Graebel W.P. Advanced Fluid Mechanics
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24 Fundamentals
Thus, after choosing the two neighboring points that we wish to describe (i.e., select-
ing dr) to find the rate at which the distance between the points change, we need
to know the local values of the six components of the rate of deformation—that is,
d
xx
d
yy
d
zz
d
xy
d
yz
d
zx
.
Example 1.7.1 Rigid-body rotation
Find the rate of deformation for rigid-body rotation as given by the velocity field
−y x 0.
Solution. From the definition of rate of deformation, d
xx
= d
yy
= d
zz
= 0d
xy
=
d
yz
= d
xz
= 0. The absence of rates of deformation confirms that the fluid is behaving
as a rotating rigid body.
Example 1.7.2 Vortex motion
Find the rate of deformation for a line vortex with velocity v =
−yG
x
2
+ y
2
xG
x
2
+ y
2
0
.
Solution. Again from the definition,
d
xx
=−d
yy
=
2xyG
x
2
+y
2
2
d
zz
=0d
xy
=
xyG
x
2
+y
2
2
d
xz
=d
yz
=0
1.8 Constitutive Relations
Although the physical laws presented so far are of general validity, very little can be said
concerning the behavior of any substance, as can be ascertained by noting that at this
point there are many more unknown quantities than there are equations. The “missing”
relations are those that describe the connection between how a material is made up,
or constituted, and the relation between stress and the geometric and thermodynamic
variables. Hooke’s law and the state equations of an ideal gas are two familiar examples
of constitutive equations. Whereas in a few cases constitutive relations can be derived
from statistical mechanics considerations using special mathematical models for the
molecular structure, the usual procedure is to decide, based on experiments, which
quantities must go into the constitutive equation and then formulate from these a set
of equations that agree with fundamental ideas, such as invariance with respect to the
observer and the like.
There is much inventiveness in proceeding along the path of determining constitutive
equations, and much attention has been given to the subject in past decades. The mental
process of generating a description of a particular fluid involves a continuous interchange
between theory and practice. Once a constitutive model is proposed, mathematical
predictions can be made that then, it is hoped, can be compared with the experiment.
Such a procedure can show a model to be wrong, but it cannot guarantee that it will
always be correct, since many models can predict the same velocity field for the very
simple flows used in viscometry (tests used in measuring the coefficient of viscosity)
and rheogoniometry (measuring the properties of complicated molecular structures such
as polymers). As an example, if a fluid is contained between two large plane sheets
placed parallel to each other, and the sheets are allowed to move parallel to one another
at different velocities, many constitutive models will predict a fluid velocity that varies
1.8 Constitutive Relations 25
linearly with the distance from one plate, with all models giving the same shear stress.
The various models, however, usually predict quite different normal stresses.
The familiar model presented by Newton and elaborated on by Navier and Stokes
has withstood many of these tests for fluids of relatively simple molecular structure and
holds for many of the fluids that one normally encounters. This model is not valid for
fluids such as polymers, suspensions, or many of the fluids encountered in the kitchen,
such as cake batter, catsup, and the like. A good rule of thumb is that if the molecular
weight of the fluid is less than a half million or so, and if the distance between molecules
is not too great (as in rarefied gases), a Newtonian model is very likely to be valid.
We will not delve deeper into a rigorous justification of a particular constitutive
equation here. We simply put down the minimum requirements that we expect of our
constitutive law and give a partial justification of the results.
Considering a fluid of simple molecular structure, such as water or air, experience
and many experiments suggest the following:
1. Stress will depend explicitly only on pressure and the rate of deformation.
Temperature can enter only implicitly through coefficients such as viscosity.
2. When the rate of deformation is identically zero, all shear stresses vanish, and
the normal stresses are each equal to the negative of the pressure.
3. The fluid is isotropic. That is, the material properties of a fluid at any given
point are the same in all directions.
4. The stress must depend on rate of deformation in a linear manner, according to
the original concepts of Newton.
The most general constitutive relation satisfying all of the above requirements is
xx
=−p +
·v +2d
xx
yy
=−p +
·v +2d
yy
zz
=−p +
·v +2d
zz
(1.8.1)
xy
=
yx
=2d
xy
xy
=
yx
=2d
xz
yz
=
zy
=2d
yz
where we have used the abbreviation
d
xx
+d
yy
+d
zz
=
v
x
x
+
v
y
y
+
v
z
z
= ·v
Here, is the viscosity and
is the second viscosity coefficient. Both of these viscosities
can depend on temperature and even pressure. The fluid described by equation (1.8.1)
is called a Newtonian fluid, although the term Navier-Stokes fluid is also used.
Up until now, pressure has deliberately been left undefined. The definition of
pressure varies in different instances. For instance, in elementary thermodynamics texts,
the term pressure is commonly used for the negative of mean normal stress. Summing
our constitutive equations gives
Mean normal stress =
sum of the total normal stress components
3
=−p +
xx
+
yy
+
zz
3
From equation (1.8.1), however, we see that
xx
+
yy
+
zz
3
=
+
2
3
·v (1.8.2)
The coefficient
+
2
3
is called the bulk viscosity,orvolume viscosity, since it
represents the amount of normal stress change needed to get a unit specific volume rate
26 Fundamentals
change. The second law of thermodynamics can be used to show that
+2/3 ≥ 0.
If we are to have the mean normal stress equal to the negative of pressure, it follows
that the bulk viscosity must be zero. At one time Stokes suggested that this might in
general be true but later wrote that he never put much faith in this relationship. Since
for most flows the term is numerically much smaller than p, Stokes’s assumption is still
widely used.
Usually one thinks of p as being the thermodynamic pressure, given by an equation
of state (e.g., p = RT for an ideal gas). In such a case, the bulk viscosity is not
necessarily zero, and so the thermodynamic pressure generally must differ from the
mean normal stress. Values of
for various fluids have been determined experimentally
in flows involving very high-speed sound waves,
2
but the data is quite sparse. Statistical
mechanics tells us that for a monatomic gas, the bulk viscosity is zero. In any case, in
flows where both the dilatation and the bulk viscosity tend to be very small compared
to pressure, the effects of the bulk viscosity can usually be neglected.
Some elaboration of the four postulates for determining our constitutive equation
are in order. We have said that the only kinematic quantity appearing in the stress is
the rate of deformation. What about strain or rate of rotation of fluid elements?
The type of fluid we are considering is completely without a sense of history.
(One class of fluids with a sense of history of their straining is the viscoelastic fluids.
For these fluids strain and strain rates appear explicitly in their constitutive equations.)
A Newtonian fluid is only aware of the present. It cannot remember the past—even the
immediate past—so strain cannot enter the model. Although such a model predicts most
of the basic features of flows, it does have its disturbing aspects, such as infinite speed
of propagation of information if compressibility is ignored. For most flows, however, it
seems a reasonable assumption.
Our assumption that when the rate of deformation is zero the stresses reduce to the
pressure is simply a reaffirmation of the principles of hydrostatics and is a basic law
used for practically all materials.
The isotropy of a fluid is a realistic assumption for a fluid of simple molecular
structure. If we had in mind materials made up of small rods, ellipsoids, or complicated
molecular chains, all of which have directional properties, other models would be called
for, and this constraint would have to be relaxed.
The linearity assumption can be justified only by experiment. The remarkable thing
is that it quite often works! If we relax this point but retain all the other assumptions,
the effect is to add only one term to the right side of equation (1.8.1), making stress
quadratic in the rate of deformation. Also, the various viscosities could also depend on
invariant combinations of the rate of strain, as well as on the thermodynamic variables.
While this adds to the mathematical generality, no fluids are presently known to behave
according to this second-order law.
We have already remarked that a state equation is a necessary addition to the
constitutive description of our fluids. Examples frequently used are incompressibility
D/Dt =0 and the ideal gas law p =RT. Additionally, information on the heat flux
and internal energy must be added to the list. Familiar laws are Fourier’s law of heat con-
duction, where the heat flux is proportional to the negative of the temperature gradient, or
q =−kT (1.8.3)
2
See, for example, Lieberman (1955).
1.9 Equations for Newtonian Fluids 27
and the ideal gas relation for internal energy,
U =UT (1.8.4)
The latter is frequently simplified further by the assumption of the internal energy
depending linearly on temperature so that
U =U
0
+c
T −T
0
(1.8.5)
c
being the specific heat at constant density or volume.
Knowledge of the constitutive behavior of non-Newtonian fluids is unfortunately
much sparser than our knowledge of Newtonian fluids. Particularly with the ever-
increasing use of plastics in our modern society, the ability to predict the behavior of
such fluids is of great economic importance in manufacturing processes. Although many
theoretical models have been put forth over the last century, the situation in general is
far from satisfactory. In principle, from a few simple experiments the parameters in a
given constitutive model can be found. Then predictions of other flow geometries can
be made from this model and compared with further experiments. The result more often
than not is that the predictions may be valid only for a very few simple flows whose
nature is closely related to the flows from which the parameters in the constitutive model
were determined. There are many gaps in our fundamental understanding of these fluids.
1.9 Equations for Newtonian Fluids
Substitution of equation (1.8.1) into equation (1.6.5) gives the result
Dv
x
Dt
=−
p −
·v
x
+g
x
+
x
2
v
x
x
+
y
v
x
y
+
v
y
x
+
z
v
x
z
+
v
z
x
Dv
y
Dt
=−
p −
·v
y
+g
y
+
x
v
y
x
+
v
x
y
+
y
2
v
y
y
+
z
v
y
z
+
v
z
y
Dv
z
Dt
=−
p −
·v
z
+g
z
+
x
v
z
x
+
v
x
z
+
y
v
z
y
+
v
y
z
+
z
2
v
z
z
(1.9.1)
When and are constant, and for incompressible flows, this simplifies greatly
with the help of the continuity condition ·v = 0 to the vector form
Dv
Dt
=−p +g +
2
v (1.9.2)
where
2
≡
2
x
2
+
2
y
2
+
2
z
2
is called the Laplace operator.
28 Fundamentals
Equation (1.9.2) can be written in component notation as
v
x
t
+v
x
v
x
x
+v
y
v
x
y
+v
z
v
x
z
=−
p
x
+g
x
+
2
v
x
v
y
t
+v
x
v
y
x
+v
y
v
y
y
+v
z
v
y
z
=−
p
y
+g
y
+
2
v
y
(1.9.3)
v
z
t
+v
x
v
z
x
+v
y
v
z
y
+v
z
v
z
z
=−
p
z
+g
z
+
2
v
z
Either form (1.9.2) or (1.9.3) is referred to as the Navier-Stokes equation.
The last form (1.9.3) is the form most useful for problem solving. Form (1.9.2)
is useful for physical understanding and other manipulations of our equations. The
Navier-Stokes equations in non-Cartesian coordinate systems are given in the Appendix.
1.10 Boundary Conditions
To obtain a solution of the Navier-Stokes equations that suits a particular problem, it is
necessary to add conditions that need to be satisfied on the boundaries of the region of
interest. The conditions that are most commonly encountered are the following:
1. The fluid velocity component normal to an impenetrable boundary is always
equal to the normal velocity of the boundary. If n is the unit normal to the boundary, then
n ·v
fluid
−v
boundary
=0 (1.10.1)
on the boundary. If this condition were not true, fluid would pass through the boundary.
This condition must hold true even in the case of vanishing viscosity (“inviscid flows”).
If the boundary is moving, as in the case of a flow with a free surface or moving body,
then, with Fxt=0 as the equation of the bounding surface, (1.10.1) is satisfied if
DF
Dt
=0 on the surface F =0 (1.10.2)
This condition is necessary to establish that F = 0isamaterial surface—that is, a
surface moving with the fluid that always contains the same fluid particles. An important
special case of the material surface is the free surface, a constant pressure surface,
typically the interface between a liquid and a gas.
2. Stress must be continuous everywhere within the fluid. If stress were not
continuous, an infinitesimal layer of fluid with an infinitesimal mass would be acted
upon by a finite force, giving rise to infinite acceleration of that layer.
At interfaces where fluid properties such as density are discontinuous, however, a
discontinuity in stress can exist. This stress difference is related to the surface tension of
the interface. Write the stress in the direction normal to the interface as
n
and denote
the surface tension by (a force per unit length). By summing forces on an area of the
interface, if the surface tensile force acts outwardly along the edge of S in a direction
locally tangent to both S and C, the result is
S
n
lower fluid
−
n
upper fluid
dS =
t ×n ds
where t and n are the unit tangent and normal vectors, respectively.
1.11 Vorticity and Circulation 29
A variation of the curl theorem allows us to write the line integral as a surface integral.
Letting
n
=
n
lower fluid
−
n
upper fluid
we have
s
n
dS =
S
− +nn · + ·ndS =0 (1.10.3)
But n · is zero, since is defined only on the surface S and n · is perpendicular
to S. Also, ·n =−
1
R
1
+
1
R
2
, where R
1
and R
2
are the principal radii of curvature of
the surfaces. (For information on this, see McConnell (1957), pp. 202–204.) By taking
components of this equation in directions locally normal and tangential to the surface,
equation (1.10.3) can be conveniently split into
n ·
n
=
1
R
1
+
1
R
2
t ·
n
=−t ·
(1.10.4)
where n is the unit normal and t is a unit tangent to the surface. In words, if surface
tension is present, the difference in normal stress is proportional to the local surface
curvature. If gradients in the surface tension can exist, shear stress discontinuities can
also be present across an interface.
3. Velocity must be continuous everywhere. That is, in the interior of a fluid,
there can be no discrete changes in v. If there were such changes, it would give rise
to discontinuous deformation gradients and, from the constitutive equations, result in
discontinuous stresses.
The velocity of most fluids at a solid boundary must have the same velocity tangential
to the boundary as the boundary itself. This is the “no-slip” condition that has been
observed over and over experimentally. The molecular forces required to peel away fluid
from a boundary are quite large, due to molecular attraction of dissimilar molecules.
The only exceptions observed to this are in extreme cases of rarified gas flow, when
the continuum concept is no longer completely valid.
3
1.11 Vorticity and Circulation
Any motion of a small region of a fluid can be thought of as a combination of translation,
rotation, and deformation. Translation is described by velocity of a point. Deformation
is described by rates of deformation, as in Section 1.7. Here, we consider the rotation
of a fluid element.
In rigid-body mechanics, the concept of angular rotation is an extremely important
one—and rather intuitive. Along with translational velocity, it is one of the basic
descriptors of the motion. In fluid mechanics we can introduce a similar concept in the
following manner.
3
Those interested in the history of this once-controversial condition should read the note at the end of
Goldstein’s Modern Developments in Fluid Dynamics (1965) for an interesting account.
30 Fundamentals
Consider again the two-dimensional picture shown in Figure 1.7.1. In Section 1.7,
we saw that the instantaneous rates of rotation of lines AB and BC were
˙
1
and
˙
2
,
given by equations (1.7.5) and (1.7.6).
Since equations (1.7.5) and (1.7.6) generally will differ, it is seen that the angular
velocity of a line depends on the initial orientation of that line. To develop our analogy
of angular velocity, we want a definition that is independent of orientation and direction
at a point, and depends only on local conditions at the point itself. In considering the
transformation of ABC into A
B
C
, it is seen that two things have happened: The angle
has changed, or deformed, by an amount d
1
+d
2
, and the bisector of the angle ABC
has rotated an amount 05d
1
−d
2
. Considering only this rotation, the rate of rotation
of the bisector is seen to be
1
2
v
y
x
−
v
x
y
(1.11.1)
Writing the curl of v in Cartesian coordinates, we have
curl v = ×v = i
v
z
y
−
v
y
z
+j
v
x
z
−
v
z
x
+k
v
y
x
−
v
x
y
We see by comparison that equation (1.11.1) is one-half the z component of the curl
of v.
If we were to consider similar neighboring points in the yz and xz planes, we
would find that similar arguments would yield the x and y components of one-half the
curl of the velocity. We therefore define the vorticity vector as being the curl of the
velocity—that is,
=curl v = ×v (1.11.2)
and note that the vorticity is twice the local angular rotation of the fluid. (Omitting the
one-half from the definition saves us some unimportant arithmetic and does not obscure
the physical interpretation of the concept.) Since vorticity is a vector, it is independent
of the coordinate frame used. Our definition agrees with the usual “right-hand-rule” sign
convention of angular rotation.
Vorticity also can be represented as a second-order tensor. Writing
xx
=
yy
=
zz
=0
xy
=
1
2
v
x
y
−
v
y
x
xz
=
1
2
v
x
z
−
v
z
x
yz
=
1
2
v
y
z
−
v
z
y
(1.11.3)
note that ×v = 2
zy
i +
xz
j+
yx
k
. Thus, the vector and the second-order tensor
contain the same information.
Since interchanging the indices in the definitions in equation (1.11.3) only changes
the sign, the vorticity (second-order) tensor is said to be skew-symmetric.
If we represent the rate of deformation and vorticity in index notation, we can
summarize some of our findings as follows:
d
ij
=
1
2
v
i
x
j
+
v
j
x
i
=d
ji
ij
=
1
2
v
i
x
j
−
v
j
x
i
=−
ji
d
ij
+
ij
=
v
i
x
j
(1.11.4)
1.11 Vorticity and Circulation 31
Flows with vorticity are said to be rotational flows; flows without vorticity are said
to be irrotational flows. Note for later use that, from a well-known vector identity, it
follows that
div = · ×v = 0 (1.11.5)
You might wonder whether vorticity should enter the constitutive equation along
with the rates of deformation. What has been called the “principle of material objectivity”
or “principle of isotropy of space” or “material frame indifference,” among other things,
states that all observers, regardless of their frame of reference (inertial or otherwise),
must observe the same material behavior. Therefore, an observer stationed on a rotating
platform, for example, sees the same fluid behavior as an observer standing on the floor
of the laboratory. As we have seen in the last section, vorticity is not satisfactory in this
regard in that it is sensitive to rigid rotations. (If you find the idea of material frame
indifference unsettling, see Truesdell (1966), page 6. Truesdell was one of the first
expounders of this concept, but he apparently had many doubts on the same question
initially.) Many constitutive equations that have been proposed violate this principle
(both intentionally and unintentionally). Present work in constitutive equations tends to
obey the principle religiously, although doubts are still sometimes expressed.
Example 1.11.1 Rigid body rotation.
Find the vorticity associated with the velocity field v = −y x 0.
Solution. This is the case of rigid body rotation, with the speed being given by
times the distance of the point from the origin and the stream lines being concentric
circles. Taking the curl of the velocity we have
= ×−y i + x j = 2 k
Thus, the flow everywhere has the same vorticity.
Example 1.11.2 Vortex motion
Find the vorticity associated with the velocity field v =
−yG
x
2
+y
2
xG
x
2
+y
2
0
.
Solution. This is a velocity field again with streamlines that are concentric circles
but with the speed now being proportional to the reciprocal distance from the origin.
This flow is called a line vortex. Taking the curl of this velocity, we have
= ×
−yG
x
2
+y
2
i +
xG
x
2
+y
2
j
=0.
We see that the vorticity is zero, except perhaps at the origin, where the derivatives
become infinite, and a more careful examination must be made.
The preceding two examples point out what vorticity is and what it is not. In both
flows, the streamlines are concentric circles, and a fluid particle travels around the origin
of the coordinate system. In the rigid-body rotation case, particles on two neighboring
streamlines travel at slightly different velocities, the particle on the streamline with the
greater radius having the greater velocity. An arrow connecting the two particles on
different streamlines will travel around the origin, as shown in Figure 1.11.1.
32 Fundamentals
Direction of
Circulation
Figure 1.11.1 Rigid body rotation
Direction of
Circulation
Figure 1.11.2 Line vortex
Considering a similar pair of neighboring streamlines in the line vortex case, the
outer streamline has a slower velocity and the outer particle will lag behind the inner
one (Figure 1.11.2). An arrow connecting the two particles in the limit of zero distance
will always point in the same direction, much as the needle of a compass would. It is
this rotation that vorticity deals with and not the rotation of a point about some arbitrary
reference point, such as the origin.
Since vorticity is a vector, many of the concepts encountered with velocity and
stream functions can be carried over. Thus, vortex lines can be defined as being lines
instantaneously tangent to the vorticity vector, satisfying the equations
dx
x
=
dy
y
=
dz
z
(1.11.6)
Vortex sheets are surfaces of vortex lines lying side by side. Vortex tubes are closed
vortex sheets with vorticity entering and leaving through the ends of the tube.
Analogous to the concept of volume flow through an area,
S
v · dA, the vorticity
flow through an area, termed circulation, is defined as
circulation = =
C
v ·ds =
S
·dS (1.11.7)
1.11 Vorticity and Circulation 33
x
y
(1, –1, 0)
(–1, –1, 0)
(–1, 1, 0)
(1, 1, 0)
Figure 1.11.3 Calculation of circulation about a square
The relation between the line and surface integral forms follow from Stokes theorem,
where C is a closed path bounding the area S.
Example 1.11.3 Circulation for a rigid rotation
Find the circulation through the square with corners at +1 +1 0 −1 +1 0
−1 −1 0 +1 −1 0 for rigid-body rotation flow shown in Figure 1.11.3.
Solution. Starting first with the line integral form of the definition in equation
(1.11.7), we see that
=
C
v ·ds =
−1
+1
v
x
y=1
dx +
−1
+1
v
y
x=−1
dx +
+1
+1
v
x
y=−1
dx +
+1
+1
v
y
x=1
dy
=
−1
+1
−dx +
−1
+1
−dx +
+1
+1
dx +
+1
+1
dy
=−−2 −−2 +2 +2 = 8
Of course, we could have obtained this result much faster by using the area integral
form of the definition (1.11.7),
=
S
·dA
Since the integrand is constant 2 and the area is 4, the result for the circulation
follows from a simple arithmetic multiplication.
Example 1.11.4 Circulation for a vortex motion
Using the square given in Example 1.11.3, find the circulation for the vortex flow given
in Example 1.5.2—that is, v =
−yG
x
2
+y
2
xG
x
2
+y
2
0
.