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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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D. Conservation of Energy and the Entropy Inequality
right-hand side after first using (2–29) to express t · u as n · (T · u). With all terms then
expressed as volume integrals over the arbitrary material control volume Vm (t), we obtain
the differential form of the energy conservation principle:
ρ
D
Dt
u2
+ e = ∇ · (T · u) + ρg · u − ∇ · q.
2
(2–45)
It appears from (2–45) that contributions from any of the terms on the right-hand side
will lead to a change in the sum of kinetic and internal energy, but may not contribute
separately to one or the other of these energy terms. However, this is not true as we may see
by further examination. First, we may note that the Cauchy’s equation of motion provides
an independent relationship for the rate of change of kinetic energy. In particular, if we take
the inner product of (2–32) with u, we obtain
ρ Du 2
= (ρg) · u + u · (∇ · T).
2 Dt
(2–46)
This relationship is known as the mechanical energy balance and is a direct consequence
of Newton’s second law. Substituting for Du 2 /Dt in (2–45) using (2–46) and recalling that
T is symmetric, we obtain the so-called thermal energy balance:
ρ
De
= T : E − ∇ · q.
Dt
(2–47)
The second-order tensor E that appears in (2–47) is defined in terms of u as
E≡
1
∇u + ∇uT
2
(2–48)
and is known as the rate-of-strain tensor. We shall later see the origins of this name. For
now, we simply note that E is the symmetric part of the velocity gradient tensor, ∇u, that
is,
∇u ≡
1
1
∇u + ∇uT + ∇u − ∇uT = E + Ω.
2
2
symmetric
(2–49)
antisymmetric
The antisymmetric contribution to ∇u, which we have denoted in (2–49) as Ω, is known as
the vorticity tensor. Again, more is said about the vorticity tensor later in this chapter.
Returning to (2–47), the term T : E represents a contribution to the internal energy of
the fluid because of the presence of mean motion (note that E ≡ 0 if ∇u ≡ 0); that is, it
represents a conversion from kinetic energy of the velocity field u to internal energy of the
fluid – a process that is termed dissipation of kinetic energy to internal energy (or heat).
The local rate of working that is due to body forces and surface forces may be seen from
(2–46) to contribute directly to kinetic energy, but to lead to changes in internal energy only
through dissipation. On the other hand, the surface energy (or heat) flux contribution to the
total energy balance contributes directly to the change of internal energy, but only indirectly
to the kinetic energy.
An alternative is to express the thermal energy balance, (2–47), in terms of the specific
enthalpy h:
h ≡ e + ( p/ρ).
(2–50)
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Basic Principles
It can be seen from the definition (2–50) that
ρ
De
Dh
Dp
p Dρ
Dh
Dp
=ρ
−
+
=ρ
−
− p∇ · u.
Dt
Dt
Dt
ρ Dt
Dt
Dt
Hence, it follows that (2–47) can be expressed in terms of the specific enthalpy as
ρ
Dp
Dh
= T:E − ∇ · q +
+ p∇ · u.
Dt
Dt
(2–51)
Although (2–47) and (2–51) are equivalent, it is generally more convenient for a flowing
system to deal with the enthalpy rather than with the internal energy.
We may note that the energy conservation principle (or, equivalently, the first law of
thermodynamics) has not improved the balance between the number of unknown, independent variables and differential relationships between them. Indeed, we have obtained a
single independent scalar equation, either (2–47) or (2–51), but have introduced several new
unknowns in the process, the three components of q and either the specific internal energy
e or enthalpy h. A relationship between e or h and the thermodynamic state variables, say,
pressure p and temperature θ , can be obtained provided that equilibrium thermodynamics
is assumed to be applicable to a fluid element that moves with a velocity u. In particular, a
differential change in θ or p leads to a differential change in h for an equilibrium system:
dh = C p dθ +
1
∂(1/ρ)
−θ
ρ
∂θ
d p.
p
Hence, for a fluid element moving with the fluid,
Dh
Dθ
= Cp
+
Dt
Dt
1
∂(1/ρ)
−θ
ρ
∂θ
Dp
,
Dt
p
and (2–51) can be expressed in terms of θ rather than h in the form
ρC p
Dθ
θ
= T : E + p∇ · u − ∇ · q −
Dt
ρ
∂ρ
∂θ
p
Dp
.
Dt
(2–52)
An alternative form for (2–52) can be written in terms of the heat capacity at constant
volume by means of the general thermodynamic relationship
Cv = C p +
θ
ρ2
∂p
∂θ
ρ
∂ρ
∂θ
.
p
However, this is less useful than (2–52) because it contains terms such as (∂ p/∂θ )ρ , which
are not small and are difficult to evaluate.
We shall see that the sum p(∇ · u) + T : E on the right-hand side of (2–52) represents
the conversion of kinetic energy to heat, due to the internal friction within the fluid and
is known as the viscous dissipation term. The last term on the left-hand side of (2–52) is
related to the work required for compressing the fluid. Although this term is identically zero
only for constant-pressure conditions (that is, the material is a solid or it is stationary so that
Dp/Dt = 0), it is frequently small compared with other terms in (2–52) because the density
at constant pressure is only weakly dependent on the temperature, and we shall generally
adopt this approximation in the analyses of nonisothermal systems in later chapters.
We have seen that the energy conservation principle, applied to a material control volume
of fluid, is equivalent to the first law of thermodynamics. A natural question, then, is whether
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D. Conservation of Energy and the Entropy Inequality
any additional useful information can be obtained from the second law of thermodynamics.
In its usual differential form the second law states
dQ
dS ≥
,
θ
where d S is the entropy change for the thermodynamic system of interest, d Q is the change in
its total heat content caused by heat exchange with the surroundings, and θ is its temperature.
When applied to a material control volume of fluid, this principle can be expressed in the
form
D
Dt
(ρs)d V +
V m(t)
Am(t)
n·q
d A ≥ 0,
θ
(2–53)
where s is the entropy per unit mass of the fluid. The only mechanism for heat transfer from
the surrounding fluid is molecular transport represented by the heat flux vector q. The sign
in front of the second term is a consequence of the fact that n is the outer unit normal. We
easily obtain a differential form of the inequality (2–53) by applying the Reynolds transport
theorem to the first term and the divergence theorem to the second term to show that
ρ
Vm (t)
Ds
q
+∇ ·
Dt
θ
d V ≥ 0.
This inequality can be satisfied for an arbitrary material control volume Vm (t) only if
ρ
Ds
q
+∇ ·
≥ 0.
Dt
θ
(2–54)
We can obtain an inequality that is equivalent to (2–54) by using thermodynamics to
express Ds/Dt in the form
Ds
De
p Dρ
θρ
=ρ
−
Dt
Dt
ρ Dt
and then substituting for De/Dt from the thermal energy balance (2–47). The result for
Ds/Dt is
ρ
Ds
1
= [T : E + p(∇ · u) − ∇ · q].
Dt
θ
(2–55)
Then, because
∇·
q
1
1
= ∇ · q − 2 q · ∇θ,
θ
θ
θ
the inequality (2–54) can be combined with (2–55) to obtain
1
q · ∇θ
≥ 0.
(T : E + p(∇ · u)) −
θ
θ2
(2–56)
Although there is no immediately useful information that we can glean from (2–56), we
shall see that it provides a constraint on allowable constitutive relationships for T and q.
In this sense, it plays a similar role to Newton’s second law for angular momentum, which
led to the constraint (2–41) that T be symmetric in the absence of body couples. In solving
fluid mechanics problems, assuming that the fluid is isothermal, we will use the equation
of continuity, (2–5) or (2–20), and the Cauchy equation of motion, (2–32), to determine the
velocity field, but the angular momentum principle and the second law of thermodynamics
will appear only indirectly as constraints on allowable constitutive forms for T. Similarly,
for nonisothermal conditions, we will use (2–5) or (2–20), (2–32), and either (2–51) or
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Basic Principles
(2–52) to determine the velocity and temperature distributions, but neither the generalization
of Newton’s second law to angular momentum nor the second law of thermodynamics will
appear directly. However, we are getting ahead of our story.
So far, we have seen that the basic macroscopic principles of continuum mechanics lead
to a set of five scalar DEs – sometimes called the field equations of continuum mechanics –
namely, (2–5) or (2–20), (2–32), and (2–51) or (2–52). On the other hand, we have identified
many more unknown variables, u, T, θ , p, and q, plus various fluid or material properties such
as ρ, Cp (or Cv ), (∂ρ/∂θ) p , [or (∂ρ/∂θ ) p ], which generally require additional equations of
state to be determined from p and θ if the latter are adopted as the thermodynamic state
variables. Let us focus just on the independent variables u, T, θ, p, and q. Taking account
of the symmetry of T, these comprise 14 unknown scalar variables for which we have so far
obtained only the five independent “field” equations that were just listed. It is evident that
we require additional equations relating the various unknown variables if we are to achieve a
well-posed problem from a mathematical point of view. Where are these equations to come
from? Why is it that the fundamental macroscopic principles of continuum physics do not,
in themselves, lead to a mathematical problem with a closed set of equations?
E. CONSTITUTIVE EQUATIONS
We have seen that the basic field equations of continuum mechanics are not sufficient
in number to provide a mathematical problem from which to determine solutions for the
independent field variables u, T, θ , p, and q. It is apparent that additional relationships must
be found, hopefully without introducing more independent variables. In the next several
sections, we discuss the origin and form of the so-called constitutive equations that provide
the necessary additional relationships.
We begin with some general observations. In the first place, the idea that additional
equations are necessary has so far been based on the purely mathematical statement that
the field equations by themselves do not lead to a problem with a closed set of equations.
Although this argument is powerful and certainly persuasive, it is also instructive to think
about the problem from a more heuristic, physical point of view. In particular, if we first
restrict ourselves to isothermal, incompressible conditions for which the relevant field equations are continuity, in the form of (2–20), and the Cauchy equations of motion, (2–32), we
see that the only material property that appears explicitly is the density ρ. That is, according
to (2–20) and (2–32) in the form that they stand, it appears that the only material property
that distinguishes the motion of one fluid from another is the density. This is clearly at
odds with experimental observation – we can find (or create by blending) a variety of fluids
that have the same density within experimental error yet clearly demonstrate differences
in flow properties. Consider, for example, the many grades of silicon oils that are sold
commercially. These various grades differ in molecular weight, but their densities are all
very nearly equal. Yet, if we were to simply pour a low- and a high-grade silicon oil from
one container to another, we could not help but note a remarkable difference in the ease
with which the fluids flow. The lowest grades would appear visually somewhat like water,
whereas the highest grades would be more nearly akin to something like corn syrup. Quite
apparently, there is something of the basic physics that is missing from the field equations
alone. Similarly, if we consider a nonisothermal system in the absence of any mean motion,
that is, u ≡ 0, the thermal energy, (2–52), reduces to the form
ρC p
36
∂θ
= −∇ · q.
∂t
(2–57)
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F. Fluid Statics – The Stress Tensor for a Stationary Fluid
Not only are there more independent variables (θ and the three components of q) than
equations (one!), but it would appear that the only material property relevant to energy
transfer is ρC p . Once again, simple observations would show that this is not enough to
characterize the energy transfer processes in real materials.
Why is it that the basic conservation principles of continuum mechanics do not provide
a complete problem statement, either from a mathematical or a physical point of view? The
answer is that the fluids or materials that we wish to consider are not actually indivisible and
homogeneous as presumed in continuum mechanics, but rather they have a definite molecular structure. Although this structure is not directly evident at the scale of resolution relevant
to continuum mechanics, we have seen in the derivation of the basic field equations that it
cannot be ignored altogether even in a purely continuum mechanical formulation. Instead,
the differences between the continuum velocity (which we have seen is really an average of
the molecular velocities “at” a point) and the instantaneous, local molecular velocities are
manifest as apparent surface force or stress, and surface energy or heat flux contributions to
the basic Newton’s second law and principles of energy conservation. Indeed, in the absence
of the stress tensor T and the heat flux vector q, as would be appropriate for a material with
a completely continuous and homogeneous structure down to the finest possible scale of
resolution, the basic field equations are completely adequate in number to determine all
of the remaining independent field variables, u, θ , and p. It is the presence of T and q,
reflecting the existence of transport processes at the molecular scale, that causes the field
equations to contain more independent variables than there are equations. In view of this,
we may anticipate that a full statement of the physics relevant to flowing fluids, whether
isothermal or not, will require additional relationships between the surface stress and/or
heat flux (representing molecular transport processes) and the macroscopic (or continuum)
velocity and temperature fields. These relationships are known as the constitutive equations
for the fluid.
But where do we get these additional equations? Because the underlying mechanisms
responsible for the appearance of surface stress or surface heat flux in the continuum
description are molecular, it is evident that continuum mechanics, by itself, can offer no
basis to deduce what form these relationships should take. Thus, if we insist on a purely
continuum mechanical approach, we must generally guess at the appropriate constitutive
equations and then judge the correctness of our guess by comparisons between theoretically
predicted velocity, temperature, or pressure fields and experimental measurements of the
same quantities.13 This is, in fact, the approach that was historically taken, and, in some
ways, it is still the most successful approach. Fortunately, just about the simplest possible
guess of equations relating T and u, or q and θ , turn out to provide an extremely good
approximation for the large class of fluids (many liquids and all gases) that we know as
Newtonian. We discuss the constitutive model for this class of fluids in more detail in Section
G of this chapter. Regardless of the success of a particular constitutive equation, however,
it is obvious that the status of constitutive equations in continuum mechanics is entirely
different from the field equations that we derived in previous sections. The latter represent
a deductive consequence of the basic laws of Newtonian mechanics and thermodynamics,
whereas the former are never more than a guess, no matter how educated, in the absence of
a fundamental molecular, statistical mechanical theory.14
F.
FLUID STATICS – THE STRESS TENSOR FOR A STATIONARY FLUID
Let us begin our quest for specific constitutive equations by considering the special case
of a stationary fluid (u ≡ 0). In this case, the acceleration of a fluid element is zero, and
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Basic Principles
the linear momentum equation, (2–32), reduces to a balance between body and surface
forces,
∇ · T + ρg = 0,
(2–58)
whereas the thermal energy equation reduces to the form (2–57). Although the equations are
thus considerably simplified for a stationary fluid, the basic problem of requiring constitutive
equations for T and q remains.
In this section, we consider an isothermal, stationary fluid. In this case, from thermodynamics, we know that the only surface force is the normal thermodynamic pressure, p.
The pressure at a point P acts normal to any surface through P with a magnitude that is
independent of the orientation of the surface. That is, for a surface with orientation denoted
by the unit normal vector n, the surface-force vector t(n) takes the form
t(n) = −n p.
(2–59)
The minus sign in this equation is a matter of convention: t(n) is considered positive
when it acts inward on a surface whereas n is the outwardly directed normal, and p is taken
as always positive. The fact that the magnitude of the pressure (or surface force) is independent of n is “self-evident” from its molecular origin but also can be proven on purely
continuum mechanical grounds, because otherwise the principle of stress equilibrium,
(2–25), cannot be satisfied for an arbitrary material volume element in the fluid. The form
for the stress tensor T in a stationary fluid follows immediately from (2–59) and the general
relationship (2–29) between the stress vector and the stress tensor:
T = − pI.
(2–60)
In other words, in this case T is strictly diagonal:
⎛
−p
T=⎝ 0
0
0
−p
0
⎞
0
0⎠ .
−p
Equation (2–60) is the constitutive equation for the stress in a stationary fluid.
Substituting (2–60) into the force balance (2–58), and noting that
∇ · T = ∇ · (− pI) = −∇ p,
we obtain the fundamental equation of fluid statics:
ρg − ∇ p = 0.
(2–61)
It follows that the presence of a body force leads to a nonzero gradient of pressure parallel to
the body force even in a stationary fluid. Indeed, it is well know that the pressure increases
with depth under the action of gravity. Provided the fluid density remains constant, the
pressure increases linearly with depth
p(z) = p0 + ρgz
(2–62)
where p0 is a reference pressure at the vertical position, z = 0, and z increases with depth.
If we consider any arbitrary volume element from within a larger body of stationary fluid, it,
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