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Basic Principles
entirely a consequence of the crude scale of resolution inherent in the continuum approximation, coupled with the discrete nature of any real fluid. Indeed, these latter contributions
would be zero if the material were truly an indivisible continuum. We shall see that the main
difficulties in obtaining pointwise DEs of motion from (2–22) all derive from the necessity
for including surface forces (whether real or effective) whose molecular origin is outside
the realm of the continuum description.
With the necessity for body and surface forces thus identified, we can complete the
mathematical statement of Newton’s second law for our material control volume:
D
Dt
ρud V =
Vm (t)
ρgd V +
Vm (t)
tdA.
(2–23)
Am (t)
The left-hand side is just the time rate of change of linear momentum of all the fluid within
the specified material control volume. The first term on the right-hand side is the net body
force that is due to gravity (other types of body forces are not considered in this book).
The second term is the net surface force, with the local surface force per unit area being
symbolically represented by the vector t. We call t the stress vector. It is the vector sum of
all surface-force contributions per unit area acting at a point on the surface of Vm (t).
Before proceeding further, let us return briefly to the derivation based upon a fixed control
volume and “conservation of linear momentum.” In this alternative approach, momentum
is transported through the surface of the control volume by convection at a rate ρu(u · n)
at each point, and this is treated as an additional contribution to the rate at which linear
momentum is accumulated or lost from the control volume. Of course, there is no term in
(2–23) that corresponds to a flux of momentum across the surface of the material (control)
volume. Because all points within Vm (t) and on its surface Am (t) are material points, they
move precisely with the local continuum velocity u and there is no flux of mass or momentum
across the surface that is due to convection.
We may now attempt to simplify (2–23) to a differential form, as we did for the mass
conservation equation, (2–6). The basic idea is to express all terms in (2–23) as integrals
over Vm (t), leading to the requirement that the sum of the integrands is zero because Vm (t)
is initially arbitrary. However, it is immediately apparent that this scheme will fail unless
we can say more about the surface-stress vector t. Otherwise, there is no way to express the
surface integral of t in terms of an equivalent volume integral over Vm (t).
We note first that t is not only a function of position and time, t(x, t), as is the case
with u, but also of the orientation of the differential surface element through x on which
it acts. The reader may well ask how this is known in the absence of a direct molecular
derivation of a theoretical expression for t (the latter being outside the realm of continuum
mechanics, even if it were possible in principle). The answer is that we can either deduce
or derive certain general properties of t, including its orientation dependence, from (2–23)
by considering the limit as we decrease the material control volume progressively toward
zero while holding the geometry (shape) of Vm constant. Let us denote a characteristic
linear dimension of Vm as , with 3 defined to be equal to Vm . An estimate for each of the
integrals in (2–23) can be obtained in terms of by use of the mean-value theorem. A useful
preliminary step is to apply the Reynolds transport theorem to the left-hand side. Although
this might, at first sight, seem to present new difficulties because ρu is a vector, whereas
the Reynolds transport theorem was originally derived for a scalar, the result given by (2–9)
carries over directly, as we may see by applying it to each of the three scalar components of
ρu and then adding the results. Thus (2–23) can be rewritten in the form
Vm (t)
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∂(ρu)
+ ∇ · (ρuu) − ρg d V =
∂t
td A.
Am (t)
(2–24)
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C. Newton’s Laws of Mechanics
3
C
Figure 2–5. A tetrahedron, ABCD, is centered
about the point x. The surface ABC is arbitrarily oriented with respect to the Cartesian axes
123, and its area is denoted as An . The unit
outer normal to ABC is n. The areas of surfaces
BCD, ACD, and ABD are denoted, respectively,
as A1 , A2 , and A3 , with outer unit normals
e1 , e2 , and e3 each parallel to the opposing coordinate axis but oriented in the negative direction.
Interior
point x
n
e1
e2
D
B
2
A
1
e3
Now, denoting the mean value over Vm or Am by the symbol
the symbolic form
3
=
2
, we can express (2–24) in
.
It is evident that, as → 0 , the volume integral of the momentum and body-force terms
vanishes more quickly than the surface integral of the stress vector. Hence, in the limit as
→ 0, (2–24) reduces to the form
td A → 0.
lim
→0
(2–25)
Am (t)
This result is sometimes called the principle of stress equilibrium, because it shows that
the surface forces must be in local equilibrium for any arbitrarily small volume element
centered at any point x in the fluid. This is true independent of the source or detailed form
of the surface forces.
Now it is clear that the stress vector at (or arbitrarily close to) a point x must depend
not only on x but also on the orientation of the surface through x on which it acts, because
otherwise the equilibrium condition (2–25) could not be satisfied. At first this dependence
on orientation may seem to suggest that we would need a triply infinite set of components
to specify t for all possible orientations of a surface through each point x. Not only is
this clearly impossible, but (2–25) shows that it is not necessary. Let us consider a surface
with completely arbitrary orientation, specified by a unit normal n, which passes near to
point x but not precisely through it. Then, using this surface as one side, we construct a
tetrahedron, illustrated in Fig. 2–5, centered around point x, whose remaining three sides
are mutually perpendicular. In the limit as the volume of this tetrahedral volume element
goes to zero, the surface-stress equilibrium principle applies, and it is obvious that the
surface-stress vector on the arbitrarily oriented surface (which now passes arbitrarily close
to x) must be expressible in terms of the surface- stress vectors acting on the three mutually
perpendicular faces (which also pass arbitrarily close to x). From this, we deduce that the
specification of the surface-force vector on three mutually perpendicular surfaces through
a point x (nine independent components in all) is sufficient to completely determine the
surface-force vector acting on a surface of any arbitrary orientation at the same point x.
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Basic Principles
We can actually go one step further than this general observation and use the stress
equilibrium principle applied to the tetrahedron to obtain a simple expression for the surfacestress vector t on an arbitrarily oriented surface at x in terms of the components of t on
the three mutually perpendicular surfaces at the point x. We denote the stress vector on a
surface with normal n as t(n). The area of the surface with unit normal n is denoted as An .
Then, applying the surface-stress equilibrium principle to the tetrahedron, we have
t(n)
An − t(e1 )
A1 − t(e2 )
A2 − t(e3 )
A3 = 0,
(2–26)
where the e’s are unit normal vectors for the three mutually perpendicular surfaces and
represents the mean value of the indicated stress vector over the surface in question.
The minus signs in (2–26) are a consequence of Newton’s third law, according to which
t(e1 ) = −t(−e1 ). Now, A1 is the projected area of An onto the plane perpendicular to
the e1 axis. Thus,
Ai =
An (n · ei ) for i = 1, 2, or 3,
and (2–26) can be expressed in the form
0 = [ t(n) − t(e1 ) (n · e1 ) − t(e2 ) (n · e2 ) − t(e3 ) (n · e3 )] An .
It follows, in the limit as → 0 (so that the tetrahedron collapses onto point x), that
t(n) = n · [e1 t(e1 ) + e2 t(e2 ) + e3 t(e3 )].
(2–27)
The quantity in square brackets is a second-order tensor, formed as a sum of dyadic
products of the vectors t(ei ) and ei for i = 1, 2, and 3. This second-order tensor is known
as the stress tensor,
T = [e1 t(e1 ) + e2 t(e2 ) + e3 t(e3 )],
(2–28)
and is denoted here by the symbol T. It follows from (2–27) and (2–28) that the surfacestress vector on any arbitrarily oriented surface through a point x is completely determined by specification of the nine independent components of the stress tensor T at that
point.
We see from (2–27) and (2–28) that the second-order tensor T is just the “linear vector
operator” that operates on the unit normal to a surface at a point P to produce the surfacestress vector acting at that point. Indeed, rewriting (2–27), we obtain
t(xp ; n)= n · T(xp ),
(2–29)
where x p is the position vector corresponding to an arbitrarily chosen point P in the flow
domain. Furthermore, the components of T are just the components of the surface-force
vectors acting on the three mutually perpendicular surfaces at xp , whose unit normal vectors
are in the direction of the three coordinate axes. For example, the 21 component of T is
the component in the 1 direction of the surface-force vector that acts on the surface with
unit normal in the e2 direction. Although the derivation of (2–29) has been carried out with
the notation of Cartesian vector and tensor analysis, it is evident that the components of T
can be defined in terms of stress vector components for any orthogonal coordinates through
point x, and the result, (2–29), is completely invariant to the choice of coordinate systems.
The stress tensor T depends only on x and t, but not on n. Because knowledge of T at a
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C. Newton’s Laws of Mechanics
point enables us to determine the surface force acting on any surface through that point, T
is said to represent the state of stress in the fluid at the given point.
With the relationship between the stress vector and the stress tensor in hand, DEs
of motion can be derived from the macroscopic form (2–24) of Newton’s second law of
mechanics. Substituting (2–29) into (2–24) and applying the divergence theorem to the
surface integral in the form
n · Td A =
Am (t)
(∇ · T)d V,
Vm (t)
we obtain
Vm (t)
∂(ρu)
+ ∇ · (ρuu) − ρg − ∇ · T d V = 0.
∂t
(2–30)
Because the initial choice for Vm (t) is arbitrary, it follows that the condition (2–30) can be
satisfied only if the integrand is equal to zero at each point in the fluid, that is,
∂(ρu)
+ ∇ · (ρuu) = ρg + ∇ · T.
∂t
(2–31)
Combining the first two terms in this equation with the continuity equation, we can also
write the DEs of motion in the form
ρ
∂u
+ u · ∇u = ρg + ∇ · T.
∂t
(2–32)
This is known as Cauchy’s equation of motion. It is clear from our derivation that it is simply
the differential form of Newton’s second law of mechanics applied to a moving fluid.
It is, perhaps, well to pause for a moment to take stock of our developments to this point.
We have successfully derived DEs that must be satisfied by any velocity field that is consistent
with conservation of mass and Newton’s second law of mechanics (or conservation of linear
momentum). However, a closer look at the results, (2–5) or (2–20) and (2–32), reveals the fact
that we have far more unknowns than we have relationships between them. Let us consider
the simplest situation in which the fluid is isothermal and approximated as incompressible.
In this case, the density is a constant property of the material, which we may assume to
be known, and the continuity equation, (2–20), provides one relationship among the three
unknown scalar components of the velocity u. When Newton’s second law is added, we do
generate three additional equations involving the components of u, but only at the cost of
nine additional unknowns at each point: the nine independent components of T. It is clear
that more equations are needed.
One possible source of additional relationships between u and T that we have not yet
considered is the generalization of Newton’s second law from linear to angular momentum.
We may state the resulting principle, for a material control volume, in the form
D
Dt
(x ∧ ρu)d V = sum of torques acting on the material
Vm (t)
control volume,
(2–33)
where x is the position vector associated with points within the material volume Vm (t). The
left-hand side of (2–33) is the rate of change of the angular momentum of the fluid inside
the material (control) volume. There are, in principle, four sources of torque acting on the
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Basic Principles
material control volume. The first two are simply the moments of the surface and body
forces that act on the fluid:
[x ∧ t(n)]d A,
(2–34a)
(x ∧ ρg)d V.
(2–34b)
Am (t)
Vm (t)
In addition, the possibility exists of body couples per unit mass c and surface couples per
unit surface area r that are independent of the moments of surface and body forces. Thus
the full statement of (2–33) takes the form
D
Dt
(x ∧ ρu)d V =
Vm (t)
[x ∧ (n · T) +r]d A +
Am (t)
[x ∧ ρg + ρc]d V.
(2–35)
Vm (t)
Clearly the presence of the surface-torque terms in (2–35) contributes additional unknowns,
and the imbalance between unknowns and equations is not improved in the most general
case by consideration of angular acceleration. However, in practice, there is no evidence of
significant surface-torque contributions in real fluids, and we shall assume that r ≡ 0. On
the other hand, some fluids do exist on which the influence of body couples is significant.
One commercially available set of examples is the so-called Ferrofluids, which are actually
suspensions of fine iron particles that have either permanent or induced magnetic dipoles.9
When such a fluid flows in the presence of a magnetic field, there is a body torque applied
to each particle, but in the continuum approximation this is described as a continuously
distributed body couple per unit mass. Thus a Ferrofluid is an example of a fluid in which
c = 0. In spite of the fact that fluids do exist in which body couples play a significant role,
however, this is not true of the vast majority of fluids and none of the liquids or gases of
common experience; water, air, oils, and so on. Let us suppose therefore that c = 0. In this
case, the angular acceleration principle reduces to the simpler form
D
Dt
(x ∧ ρu)d V =
Vm (t)
x ∧ (n · T)d A +
Am (t)
x ∧ ρgd V.
(2–36)
Vm (t)
We can explore the consequences of this equation by converting it to an equivalent differential form. To do this, we first apply Reynolds transport theorem to the left-hand side. This
gives
D
Dt
(x ∧ ρu)d V =
Vm (t)
x∧
Vm (t)
∂(ρu)
+ ∇ · (ρuu) d V.
∂t
(2–37)
Also, on application of the divergence theorem, the surface integral in (2–36) becomes
x ∧ (n · T)d A =
Am (t)
[x ∧ (∇ · T) − : T]d V,
Vm (t)
where is the third-order alternating tensor,10
⎧
⎪ +1 if (ijk) is an even permutation of (123)
⎨
εi jk ≡ −1 if (ijk) is an odd permutation of (123),
⎪
⎩
0 otherwise (some or all subscripts are equal)
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(2–38)
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D. Conservation of Energy and the Entropy Inequality
and the symbol : indicates a double inner product of this tensor with T.11 Introducing (2–37)
and (2–38) into (2–36), we thus obtain
x∧
Vm (t)
∂(ρu)
+ ∇ · (ρuu) − ∇ · T − ρg + : T d V = 0.
∂t
(2–39)
In view of the differential form of the equation of motion, Eq. (2–31), and the fact that
Vm (t) is arbitrary, we see that the angular acceleration principle, (2–33), requires that
:T = 0
(2–40)
at all points in the fluid. Condition (2–40) requires that the stress tensor must be symmetric:
T = TT .
(2–41)
Note that the condition of stress symmetry is not valid if there is a significant body couple
per unit mass c in the field. In this case, we can easily show, following the same steps that
we used in going from (2–35) to (2–40), that
: T − ρc = 0.
(2–42)
This gives a relationship between c and the off-diagonal components of T, but the stress is
clearly not symmetric. We hereafter restrict our attention to the case in which c = 0.
We see that application of the angular acceleration principle does reduce, somewhat,
the imbalance between the number of unknowns and equations that derive from the basic
principles of mass and momentum conservation. In particular, we have shown that the stress
tensor must be symmetric. Complete specification of a symmetric tensor requires only six
independent components rather than the full nine that would be required in general for a
second-order tensor. Nevertheless, for an incompressible fluid we still have nine apparently
independent unknowns and only four independent relationships between them. It is clear
that the equations derived up to now – namely, the equation of continuity and Cauchy’s
equation of motion – do not provide enough information to uniquely describe a flow system.
Additional relations need to be derived or otherwise obtained. These are the so-called
constitutive equations. We shall return to the problem of specifying constitutive equations
shortly. First, however, we wish to consider the last available conservation principle, namely,
conservation of energy.
D. CONSERVATION OF ENERGY AND THE ENTROPY INEQUALITY
We begin again by considering an arbitrary material control volume as it moves along
with the fluid, and in this case we consider the change in its total energy with respect to
time. In the simplest molecular description based on a hard-sphere gas model, this energy
would be recognized as purely kinetic in nature, associated with the intensity of individual
molecular motions. In the continuum approximation, however, we explicitly resolve only the
molecular average velocity (denoted as u in the earlier developments of this chapter), and it
is necessary to consider the total energy as consisting of two parts. First is the kinetic energy
associated with the macroscopic or continuum velocity field u, and second is a so-called
internal energy term that encompasses all additional contributions including those that are
due to the differences between the continuum velocity u at a point and the actual velocities
of the molecules that occupy the continuum averaging volume that is centered at that point.
In this description, the internal energy is a measure of the intensity of random molecular
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Basic Principles
motion relative to the mean continuum velocity. Thus, from the continuum point of view,
the total energy of an arbitrary material control volume is written as
Vm (t)
ρ(u · u)
+ ρe d V,
2
where u · u = u is the local “speed” of the continuum motion and e is the internal energy
(representing additional kinetic energy at the molecular level) per unit mass.12
The rate at which the total energy changes with time is determined by the principle of
energy conservation for the material volume element, according to which
2
D
Dt
Vm (t)
rate of work done
rate of internal energy
on the material
flux across the
ρu 2
+ ρe d V = control volume + boundaries of the
2
by external
material control
forces
volume.
(2–43)
We note that this conservation principle, for a closed system such as the material control
volume, is precisely equivalent to the first law of thermodynamics, which we can obtain
from it by integrating with respect to time over some finite time interval.
The terms on the right-hand side of (2–43) can be expressed in mathematical form, based
on the following observations. First, work can be done on the material control volume only
as a consequence of forces acting on it. In our continuum description, these are body forces
and surface forces associated with the stress vector t(n). We recall that the surface forces
appear, in part, as a consequence of our inability to fully resolve momentum transfer at the
molecular level in a continuum description. It is not surprising, therefore, that work done in
the macroscopic description may lead to changes in either the macroscopic kinetic energy or
the internal energy representing changes in the intensity of motions at the molecular level.
The motivation for a term in (2–43) that is associated with energy flux across the boundaries
of the material control volume is very similar to that associated with the appearance of a
surface force (or stress) in the linear momentum principle. In particular, there would be no
local flux of kinetic or internal energy across the surface of a material control volume if the
fluid were actually a continuous, infinitely divisible, and homogeneous medium, because
the material control volume is defined as moving and deforming with the fluid in such a way
that the local flux of mass across its surface is zero. However, random motions of molecules
(which are not resolved explicitly in the continuum description) can contribute a net flux
of internal energy across the surface, and this can only be included in the continuum
energy balance (2–43) by the assumed existence of a surface energy flux vector q. This
surface energy flux is usually called the heat flux vector, in recognition of the fact that it is
internal energy (or average intensity of molecular motion) that is being transferred across
the surface by random molecular motion. Incorporating the rate of working terms that are
due to surface and body forces, as well as a surface flux of energy term, we can write (2–43)
in the mathematical form:
D
Dt
Vm (t)
ρu 2
+ ρe d V =
2
[t(n) · u]d A +
Am (t)
(ρg) · ud V −
Vm (t)
q · nd S.
Am (t)
(2–44)
Here we have adopted the convention that a flux of heat into the material control volume
is positive. The negative sign in the last term appears because n is the outer normal to the
material control volume.
To obtain a pointwise DE from (2–44), we follow the usual procedure of applying
the Reynolds transport theorem to the left-hand side and the divergence theorem to the
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