C. NEWTON’S LAWS OF MECHANICS

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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)



26



∂(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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