count them as independent parameters.) When n — 1 and yu^m* = /x,
the power-law fluid reduces to the incompressible Newtonian fluid (2.3.2-25). Since this
model is relatively simple, it has been used widely in calculations. Its disadvantage is that it
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2.3. Behavior of Materials
45
does not reduce to Newtonian behavior either in the limit y -> 0 or in the limit y —> oo as
we currently believe all real fluids do. For most polymers and polymer solutions, n is less
than unity. For this case, (2.3.3-6) predicts an infinite viscosity in the limit of zero rate of
deformation and a zero viscosity as the rate of deformation becomes unbounded.
Hermes and Fredrickson (1967) proposed one possible superposition of Newtonian and
power-law behavior:
„
(2.3.3-8)
where a\/ zo, and n are experimentally determined parameters. (Since a* and s0 appear only
in combination, we do not consider them as independent parameters.) If we assume that n is
less than unity, it predicts a lower-limiting viscosity /z0 as y —• 0. [Equation (2.3.3-8) often
may prove to be more useful than the Ellis fluid (described below), since the stress tensor is
given as an explicit function of the rate of deformation tensor.]
The Sisko fluid (Bird 1965b; Sisko 1958) is another superposition of Newtonian and
power-law behavior:
for y < y0 : rj(y) =
(2.3.3-9)
Here /X0, yo = 1Ao» and a are parameters whose values depend upon the particular material
being described. It properly predicts a lower-limiting viscosity /AQ as y - • 0, but it cannot
be used for y > y0, since a is usually between 1 and 3 (Bird 1965b).
The Bingham plastic (Reiner 1960, p. 114) is of historical interest but of limited current
practical value. It describes a material that behaves as a rigid solid until the stress has
exceeded some critical value:
for T > T0 : rj(y) = m + ~
(2.3.3-10)
y
forr < r0 : D = 0
(2.3.3-11)
This model contains two parameters: RJO and To. It was originally proposed to represent the
behavior of paint. The idea of a critical stress R0 at which the rigid solid yielded and began
to flow probably was postulated on the basis of inadequate data in the limit r —> 0. Though
later work has failed to establish that any materials are true Bingham plastics, the concept
is firmly established in the older literature.
Alternative Form
Let us define
.S)
(2.3.3-12)
From (2.3.3-1),
r = t(y)
= n(y)Y
(2.3.3-13)
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46
2. Foundations for Momentum Transfer
We assume that rj(y) is a differentiable function, which means
dx
T(y) - r(0)
= lim
dy~ y=0
'
= lim —— = ?7(0)
y^O
(2.3.3-14)
y
Equations (2.3.3-3) and (2.3.3-14) require that
dx
dy
> 0
(2.3.3-15)
y=0
If the derivative dx/dy is continuous, it must be positive in some neighborhood of y = 0.
In this neighborhood, x(y) will be a strictly increasing function of y and, for this reason, it
will have an inverse:
Y = k(x)
(2.3.3-16)
Equation (2.3.3-16) follows from (2.3.3-1) for y sufficiently close to zero. It is possible
that x(y) ceases to increase when y exceeds some value, but such behavior has not been
observed experimentally.
From (2.3.3-13) and (2.3.3-16),
y
ir
1
X(x)
v
x'
r]{y)
1
rj(X(x))
(2 3 3 17)
This suggests that we may write (2.3.3-1) in the alternative form
2D = (p(x)S
(2.3.3-18)
where
<p(x) = - 1 —
ri(X(x))
is referred to as the fluidity.
By analogy with (2.3.3-4) and (2.3.3-5), if we assume
(p = (p(x, n o , so)
(2.3.3-19)
(2.3.3-20)
the Buckingham-Pi theorem (Brand 1957) requires that
VLo<P = <P*( — )
VMo/
The power-law fluid (2.3.3-6) can be written in this alternative form:
Mo
=
l
m~
(2.3.3-21)
V Mo
/nx(l-n)/n
(2.3.3-22)
It still does not reduce to Newtonian behavior in the limit r —> 0 or in the limit x ~* oc as
real fluids do.
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2.3. Behavior of Materials
47
The Ellis fluid (Reiner 1960, p. 246; Bird 1965b) is still another superposition of Newtonian and power-law behavior:
T
ff-1
(p(r) = — 1 + 1 -
(2.3.3-23)
Mo
where /XQ, TI/2, and a are parameters to be fixed by comparison with experimental data. It
includes the power-law fluid as a special case corresponding to the limit
Mo
—> m
Tw?
° —
\Mo/
(2.3.3-24)
For polymers and their solutions, a is usually between 1 and 3 (Bird 1965b), which means
that it properly predicts a lower-limiting viscosity /XQ as r —> 0. Equation (2.3.3-23) may be
one of the more useful three-parameter models of the class of generalized Newtonian fluids.
Summary
Many more models of the form of (2.3.3-1) and (2.3.3-18) have been proposed in addition
to those mentioned here (Reiner 1960; Bird 1965b; Bird, Armstrong, and Hassager 1987,
p. 169). They often have been published in a one-dimensional form. The reader interested
in applying to another situation a model that has been presented in this fashion should first
express the model in a form consistent with either (2.3.3-1) or (2.3.3-18).
Exercise 2.3.3-1
Show that p in (2.3.3-1) must be the mean pressure defined by (2.3.2-26).
Exercise 2.3.3-2
Starting from (2.3.3-1), derive (2.3.3-13).
Exercise 2.3.3-3 Reiner-Phiiippoff fluid One of the classic descriptions of stress-deformation behavior from the pre-1945 literature is the Reiner-Phiiippoff fluid (Philippoff 1935):
d v
,
\
_ I
_L
— Moo i
Mo — Mo
1
„
1 + (A12/T0) J
Following the usual practice of that period, we have stated the model in a form appropriate
for a one-dimensional flow in rectangular, Cartesian coordinates:
Vi = V\ ( Z 2 )
v2 = 0
v3 = 0
Here /Lt0, /Xoo, and TQ are three material parameters, constants for a given material and a given
set of thermodynamic state variables.
i) How would you generalize this model so that it could be applied to a totally different,
multidimensional flow?
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48
2. Foundations for Momentum Transfer
ii) Show that this model correctly predicts Newtonian behavior for both low and high
stresses.
Exercise 2.3.3-4 Eyring fluid The Eyring fluid is another classic description of stress-deformation
behavior (Bird et al. 1960). For a one-dimensional flow in rectangular, Cartesian coordinates,
it takes the form
5i2 = A arcsinh I i — )
\B dz2/
Here A and B are material parameters, constants for a given material and a given set of
thermodynamic state variables.
i) How would you generalize this model so that it could be applied to a totally different,
multidimensional flow?
ii) Are there any unpleasant features to this model?
Exercise 2.3.3-5 Knife in a jar of peanut butter You all will have had an experience similar to
• finding that a kitchen knife could be supported vertically in a jar of peanut butter
without the knife touching the bottom of the jar or
• finding that a screwdriver could be supported vertically in a can of grease without the
screwdriver touching the bottom of the can.
To better understand such observations, consider a knife that is allowed to slip slowly into
a Bingham plastic, until it comes to rest without being in contact with any of the bounding
surfaces of the system. Relate the depth H to which the knife is submerged and the total
weight of the knife to the properties of the Bingham plastic. You may assume that you know
both the mass density p ( w ) of the metal and the mass density p{1) of the fluid, you may
neglect end effects at the intersection of the knife with the fluid-fluid interface, and you may
approximate the knife as parallel planes ignoring its edges.
Hint: Make the same assumption about the pressure that you did in solving Exercise 2.2.3-1.
2.3.4
Noll Simple Fluid
Many commercial processes involve viscoelastic fluids, ranging from polymers and polymer
solutions to food products. (Viscoelastic is used here in the sense that, following a material
particle, the stress depends upon the history of the deformation to which the immediate
neighborhood of the material particle has been subjected. These fluids exhibit a finite relaxation time and normal stresses in viscometric flows [Coleman et al. 1966, p. 47].) The
behavior of these fluids is generally much more complex than we have suggested in Sections
2.3.2 and 2.3.3. Sometimes the simple models discussed in Section 2.3.3 are adequate for
representing the principal aspects of material behavior to be observed in a particular experiment. More often they are not. Noll and coworkers (Noll 1958; Coleman, Markovitz, and
Noll 1966; Coleman and Noll 1961; Truesdell and Noll 1965) have suggested a description of stress-deformation behavior that apparently can be used to explain all aspects of the
behavior of viscoelastic liquids that have been observed experimentally.
Before trying to say exactly what is meant by a Noll simple fluid, let us pause for a little
background. Let £ be the position at time t — s (0 < s < oo) of the material particle that at
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2.3. Behavior of Materials
49
time t occupies the position z:
Z = Xtfat-s)
(23.4-1)
We call Xt *ne relative deformation function because material particles are named or identified by their positions in the current configuration. Equation (2.3.4-1) describes the motion
that took place in the material at all times t — s prior to the time t. The gradient with respect
to z of the relative deformation function is called the relative deformation gradient (see also
Exercise 2.3.2-1):
F,(f -s) = Vx,(z, t - s)
(2.3.4-2)
The right relative Cauchy-Green strain tensor is defined as
Ct(t -s) = Fj(t - s) • F,(f - s)
(2.3.4-3)
Noll (Noll 1958, Coleman et al. 1966, Coleman and Noll 1961, Truesdell and Noll 1965)
defines an incompressible simple fluid as one for which the extra stress tensor S at the position
z and time t is specified by the history of the relative right Cauchy-Green strain tensor for
the material that is within an arbitrarily small neighborhood of z at time t:
Un oo *
S= — H (C,(t-soa))
(2.3.4-4)
Here we follow Truesdell's discussion of the dimensional indifference of the definition of
a simple material (Truesdell 1964; Truesdell and Noll 1965, p. 65). The quantity W£Lo*
is a dimensionally invariant tensor-valued functional (an operator that maps tensor-valued
functions into a tensor). The constants /xo and so are the characteristic viscosity and the
characteristic (relaxation) time of the fluid. Like any characteristic quantities introduced in
defining dimensionless variables, the definitions for /x0 and so are arbitrary. The advantages
and disadvantages of particular definitions for /xo and so have been discussed elsewhere
(Slattery 1968a).
Equation (2.3.4-4) clearly satisfies the principles of determination and local action (Section 2.3.1). That it also satisfies the principle of frame indifference is less obvious (Truesdell
1966a, pp. 39, 58, and 63).
Since the form of the functional H™=Q* is left unspecified, it is clear that the Noll simple
fluid incorporates a great deal of flexibility. It is for exactly this reason that many workers
believe the Noll simple fluid to be capable of explaining all(?) manifestations of fluid behavior
that have been observed experimentally to date. It should be viewed as representing an entire
class of constitutive equations or an entire class of fluid behaviors.
But the generality of the Noll simple fluid is also its weakness. Only two classes of flows
have been shown to be dynamically possible for every simple fluid (Coleman et al. 1966;
Coleman and Noll 1959, 1961, 1962; Truesdell and Noll 1965; Coleman 1962; Noll 1962;
Slattery 1964): the viscometric flows and the extensional flows. Most flows of engineering
interest cannot be analyzed without first specifying a particular form for the functional
<1_/OO
*
This does not mean that the Noll simple fluid is of no significance to those of us interested
in practical problems. It actually is a very simple model for fluid behavior in the sense that it
incorporates, at most, two dimensional parameters (Truesdell 1964; Truesdell and Noll 1965,
p. 65): no and S0. This dimensional simplicity, together with its capacity for representing
a wide range of fluid behaviors, makes the Noll simple fluid ideal for use in preparing
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50
2. Foundations for Momentum Transfer
Table 2.4.1 - 1 . The differential mass balance in three
coordinate systems
Rectangular Cartesian coordinates (z u z2, z3):
dp
a
a
— + — ( p v i ) + —(pv2)
dt
3zi
3z2
a
+ — (pv3) = 0
3z3
Cylindrical coordinates (r, 6, z):
dp
1 a
1a
a
T + - ^(prvr) + --^(pve) + —(pv:) = 0
dt r dr
r 80
dz
Spherical coordinates (r, 9, <p);
dp
1 a
i d
1 a
7
•r- + -7 -r-(pr2vr) + — - — (PV9 sin(9) + — - —-(pv9) = 0
3? H 3r
r sinw 30
r sin0 3^>
dimensionless correlations of experimental data. The limitation upon correlations formed in
this way is that, since material behavior in the form of the functional W^L0* has not been
fully specified, correlations of experimental data can be made for only one fluid at a time.
Although this is a serious limitation, it is not necessary to have all of the data that would be
required in order to describe the behavior of the material under study. For more on scale-ups
and data correlations for viscoelastic fluids, see Slattery (1965, 1968a).
There have been alternative descriptions for the complex behavior observed in real fluids.
Of these, Oldroyd's (1965) generalized elasticoviscous fluid has attracted perhaps the most
interest.
If you wish to learn more about the behavior of real materials and their description, you are
fortunate to have several excellent texts available (Fredrickson 1964, Lodge 1964, Truesdell
and Noll 1965, Coleman et al. 1966, Truesdell 1966a, Leigh 1968, Bird et al. 1977).
2.4
Summary
2.4.1
Differential Mass and Momentum Balances
We would like to summarize here some of the most common relationships in rectangular
Cartesian, cylindrical, and spherical coordinates.
Table 2.4.1-1 presents the differential mass balance (1.3.3-4)
p+div(pv) = 0
(2.4.1-1)
dt
for these coordinate systems.
It is generally more convenient to work with the differential momentum balance (2.2.3-4)
in terms of the viscous portion of the stress tensor S = T + PI:
[
u
"I
— + (Vv)-v = - V P + divS + pf
(2.4.1-2)
dt
J
It is the components of this equation that are presented in Tables 2.4.1-2,2.4.1-4, and 2.4.1-6.
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2.4.
Summary
5I
Table 2.4.1-2. Differential momentum balance in rectangular Cartesian coordinates
z\ component:
dv\dv\
P
+ Ul
\tit
dv\
dv\
dv\ \
3Zl
dzj
0Z3 1
dvo
dv2\d
odS \ dSn
, 35,2 , dSn
\
0Z\
\
\ PJi
u Z7
0 Zi,
dS22
dS23
z2 component:
dV2
Po(
{dt
+
+
tiv-,
"' 3z,
+
OZ~\ 1
dP
3z2
dS2\
+
3zj
P l
' 3z 2 ' 3z 3 '
z} component:
(dv3
P
+
\tit
dv3
39i>3
OZ\
0Z2
>
3
\
V
'dz3)
1
^ +
95* 3 1
95* 3 2
+
3z,
3z2
9S33
+
3z,+
P 3
Table 2.4.1-3. Differential momentum balance in rectangular Cartesian coordinates
for a Newtonian fluid with constant p and /z, the Navier-Stokes equation
Z] component:
dp ,
(d2xh
/9
(
, d2v, , 3 2
z2 component:
,
p
^
\ dt
Z3 component:
dv2
dv2
+V ^
3z
+
+ f,
+U
3;
3zi
V ^
3z
2
dv2\
+V _
3z/
+ I^
3z2
^d + d
3z 2
)=
3z3/
2
i
v2
+
;2n2
W
32l|3
32t>?
\9zi2
3z 2 2
I
+M(T^ + T 4
3z3
d2v2
3
Commonly, the only external force to be considered is a uniform gravitational field, which
we may represent as
f=-V0
(2.4.1-3)
For an incompressible fluid, (2.4.1-3) allows us to express (2.4.1-2) as
p\-1 +(Vv)•v = - V V + divS
dt
J
(2.4.1-4)
where
V = p + p(f)
(2.4.1-5)
is referred to as the modified pressure. The components of (2.4.1-4) are easily found from
Tables 2.4.1-2,2.4.1-4, and 2.4.1-6 by deleting the components of f and replacing P with V.
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52
2. Foundations for Momentum Transfer
Table 2.4.1 -4. Differential momentum balance in
cylindrical coordinates
r component:
/3vr
Vg_ 8vr
3D,.
-
dr
H
—<L .f D
r dr
r
86
dz
Y
9 component:
fdVg
\dt
P
dVg
v0 9 Vg
vrve
dr
r 39
r
+Vr
1 9
r2
lap
r 86
-
3Vg\
Vz
lc 9
r 36»
dz )
dSgz
|
|
f
dz '
P
"
z component:
P I —- + v,.
dP
dr
Vg
Jrl
dvz
+
+ Vs
T-
1 9
r dr
H
1 dS,e
r d9
+
Table 2.4.1-5. Differential momentum balance in cylindrical
coordinates for a Newtonian fluid with constant p and /X, the
Navier-Stokes equation
r component:
8vr
r
[8/18
8p
"3r
VQ 8vr
+M
37\r3r
Vg2
8v\
1 82VR
2 8VE
^"gp"
r2 9(9
\
{rVr
')
+
0 component:
' 8ve
,8ve
v d8ve
vrve
~~r~
1 8p
r 39
f 8 (\ 8
\_3r \r 8r
8v0\
+ V:
\
J
~dz )
1 32I;,
r2 862
2 8VR
r2 86
9 2 i;,l
dz2 J
: component:
8
v,
9i;,
us 9i>-
9u. \
9r
r d9
dz J
dp
[1 9 / dv,\
1 92u, 9 2 i',l
= --i-+M - r - U - — + - — + — y + p / r
9s
L r 9'' V 9 r /
' 9^"
9z
J
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2.4. Summary
53
Table 2.4.1-6. Differential momentum balance in spherical coordinates
r component:
dVr
3 iv
P
+ Vr
\3t
+
3r
8r
I
_
3D,-
89
r
r2 c
!
l>«,
3fr
!%2 + V
r
r sin $ 3ip
)
r rsin9 3
Jrr)
r
9 component:
1dvn
\dt+Vr
dv9
p
V
i
e
r 89
dr
1dP
r 39
dvB
3
J
r: sdr
'
S
Vy
p 2 COt0
8VB
r sin $ 8<p
1
' ' rw9
r
r
)
9
1
[SB0sm9) +
90'
r sin 9
COt0
component:
(d.Vy
P
\ d t
dv9
+Vr
dr
1
3P
3<p
r sin 0
>evv
3D,,
1
f
961
r sin O dtp
1 3 ;.3c j ,
3r
r
3
1
r sir r 0 30 **
r
')
i
2
9S W
r siou
In Sections 2.3.1 to 2.3.4, we pointed out the need for information beyond the differential
mass and momentum balances and the symmetry of the stress tensor, and we discussed several
possible descriptions of stress-deformation behavior. One of these was the Newtonian fluid,
(2.3.2-21). The divergence of the stress tensor for a Newtonian fluid may be written as
div T = div( -PI + A [div v]I + 2/xD)
(2.4.1-6)
If we assume that k and \I are constants with respect to position, (2.4.1-6) becomes
divT = - V F + A V ( d i v v ) + 2/xdivD
(2.4.1-7)
Since
divD = -div(Vv) + -V(divv)
(2.4.1-8)
we have
div T = - VP + (k + /x)V(div v) + M div (Vv)
(2.4.1-9)
With (2.4.1-9), the differential momentum balance (2.2.3-4) becomes for a Newtonian fluid
p
J^)l
pf
(2.4.1-10)
dt
Stress-deformation behavior of an incompressible Newtonian fluid is described by
(2.3.2-25). The differential momentum balance for this case becomes
J]l
dt
/of
(2.4.1-11)
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54
2. Foundations for Momentum Transfer
Table 2.4.1 -7. Differential momentum balance in spherical coordinates
for a Newtonian fluid with constant p and /x, the Navier-Stokes equation
r component:
dvr
dt
dvr
3r
vg dvr
r do
dv
e dvt
vv 3ur
r smd dtp
v$2 + v
r
dr
9 component:
(dVg
?,
dve
vdu¥
vr
P \ Tdt- + IVT-H
+V>'3r —r de H rsm9 1 dtp
\ >'
3r
r de
i dp
r dO
/
! V
2 dvr
r 2 de
r
ve
r sin2 r
r
2cos9 i
2
sin2 9 dtp
2
<p component:
p /3iV+ Vr3 iv
u i 9/7
rsin9
3w
vg 3
3u»
rsi
X
\
t
dtp
C
2
ay,'
r sin 9 ' r sin0 d<p
2
2
2
r
2cosfl dV
-- I + p/p
r 2 sin2 # d
where
1 3 /
2
r dr {
2 9 "
1
3 /
3 \
:in6
2
sin( 9 36» \
+
1
l ;• sin ^
2
/ 3
2
\
2
which is the Navier-Stokes equation. The components of the Navier-Stokes equation are
presented in Tables 2.4.1-3, 2.4.1-5, and 2.4.1-7.
We deal only with physical components of spatial vector fields and tensor fields when
discussing curvilinear coordinate systems in most of this text. For this reason, in Tables
2.4.1-8 through 2.4.1-10 we adopt a somewhat simpler notation for physical components
in cylindrical and spherical coordinates than that suggested in Appendix A. We denote the
physical components of spatial vector fields in cylindrical coordinates as vrj vo, and vz rather
than V{\), T>(2), and v^y, the physical components of second-order tensor field are indicated
as Drr, Dre, DOz, etc. The notation used in spherical coordinates is very similar.
Because of this change in notation, we do not employ the summation convention hereafter
with the physical components of spatial vector fields and second-order tensor fields. The
quantity Dn is a single physical component of the second-order tensor field D. When used
in context, there should be no occasion to misinterpret it as the sum of three rectangular
Cartesian components. When we have occasion to discuss physical components with respect
to other curvilinear coordinate systems, we revert to the notation introduced in Appendix A.
Exercise 2.4.1 -1 In a convenient rectangular Cartesian coordinate system, derive an expression
for </> and show that cj> is arbitrary to a constant.
Exercise 2.4.1 -2
Starting with div D, derive (2.4.1-8).
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2.4. Summary
55
Table 2.4.1 -8. Components of rate
of deformation tensor in rectangular
Cartesian coordinates
Du =
O22 =
O33 =
dv2
dv}
O 1 2 = O21 =
O,3 = O 3 1 =
O 2 3 = 032 =
K£+£)
2 \ dz^
dz\ )
1 / 9 V2
9 ^3 \
+
2fe 9^J
Table 2.4.1-9. Components of the
rate of deformation tensor in
cylindrical coordinates
dv
D r r ~ ~g7
r
o9e
1 dVg
+
V,-
r
o ,.
D,, =
Der =
D,z = D
Dgz
=
1
\
2
1 a 11,.
d
2 ydz
1 /8vg
2
r~80
]
+ —)
1 9i;2
+
r~d9 )
2.4.2 Stream Function and the Navier-Stokes Equation
In Section 1.3.7, we expressed the velocity components for a two-dimensional motion of
an incompressible fluid in terms of a stream function \jr. In this way, the differential mass
balance is automatically satisfied. Here we examine the result of the introduction of a stream
function upon the equation of motion for an incompressible Newtonian fluid, where the
external force may be expressed in terms of a potential as described in Section 2.4.1.
When the external force is representable as the gradient of a scalar potential, we may
introduce the modified pressure of Section 2.4.1 into the Navier-Stokes equation (2.4.1-11)
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56
2. Foundations for Momentum Transfer
Table 2.4.1 -10. Components of the rate of
deformation tensor in spherical coordinates
__
3D,-
" ~ 17
1 dvg
iv
v oO
v
1 dv0
r sin0 dip
iv
r
i%cot0
3
\r /
Or,
2 \_r sinw 3 ^
r d0 j
dr V r
_ 1 [sm0 d / v<p \
1 dvgl
- - [-7-gg (^—^j + — ^ " ^
to obtain
d\
p— + p(Vv) • v = - V V + n div (Vv)
(2.4.2-1)
If we take the curl of this equation, modified pressure V is eliminated to yield
9 (curl v)
+ curl([Vv] • v) = v div(V[curl v])
dt
(2.4.2-2)
where
v= -
(2.4.2-3)
is the kinematic viscosity. In any coordinate system for which the velocity vector has only
two nonzero components, Equation (2.4.2-2) has only one nonzero component. The nonzero
component expressed in terms of the stream function is presented for several situations in
Table 2.4.2-1.
The differential equations of Table 2.4.2-1 can also be derived by recognizing that in any
two-dimensional flow (2.4.2-1) will have only two nonzero components. Modified pressure
may be eliminated between these two equations by recognizing that
d2v
d2v
dx1 dxJ
dx' dx<
(2.4.2-4)
The velocity components in the resulting differential equation may be expressed in terms of
a stream function.
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2.4. Summary
57
Table 2.4.2-1. The stream function
Assumed form
Coordinate of velocity
Velocity
system0
distribution
components
Rectangular u3 = 0
Cartesian
Nonzero component
of (2.4.2-2)b
Operator
a2
£2
u, = —
OZ2
-
a2
9Z,
E4 f =
v\ — V\(z\, !<£)
9
2 +
oz22
E2(E2f)
1
V2 = V 2(Zu Z 2) V2 = - —
1 1
94
zSdzz2
4
i
Cylindrical
vz = 0
1 df
vr = 1
r 90
dt
r
d(r,9)
92
1 9
9r 2
r dr
1 92
4
V'r = Vrir, 0)
Cylindrical
r
£2
= vE \fr
V0 = Vg(r,0)
ve = -
ve = 0
Vr =
*9 \
1 1 /
3 4 I
+
1 df
£2=
r2 B92
92
1 9
r ~dz
92
_J_ dz2
vr = vr(r, z)
v, = - - —
r or
Spherical
F
vv = 0
^
vr — = - — ( E f
r2 sine d6 dt
)
Y
d(f,E2f)
d(r,9)
d(r9)
r
~
v0 =
—7 - r r sin S 9r
=
= 1 d
dr2
2E2
sin#
r22sm8
2
r sm8
x I — cos 9
\dr
ve = ve{r, 9)
2
E =
r2 sine
r
a
/
1
a
d6 \sin9 86
sin 6
r d9
vE4f
a
This table is taken from Bird et al. (1960, Table 4.2-1) and from Goldstein (1938, p. 114). Goldstein also presents relations
for axisymmetric flows with a nonzero component of velocity around the axis.
*The Jacobian notation signifies
8/
d£
dx
S.y
d(x,y)
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58
2. Foundations for Momentum Transfer
2.4.3
Interfacial Tension and the Jump Mass and Momentum Balances
To this point, when we have used a dividing surface to represent a phase interface, we have
not recognized that there might be mass, momentum, or stresses associated with the interface.
While we will continue to neglect mass and momentum associated with the interface, the
imbalance of long-range intermolecular forces at a deformed interface can be taken into
account in the jump momentum balance through the introduction of stresses that aci tangent
to the dividing surface. In the majority of problems, the magnitude of these stresses can be
described in terms of the interfacial tension y and the jump momentum balance becomes
(Slattery 1990, p. 237)
V ia) y + IHyi
+ [ - p v (v - u) . £ + T • £ ] = 0
(2.4.3-1)
Here H = (KI + K2) /2 is the mean curvature (Slattery 1990, p. 1116) of the surface; K\ and
K2 are the principal curvatures of the surface (Slattery 1990, p. 1119). The surface gradient
operator V (a) defines a gradient with respect to position (y l , y2) on the surface (Slattery
1990, p. 1075). The surface coordinates yl and y2 in general define a curvilinear coordinate
system on the surface (Slattery 1990, p. 1065). It is only in the case of a planar surface
that a rectangular Cartesian coordinate system can be introduced and the surface gradient
operation takes a familiar form. The surface gradient V((7)]/ of interfacial tension appears
because y will often be a function of position on the surface through its dependence upon
temperature and concentration.
For most problems, you will have to express the jump mass balance (Section 1.3.6)
[p(v-u) . £ ] = 0
(2.4.3-2)
and the jump momentum balance (2.4.3-1) in forms appropriate for the configuration of the
surface. Many problems involve interfaces that we are willing to describe as planes, cylinders,
spheres, two-dimensional surfaces, or axially symmetric surfaces. For these configurations,
you will find your work already finished in Tables 2.4.3-1 through 2.4.3-8.
For a further introduction to interfacial behavior, see Slattery (1990).
Exercise 2.4.3-1 Floating sphere As you can simply demonstrate for yourself, it is easy to make
a small needle float at a water-air interface. Just rub the needle with your fingers before
carefully placing it on the surface of the water. The oils from your skin ensure that the solid
surface is not wet by the water, and the needle is supported by the force of interfacial tension.
Determine the maximum diameter dmax of a solid sphere that will float at an interface,
assuming that the contact angle measured through the gas is zero:
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2.4. Summary
59
Table 2.4.3-1. Stationary plane dividing surface
viewed in a rectangular Cartesian coordinate system
Dividing surface
z3 = a constant
Surface coordinates
1 _
2
Jump mass balance
[p^ 3 ]
=0
Jump momentum balance
z\ component
13&] = 0
Z2 component
^ • + [723ft] = 0
Z3 component
-PV3£3
+
I 33fe]
=0
Table 2.4.3-2. Stationary plane dividing surface
viewed in a cylindrical coordinate system
Dividing surface
z = a constant
Surface coordinates
y2 = 6
Jump mass balance
\pv-j
r*] = o
Jump momentum balance
r component
dy
6 component
dr
1 9
^
r 96»
;z]=0
+\T 1
L ''•
r ^IT •
; 1= 0
1 •* # z ' 2
L
x
1
z component
-pvz2%
z
4-r
z I = 0
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60
2. Foundations for Momentum Transfer
Table 2.43-3. Alternative form for stationary plane
dividing surface viewed in a cylindrical coordinate system
Dividing surface
9 = a constant
Surface coordinates
2
Jump mass balance
[pv>ei;e] = 0
Jump momentum balance
r component
dr
9 component
L offl]
''
[-pi
= 0
o Ho 1
==
0
z component
dz
els] = 0
I
Table 2.4.3-4. Cylindrical dividing surface viewed in
a cylindrical coordinate system
Dividing surface
r =
Surface coordinates
y1 = 9
2
y =z
Unit normal
?,=
1
//=
1
~2R
Mean curvature
Speed of displacement of surface
dR
u
~It
Jump mass balance
rS,] = 0
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2.4. Summary
6I
Jump momentum balance
r component
~R+\
9 component
\-pv,2
L
z component
dy
Table 2.4.3-5. Spherical dividing surface viewed in a
spherical coordinate system
Dividing surface
r = R(t)
Surface coordinates
Unit normal
hr = 1
Mean curvature
1
H =
Speed of displacement of surface
dR
i
» • € = It
Jump mass balance
*]- 0
ov,
Jump momentum balance
r component
2y
r
+
9 component
1 3y
~R 39
4>
component
1
R sin$d
*]- 0
[Te,* ] - 0
[n,*]- 0
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62
2. Foundations for Momentum Transfer
Table 2.4.3-6. Two-dimensional surface viewed in a rectangular Cartesian
coordinate system
Dividing surface
Z3=/l(Zi,/)
Surface coordinates
yl = z,
Unit normal
Mean curvature
isplacement of surf ace
mass balance
[p(v • £ - u • £)] = 0
momentum balance
component
)
^
y|, [ p ( «
$)^,
n§,
,3&] = 0
z2 component
z component
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2.4. Summary
63
Table 2.4.3-7. Axially symmetric surface viewed in a cylindrical
coordinate system
Dividing surface
2 = h(r, t)
Surface coordinates
Unit normal
Mean curvature
-3/2
2r
13
3/1
/3/A
Speed of displacement of surface
-1/2
Jump momentum balance
r component
,3,7 J 3r
+ [-P(v • £ - u • £)2^
=o
(9 component
= 0
2 component
=o
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64
2. Foundations for Momentum Transfer
Table 2.4.3-8. Alternative form for axially symmetric surface viewed in a
cylindrical coordinate system
Dividing surface
r = c(z,t)
Surface coordinates
Unit normal
-1/2
9c
-1/2
Mean curvature
/ <\
/dc\
9 1c
1+
Speed of displacement of surface
€
de l 1 + /3c
U
/wmp m<355 balance
[P(V . c - u
= 0
/wmp momentum balance
r component
0 component
=o
z component
1+
= 0
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