D. CONSERVATION OF ENERGY AND THE ENTROPY INEQUALITY

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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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