Graebel W.P. Advanced Fluid Mechanics
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4 Fundamentals
1.2 Velocity, Acceleration, and the Material Derivative
A fluid is defined as a material that will undergo sustained motion when shearing forces
are applied, the motion continuing as long as the shearing forces are maintained. The
general study of fluid mechanics considers a fluid to be a continuum. That is, the fact
that the fluid is made up of molecules is ignored but rather the fluid is taken to be a
continuous media.
In solid and rigid body mechanics, it is convenient to start the geometric discussion
of motion and deformation by considering the continuum to be made up of a collection
of particles and consider their subsequent displacement. This is called a Lagrangian,
or material, description , named after Joseph Louis Lagrange (1736–1836). To illustrate
its usage, let XX
0
Y
0
Z
0
t YX
0
Y
0
Z
0
t ZX
0
Y
0
Z
0
t be the position at time
t of a particle initially at the point X
0
Y
0
Z
0
. Then the velocity and acceleration of
that particle is given by
v
x
X
0
Y
0
Z
0
t=
XX
0
Y
0
Z
0
t
t
v
y
X
0
Y
0
Z
0
t=
XX
0
Y
0
Z
0
t
t
v
z
X
0
Y
0
Z
0
t=
XX
0
Y
0
Z
0
t
t
and
(1.2.1)
a
x
X
0
Y
0
Z
0
t=
v
x
X
0
Y
0
Z
0
t
t
=
2
XX
0
Y
0
Z
0
t
t
2
a
y
X
0
Y
0
Z
0
t=
v
y
X
0
Y
0
Z
0
t
t
=
2
YX
0
Y
0
Z
0
t
t
2
a
z
X
0
Y
0
Z
0
t=
v
z
X
0
Y
0
Z
0
t
t
=
2
ZX
0
Y
0
Z
0
t
t
2
The partial derivatives signify that differentiation is performed holding X
0
Y
0
and
Z
0
fixed.
This description works well for particle dynamics, but since fluids consist of an
infinite number of flowing particles in the continuum hypothesis, it is not convenient to
label the various fluid particles and then follow each particle as it moves. Experimental
techniques certainly would be hard pressed to perform measurements that are suited to
such a description. Also, since displacement itself does not enter into stress-geometric
relations for fluids, there is seldom a need to consider using this descriptive method.
Instead, an Eularian,orspatial, description, named after Leonard Euler
(1707–1783), is used. This description starts with velocity, written as v =vxt, where
x refers to the position of a fixed point in space, as the basic descriptor rather than
displacement. To find acceleration, recognize that acceleration means the rate of change
of the velocity of a particular fluid particle at a position while noting that the particle
is in the process of moving from that position at the time it is being studied. Thus, for
instance, the acceleration component in the x direction is defined as
a
x
= lim
t→0
v
x
x +v
x
t y +v
y
t z +v
z
t t +t −v
x
x y z t
t
=
v
x
t
+v
x
v
x
x
+v
y
v
x
y
+v
z
v
x
z
1.3 The Local Continuity Equation 5
since the rate at which the particle leaves this position is v dt. Similar results can be
obtained in the y and z direction, leading to the general vector form of the acceleration as
a =
v
t
+v · v (1.2.2)
The first term in equation (1.2.2) is referred to as the temporal acceleration, and the
second as the convective, or occasionally advective, acceleration.
Note that the convective acceleration terms are quadratic in the velocity components
and hence mathematically nonlinear. This introduces a major difficulty in the solution
of the governing equations of fluid flow. At this point it might be thought that since the
Lagrangian approach has no nonlinearities in the acceleration expression, it could be
more convenient. Such, however, is not the case, as the various force terms introduced
by Newton’s laws all become nonlinear in the Lagrangian approach. In fact, these
nonlinearities are even worse than those found using the Eularian approach.
The convective acceleration term
v ·
v can also be written as
v ·
v =
1
2
v ·v
+v × ×v (1.2.3)
This can be shown to be true by writing out the left- and right-hand sides.
The operator
t
+
v ·
, which appears in equation (1.2.2), is often seen in fluid
mechanics. It has been variously called the material,orsubstantial, derivative, and
represents differentiation as a fluid particle is followed. It is often written as
D
Dt
=
t
+
v ·
(1.2.4)
Note that the operator v · is not a strictly correct vector operator, as it does not
obey the commutative rule. That is, v · = ·v. This operator is sometimes referred
to as a pseudo-vector. Nevertheless, when it is used to operate on a scalar like mass
density or a vector such as velocity, the result is a proper vector as long as no attempt
is made to commute it.
1.3 The Local Continuity Equation
To derive local equations that hold true at any point in our fluid, a volume of arbitrary
shape is constructed and referred to as a control volume. A control volume is a device
used in analyzing fluid flows to account for mass, momentum, and energy balances. It
is usually a volume of fixed size, attached to a specified coordinate system. A control
surface is the bounding surface of the control volume. Fluid enters and leaves the
control volume through the control surface. The density and velocity inside and on the
surface of the control volume are represented by and v. These quantities may vary
throughout the control volume and so are generally functions of the spatial coordinates
as well as time.
The mass of the fluid inside our control volume is
V
dV. For a control volume
fixed in space, the rate of change of mass inside of our control volume is
d
dt
V
dV=
V
t
dV (1.3.1)
The rate at which mass enters the control volume through its surface is
S
v ·n dS (1.3.2)
6 Fundamentals
where v ·ndS is the mass rate of flow out of the small area dS. The quantity v ·n is
the normal component of the velocity to the surface. Therefore, a positive value of v ·n
means the v ·n flow locally is out of the volume, whereas a negative value means that
it is into the volume.
The net rate of change of mass inside and entering the control volume is then found
by adding together equations (1.3.1) and (1.3.2) and setting the sum to zero. This gives
V
t
dV +
S
v ·n dS = 0 (1.3.3)
To obtain the local continuity equation in differential equation form, use (1.1.1) to
transform the surface integral to a volume integral. This gives
S
v ·n dS =
V
·vdV
Making this replacement in (1.3.3) gives
V
t
dV +
V
·vdV = 0
Rearranging this to put everything under the same integral sign gives
V
t
+ ·v
dV = 0 (1.3.4)
Since the choice of the control volume was arbitrary and since the integral must vanish
no matter what choice of control volume was made, the only way this integral can
vanish is for the integrand to vanish. Thus,
t
+ ·v = 0 (1.3.5)
Equation (1.3.5) is the local form of the continuity equation. An alternate expression
of it can be obtained by expanding the divergence term to obtain
t
+v · + ·v =0
The first two terms represent the material derivative of . It is the change of mass density
as we follow an individual fluid particle for an infinitesimal time. Writing
D
Dt
≡
t
+v·,
equation (1.3.5) becomes
D
Dt
+ ·v =0 (1.3.6)
The first term in equation (1.3.6) represents the change in density following a particle,
and the second represents (as shall be shown later) mass density times the change in
volume per unit volume as a mass particle is followed.
An incompressible flow is defined as one where the mass density of a fluid particle
does not change as the particle is followed. This can be expressed as
D
Dt
=0 (1.3.7)
Thus, the continuity equation for an incompressible flow is
·v = 0 (1.3.8)
1.4 Path Lines, Streamlines, and Stream Functions 7
1.4 Path Lines, Streamlines, and Stream Functions
A path line is a line along which a fluid particle actually travels. Since it is the time
history of the position of a fluid particle, it is best described using the Lagrangian
description. Since the particle incrementally moves in the direction of the velocity
vector, the equation of a path line is given by
dt =
dX
V
x
=
dY
V
y
=
dZ
V
z
(1.4.1)
the integration being performed with X
0
Y
0
, and Z
0
held fixed.
A streamline is defined as a line drawn in the flow at a given instant of time such
that the fluid velocity vector at any point on the streamline is tangent to the line at that
point. The requirement of tangency means that the streamlines are given by the equation
dx
v
x
=
dy
v
y
=
dz
v
z
(1.4.2)
While in principle the streamlines can be found from equation (1.4.2), it is usually easier
to pursue a method utilizing the continuity equation and stream functions described in
the following.
A stream surface (or stream sheet) is a collection of adjacent streamlines, providing
a surface through which there is no flow. A stream tube is a tube made up of adjoining
streamlines.
For steady flows (time-independent), path lines and streamlines coincide. For
unsteady flows (time-dependent), path lines and streamlines may differ. Generally path
lines are more difficult to find analytically than are streamlines, and they are of less use
in practical applications.
The continuity equation imposes a restriction on the velocity components. It is a
relation between the various velocity components and mass density. It is not possible
to directly integrate the continuity equation for one of the velocity components in terms
of the others. Rather, this is handled through the use of an intermediary scalar function
called a stream function.
Stream functions are used principally in connection with incompressible flows—
that is, flows where the density of individual fluid particles does not change as the
particle moves in the flow. In such a flow, equation (1.3.8) showed that the continuity
equation reduces to
v
x
x
+
v
y
y
+
v
z
z
=0 (1.4.3)
Because the equations that relate stream functions to velocity differ in two and three
dimensions, the two cases will be considered separately.
1.4.1 Lagrange’s Stream Function for Two-Dimensional Flows
For two-dimensional flows, equation (1.4.3) reduces to
v
x
x
+
v
y
y
=0 (1.4.4)
8 Fundamentals
indicating that one of the velocity components can be expressed in terms of the other.
To attempt to do this directly by integration, the result would be
v
y
=−
v
x
x
dy
Unfortunately, this cannot be integrated in any straightforward manner. Instead, it is
more convenient to introduce an intermediary, a scalar function called Lagrange’s
stream function, that allows the integration to be carried out explicitly.
Let
v
x
=
y
(1.4.5)
Then equation (1.4.4) becomes
v
y
y
+
2
xy
=0
which can be integrated with respect to y to give
v
y
=−
x
(1.4.6)
Therefore, expressing the two velocity components in terms of in the manner of
equations (1.4.5) and (1.4.6) guarantees that continuity is satisfied for an incompress-
ible flow.
In two dimensions, the tangency requirement equation (1.4.2) reduces to dx/v
x
=
dy/v
y
on the streamline. Using equations (1.4.5) and (1.4.6) reduces this to
dx
y
=
dy
−
x
,
or, upon rearrangement,
0 =
x
dx +
y
dy =d (1.4.7)
Equation (1.4.7) states that along a streamline d vanishes. In other words, on a
streamline, is constant. This is the motivation for the name stream function for .
Sometimes it is more convenient in a given problem to use cylindrical polar coor-
dinates rather than Cartesian coordinates. The relation of cylindrical polar coordinates
to a Cartesian frame is shown in Figure 1.4.1, with the appropriate tangent unit vectors
shown at point P. The suitable expressions for the velocity components in cylindrical
polar coordinates using Lagrange’s stream function are
v
r
=
r
and v
=−
r
(1.4.8)
Note from either equation (1.4.5), (1.4.6), or (1.4.8) that the dimension of is length
squared per unit time.
A physical understanding of can also be found by looking at the discharge through
a curve C as seen in Figure 1.4.2. The discharge through the curve C per unit distance
into the paper is
Q =
B
A
v
x
dy −v
y
dx (1.4.9)
When the velocity components are replaced by their expressions in terms of , equa-
tion (1.4.9) becomes
Q =
B
A
y
dy −
−
x
dx
=
B
A
d =
B
−
A
(1.4.10)
1.4 Path Lines, Streamlines, and Stream Functions 9
θ
z
r
P
z
x
y
Figure 1.4.1 Cylindrical coordinate conventions
ds
dx
Curve C
A
B
dy
v
y
v
x
Figure 1.4.2 Discharge through a curve
Thus, the discharge through the curve C is equal to the difference of the value of at
the endpoints of C.
Example 1.4.1 Lagrange stream function
Find the stream function associated with the two-dimensional incompressible flow:
v
r
=U
1−
a
2
r
2
cos
v
=−U
1+
a
2
r
2
sin
Solution. Since v
r
=/r, by integration of v
r
with respect to , find
=
U
1−
a
2
r
2
cosdr=U
r −
a
2
r
sin +fr
The “constant of integration” f here possibly depends on r, since we have integrated a
partial derivative that had been taken with respect to , the derivative being taken with
r held constant in the process.
10 Fundamentals
Differentiating this with respect to r gives
r
=−v
=U
1+
a
2
r
2
sin +
dfr
dr
Comparing the two expressions for v
, we see that df/dr must vanish, so f must be a
constant. We can set this constant to any convenient value without affecting the velocity
components or the discharge. Here, for simplicity, we set it to zero. This gives
= U
r −
a
2
r
sin
Example 1.4.2 Path lines
Find the path lines for the flow of Example 1.4.1. Find also equations for the position
of a fluid particle along the path line as a function of time.
Solution. For steady flows, path lines and streamlines coincide. Therefore, on
a path line, is constant. Using the stream function from the previous example,
we have
= U
r −
a
2
r
sin = constant =B (say).
The equations for a path line in cylindrical polar coordinates are
dt =
dr
v
r
=r
d
v
Since on the path line sin =
0
r−
a
2
r
, then
v
r
=U
1−
a
2
r
2
1−sin
2
=
U
r
2
r
2
−a
2
2
−
0
r
2
If for convenience we introduce new constants
F
2
=a
2
+
2
0
2
+
0
a
2
+
2
0
4
G
2
=a
2
+
2
0
2
−
0
a
2
+
2
0
4
k=
F
G
and a new variable s =r/F . Then, since a
4
=FG
2
and 2a
2
+
0
2
=F
2
+G
2
,
v
r
=
U
F
2
s
2
F
4
s
4
−F
2
+G
2
F
2
s
2
+F
2
G
2
=
UG
F
2
s
2
1−s1−k
2
s
2
Consequently, we have
dt =
F
2
s
2
UG1−s1−k
2
s
2
ds
Upon integration this gives
t −t
0
=
F
2
UG
r/F
r
0
/F
s
2
1−s1−k
2
s
2
ds where r
0
is the value of r at t
0
1.4 Path Lines, Streamlines, and Stream Functions 11
The integral in the previous expression is related to what are called elliptic integrals.
Its values can be found tabulated in many handbooks or by numerical integration. Once
r is found as a function of t on a path line (albeit in an inverse manner, since we have
t as a function of r), the angle is found from = sin
−1
0
r−
a
2
r
. From this we see that
the constant
0
can also be interpreted as
0
=
r
0
−
a
2
r
0
sin
0
. Note that calculation of
path lines usually is much more difficult than are calculations for streamlines.
1.4.2 Stream Functions for Three-Dimensional Flows, Including
Stokes Stream Function
For three dimensions, equation (1.4.3) states that there is one relationship between the
three velocity components, so it is expected that the velocity can be expressed in terms
of two scalar functions. There are at least two ways to do this. The one that retains the
interpretation of stream function as introduced in two dimensions is
v = × (1.4.11)
where and are each constant on stream surfaces. The intersection of these stream
surfaces is a streamline. Equations (1.4.4) and (1.4.5) are in fact a special case of this
result with =z.
Although equation (1.4.11) preserves the advantage of having the stream functions
constant on a streamline, it has the disadvantage that v is a nonlinear function of
and . Introducing further nonlinearities into an already highly nonlinear problem is not
usually helpful!
Another possibility for solving the continuity equation is to write
v = ×A (1.4.12)
which guarantees ·v =0 for any vector A. For the two-dimensional case, A =0 0
corresponds to our Lagrange stream function. Since only two scalars are needed, in three
dimensions one component of A can be arbitrarily set to zero. (Some thought must be
used in doing this. Obviously, in the two-dimensional case, difficulty would be encoun-
tered if one of the components of A that we set to zero was the z component.) The form
of equation (1.4.12), while guaranteeing satisfaction of continuity, has not been much
used, since the appropriate boundary conditions to be imposed on A can be awkward.
A particular three-dimensional case in which a stream function is useful is that of
axisymmetric flow. Taking the z-axis as the axis of symmetry, either spherical polar
coordinates,
R =
x
2
+y
2
+z
2
=cos
−1
z
R
= tan
−1
y
z
(1.4.13)
(see Figure 1.4.3, with the appropriate tangent unit vectors shown at point P), or
cylindrical polar coordinates with
r =
x
2
+y
2
= tan
−1
y
x
z=z (1.4.14)
(see Figure 1.4.1) can be used. The term axisymmetry means that the flow appears the
same in any = constant plane, and the velocity component normal to that plane is
zero. (There could in fact be a swirl velocity component v
without changing anything
12 Fundamentals
θ
β
R
P
x
z
y
Figure 1.4.3 Spherical coordinates conventions
we have said. It would not be related to or determined by .) Since any plane given by
equal to a constant is therefore a stream surface, we can use equation (1.4.11) with
= , giving
v
R
=
1
R
2
sin
v
=
−1
R sin
R
(1.4.15)
in spherical coordinates with = R ,or
v
r
=−
1
r
z
v
z
=
1
r
r
(1.4.16)
in cylindrical coordinates, with =r z. This stream function in either a cylindrical
or spherical coordinate system is called the Stokes stream function. Note that the
dimensions of the Stokes stream function is length cubed per unit time, thus differing
from Lagrange’s stream function by a length dimension.
The volumetric discharge through an annular region is given in terms of the Stokes
stream function by
Q =
B
A
2rv
z
dr −v
r
dz =
B
A
2r
1
r
r
dr −
−1
r
z
dz
=2
B
A
r
dr +
z
dz
=2
B
A
d = 2
B
−
A
(1.4.17)
Example 1.4.3 Stokes stream function
A flow field in cylindrical polar coordinates is given by
v
r
=−
15Ua
3
rz
r
2
+z
2
5/2
v
z
=
Ua
3
r
2
−2z
2
2r
2
+z
2
5/2
Find the Stokes stream function for this flow.
Solution. Since v
r
=
1
r
z
, integration of v
r
with respect to z gives
=−15Ua
3
rz
r
2
+z
2
5/2
rdz =−Ua
3
r
2
2r
2
+z
2
3/2
+fr
1.5 Newtons Momentum Equation 13
Differentiating this with respect to r, we have
r
=rv
z
=05Ua
3
rr
2
−2z
2
r
2
+z
2
5/2
+
dfr
dr
Comparing this with the preceding expression for v
z
, we see that dfr/dr = rU , and
so df/dr = 05Ur
2
. Thus, finally
=−
Ua
3
r
2
2r
2
+z
2
3/2
+05Ur
2
=
Ur
2
2
1−
a
3
r
2
+z
2
3/2
The preceding derivations were all for incompressible flow, whether the flow is steady or
unsteady. The derivations can be extended to steady compressible flow by recognizing
that since for these flows the continuity equation can be written as ·
v
= 0, the
previous stream functions can be used simply by replacing v by v.
A further extension, to unsteady compressible flows, is possible by regarding
time as a fourth dimension and using the extended four-dimensional vector V =
v
x
v
y
v
z
together with the augmented del operator
a
=
x
y
z
t
. The
continuity equation (1.3.5) can then be written as
a
·V =0 (1.4.18)
The previous results can be extended to this general case in a straightforward manner.
1.5 Newtons Momentum Equation
Next apply Newton’s momentum equation to the control volume. The momentum in
the interior of the control volume is
V
vdV . The rate at which momentum enters
the control volume through its surface is
S
vv·n dS. The net rate of change of
momentum is then
d
dt
V
v dV +
S
vv·n dS =
V
v
t
dV +
V
·vvdV
=
V
v
t
+v ·v +
v ·
v
dV (1.5.1)
=
V
v
t
+ ·v
+
v
t
+v ·v
dV =
V
Dv
Dt
dV
The first bracket in the third line of equation (1.5.1) vanishes by virtue of equation
(1.3.5), the continuity equation. The second bracket represents the material derivative
of the velocity, which is the acceleration.
Note that in converting the surface integral to a volume integral using equa-
tion (1.1.1) the “vector” was vv, which is a product of two vectors but is neither the dot
nor cross product. It is sometimes referred to as the indefinite product. This usage is in
fact a slight generalization of equation (1.1.1), which can be verified by writing out the
term in component form.
The forces applied to the surface of the control volume are due to pressure and
viscous forces on the surface and gravitational force distributed throughout the volume.
The pressure force is normal to the surface and points toward the volume. On an