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B. Conservation of Mass – The Continuity Equation
There is no reason, at this point, to assume that the fluid density ρ is necessarily constant.
Indeed, conservation of mass requires the density inside the volume element to change
with time in such a way that any imbalance in the mass flux in and out of the volume
element is compensated for by an accumulation of mass inside. Expressing this statement
in mathematical terms we obtain
V
∂ρ
dV = −
∂t
ρu · nd A,
(2–3)
A
where V denotes the arbitrarily chosen volume element of fixed position and shape and A
denotes its (closed) surface. Equation (2–3) is an integral constraint on the velocity and
density fields within a given closed volume element of fluid. Because this volume element
was chosen arbitrarily, however, an equivalent differential constraint at each point in the
fluid can be derived easily. First, the well-known divergence theorem5 is applied to the
right-hand side of (2–3), which thus becomes
V
∂ρ
+ ∇ · (ρu) d V = 0.
∂t
(2–4)
Then we note that this integral condition on ρ and u can be satisfied for an arbitrary volume
element only if the integrand is identically zero, that is,
∂ρ
+ ∇ · (ρu) = 0.
∂t
(2–5)
This is the continuity equation, which we now recognize as the pointwise constraint on
ρ and u that is required by conservation of mass. To justify (2–5), we note that the only
other way, in principle, to satisfy (2–4) would be if the integrand were positive within some
portion of V and negative elsewhere in such a way that the nonzero contributions to the
volume integral cancel. However, if this were the case, the freedom to choose an arbitrary
volume element would lead to a contradiction. In particular, instead of the original choice
of V, we could simply choose a new volume element that lies entirely within the region
where the integrand is positive (or negative). Evidently, (2–4) is then violated, leading us to
conclude that (2–5) must hold everywhere.
Although the derivation of the continuity equation by use of a fixed control volume is
perfectly satisfactory, it is less obvious how to apply Newton’s laws of mechanics in this
framework. The familiar use of these principles from coursework in classical mechanics is
that they are applied to describe the motion of a specific “body” subject to various forces
or torques. To apply these same laws to a fluid (i.e., a liquid or a gas), we introduce the
concepts of material points and a material volume (or material control volume) that we
denote as Vm (t). Now a material point is a continuum point that moves with the local
continuum velocity of the fluid. A material volume Vm (t), is a macroscopic control volume
whose shape at some initial instant, t = 0, is arbitrary, that contains a fixed set of material
points. Because the material volume contains a fixed set of such points, it must move with
the local continuum velocity of the fluid at every point. Hence, as illustrated in Fig. 2–3,
it must deform and change volume in such a way that the local flux of mass through all
points on its surface is identically zero for all time (though, of course, there may still be
exchange of molecules due to random molecular motion). Because mass is neither created
nor destroyed according to the principle of mass conservation, the total mass contained
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Basic Principles
n
Vm (0)
n
u(x)
Time increment of t
Vm (t)
Sm (0)
Usurface = u(x)
Sm (t)
Figure 2–3. An arbitrarily chosen material control volume shown at some initial moment t = 0 and at a
later time t, after which it has translated and distorted in shape because each point on its surface moves
with the local fluid velocity u. Equation (2–6) represents a statement of mass conservation for this material
volume.
within the material volume is constant, independent of time. This may be expressed in the
mathematical statement
D
Dt
ρd V = 0.
(2–6)
Vm (t)
Here, the symbol D/Dt stands for the convected or material time derivative, which
we shall subsequently discuss in some detail. In the context of (2–6) it is clear that D/Dt
represents the time derivative of the total mass of material in the material volume Vm (t).
Alternatively, we could say that it is the time derivative of the total mass associated with the
fixed set of material points that comprise Vm (t). We shall see shortly that Eq. (2–6), which
derives directly from the definition of a material volume for a fluid that conserves mass, is
entirely equivalent to (2–3) or (2–4) and leads precisely to the pointwise continuity equation,
(2–5). However, this cannot be seen easily without further discussion of the convected or
material time derivative.
The first question that we may ask is the form of the relationship between D/Dt and
the ordinary partial time derivative ∂/∂t. The so-called “sky-diver” problem illustrated in
Fig. 2–4 provides a simple physical example that may serve to clarify the nature of this
relationship without the need for notational complexity. A sky diver leaps from an airplane
at high altitude and begins to record the temperature T of the atmosphere at regular intervals
of time as he falls toward the Earth. We denote his velocity as −Udiver iz , where iz is a unit
vector in the vertical direction, and the time derivative of the temperature he records as
D ∗ T /Dt ∗ . Here, D ∗ /Dt ∗ represents the time rate of change (of T) measured in a reference
frame that moves with the velocity of the diver. Evidently there is a close relationship
between this derivative and the convected derivative that was introduced in the preceding
paragraph. Let us now suppose, for simplicity, that the temperature of the atmosphere varies
with the distance above the Earth’s surface but is independent of time at any fixed point,
say, z = z ∗ . In this case, the partial time derivative ∂ T /∂t is identically equal to zero.
Nevertheless, in the frame of reference of the sky diver, D ∗ T /Dt ∗ is not zero. Instead,
D∗ T
∂T
= −Udiver
.
∗
Dt
∂z
(2–7)
The temperature is seen by the sky diver to vary with time because he is falling relative
to the Earth’s surface with a velocity Udiver , while T varies with respect to position in the
direction of his motion.
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B. Conservation of Mass – The Continuity Equation
T(z)
Udiver
Temp.
T(z*)
T(z*)
T(z(t))
Time
z
T(z(t))
Temp.
y
Time
x
Figure 2–4. A sky diver falls with velocity Udiver from a high altitude carrying a thermometer and a
recording device that plots the instantaneous temperature, as shown in the lower left-hand corner. During
the period of descent, the temperature at any fixed point in the atmosphere is independent of time (i.e.,
the partial time derivative ∂ T /∂t ≡ 0). However, the sky diver is in an inversion layer and the temperature
decreases with decreasing altitude. Thus the recording of temperature versus time obtained by the sky diver
shows that the temperature decreases at a rate DT /Dt ∗ = Udiver ∂ T /∂z.. This time derivative is known as
the Lagrangian derivative for an observer moving with velocity Udiver .
The relatively simple concept represented by the sky-diver example is easily generalized
to provide a relationship between the convected derivative of any scalar quantity B associated
with a fixed material point and the partial derivatives of B with respect to time and spatial
position in a fixed (inertial) reference frame. Specifically, B changes for a moving material
point both because B may vary with respect to time at each fixed point at a rate ∂ B/∂t and
because the material point moves through space and B may be a function of spatial position
in the direction of motion. The rate of change of B with respect to spatial position is just
∇ B. The rate at which B changes with time for a material point with velocity u is then just
the projection of ∇ B onto the direction of motion multiplied by the speed, which is u · ∇ B.
It follows that the convected time derivative of any scalar B can be expressed in terms of
the partial derivatives of B with respect to time and spatial position as
DB
∂B
=
+ u · ∇ B.
Dt
∂t
(2–8)
It may now be evident why the convected derivative D/Dt is also known as the material
time derivative. D B/Dt is, in fact, the time derivative of the quantity B for a fixed material
point. Material points are often specified by the position vector x0 corresponding to their
position at t = 0. The position vector of the material point x0 at an arbitrary time t > 0 is
thus
t
x = x(x0 , t) ≡ x0 +
u(τ, x0 )dτ
(2–9)
0
Any property of the fluid, say B, that is specified as a function of time for an arbitrary
material point x0 ,
B(x0 , t),
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Basic Principles
can be turned into the corresponding spatial description by use of (2–9):
B[x(x0 , t), t].
It is now clear that there are two distinct time derivatives:
∂
≡
∂t
∂
∂t
D
≡
Dt
and
x
∂
∂t
.
x0
Furthermore, the relationship between them can be obtained formally by application of the
chain rule6 :
DB
∂
=
(B(x0 , t))
Dt
∂t
∂B
=
∂ xi
∂ xi
∂t
=
x0
∂
(B(x(x0 , t), t))
∂t
∂B
∂t
+
x0
x0
,
x
DB
∂B
∂B
+
= ui
.
Dt
∂ xi
∂t
This is identical to (2–8), though now expressed in Cartesian component notation.6 Time
derivatives at fixed spatial position are often called the Eulerian time derivatives, whereas
those taken at a fixed material point are known as Lagrangian. Although we have derived
a simple relationship relating the convected or material derivative to the ordinary partial
derivative at a fixed point, this cannot be applied directly to (2–6) without derivation of a
general relationship, known as the Reynolds transport theorem.
Let us consider any scalar quantity B(x, t) that is associated with a moving fluid. Then
the Reynolds transport theorem says
D
Dt
B(x, t)d V =
Vm (t)
Vm (t)
∂B
+ ∇ · (Bu) d V.
∂t
(2–10)
This is essentially a generalization of Leibnitz rule for differentiation of a one-dimensional
integral with respect to some variable when both the integrand and the limits of integration
depend on that variable. The proof of (2–10) is straightforward.7 We first note that every
point x(t) within a material control volume is a material point whose position is prescribed
by (2–9). Hence, once the (arbitrary) initial shape of the material control volume is chosen
(so that all initial values of x0 are specified), a scalar quantity B associated with any point
within the material control volume can be completely specified as a function of time only,
that is, B[x(t), t]. Thus the usual definition of an ordinary time derivative can be applied to
the left-hand side of (2–10), and we write
D
Dt
B[x(t), t]d V
≡ lim
δt→0
Vm (t)
1
δt
B(t + δt) d V −
B(t)d V
Vm (t+δt)
.
Vm (t)
(2–11)
To the quantity on the right-hand side of (2–11), we now add and subtract the term
Vm (t) B(t + δt)d V :
D
Dt
Bd V = lim
Vm (t)
δt→0
+
22
1
δt
1
δt
B(t + δt)d V −
Vm (t+δt)
B(t + δt)d V −
Vm (t)
B(t + δt)d V
Vm (t)
B(t)d V
Vm (t)
.
(2–12)
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B. Conservation of Mass – The Continuity Equation
The second term is just
lim
δt→0
Vm (t)
B(t + δt) − B(t)
dV ≡
δt
Vm (t)
∂B
d V.
∂t
(2–13)
The first term is simply rewritten as
lim
δt→0
1
δt
B(t + δt)d V ,
(2–14)
Vm (t+δt)−Vm (t)
which shows that it is the integral of B(t) over the differential volume element Vm (t + δt) −
Vm (t). To evaluate (2–14), we notice that any differential element of surface dAm of Vm (t) will
move a distance u · nδt in a time interval δt. Thus, for sufficiently small δt, the differential
volume element d V in (2–14) can be approximated as (u · n)δtd Am and the volume integral
over Vm (t + δt) − Vm (t) then converted to an integral over the surface of Vm (t). Thus,
lim
δt→0
1
δt
= lim
δt→0
B(t + δt)d V
Vm (t+δt)−Vm (t)
1
δt
B(t + δt)u · nδtd A =
Am (t)
B(t)u · nd A,
(2–15)
Am (t)
where Am (t) is the surface of the material volume, and
D
∂B
Bd V =
B(u · n)d A.
dV +
Dt Vm (t)
Vm (t) ∂t
Am (t)
(2–16)
The proof of the Reynolds transport theorem in the form (2–10) is completed by application
of the divergence theorem to the surface integral in (2–16). The physical interpretation of the
Reynolds transport theorem, seen from (2–16), is that the total accumulation of a quantity B
in a material control volume is the sum of the volume integral of the local accumulation of
B at each fixed point in space, plus the total rate of entry through the surface of the control
volume that is due to its motion through space. We may note that if we were to consider a
volumetric region moving through space with some velocity u∗ that differs from the fluid
velocity u, the accumulation of α within that region would take the same form as (2–16),
but with u replaced with u∗ :
D∗
∂B
D∗
∂
Bd V =
B(u∗ · n)d A, where
≡
dV +
+ u∗ · ∇.
∗
∗ (t) ∂t
∗ (t)
Dt V ∗ (t)
Dt ∗
∂t
V
A
We can now apply the Reynolds transport theorem in the form (2–10) to (2–6). In this
case, the scalar property B(x, t) is just the fluid density, ρ(x, t). Thus, the mass conservation
principle, (2–6), can be reexpressed in the form
Vm (t)
∂ρ
+ ∇ · (ρu) d V = 0.
∂t
(2–17)
Because the initial choice of Vm (t) is arbitrary, we obtain the same differential form for
the continuity equation, (2–5), that we derived earlier by using a fixed control volume. Of
course, the fact that we obtain the same form for the continuity equation is not surprising.
The two derivations are entirely equivalent. In the first, conservation of mass is imposed
by the requirement that the time rate of change of mass in a fixed control volume be
exactly balanced by a net imbalance in the influx and efflux of mass through the surface. In
particular, no mass is created or destroyed. In the second approach, we define the material
volume element so that the mass flux through its surface is everywhere equal to zero. In this
case, the condition that mass is conserved means that the total mass in the material volume
element is constant. The differential form (2–5) of the statement of mass conservation,
which we have called the continuity equation, is the main result of this section.
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Basic Principles
Before leaving the continuity equation and mass conservation, there are a few additional
remarks that we can put to good use later. The first is that (2–5) can be expressed in a precisely
equivalent alternative form:
1 Dρ
+ ∇ · u = 0.
ρ Dt
(2–18)
Or, because the specific volume of the fluid is V ≡ 1/ρ, we can also write (2–18) as
1 DV
= ∇ · u.
V Dt
(2–19)
The left-hand side of (2–19) is sometimes referred to as the rate of expansion or the
rate of dilation of the fluid and provides a clear physical interpretation of the quantity
∇ · u(or div u).
The forms of the continuity equation (2–18) or (2–19) also lead directly to a simpler
statement of the mass conservation principle that applies if it can be assumed that the density
is constant, so that Dρ/Dt = 0. In this case, the fluid is said to be (i.e., is approximated
as) incompressible. In general, the density is related to the temperature and pressure by
means of an equation of state, ρ = ρ( p, T ). In an isothermal fluid, the incompressibility
approximation is therefore a statement that the density is independent of the pressure. No
fluid is truly incompressible in this sense. However, experience has shown that it is a good
approximation if a dimensionless parameter, known as Mach number, M, is small:
M≡
|u|
u sound
1.
Here, |u| represents a characteristic velocity of the flow and u sound is the speed of sound in
the fluid at the same temperature and pressure. It may be noted that u sound for air at room
temperature and atmospheric pressure is approximately 300 m/s, whereas the same quantity
for liquids such as water at 20◦ C is approximately 1500 m/s. Thus the motion of liquids
will, in practice, rarely ever be influenced by compressibility effects. For nonisothermal
systems, the density will vary with the temperature, and this can be quite important because
it is the source of buoyancy-driven motions, which are known as natural convection flows.
Even in this case, however, it is frequently possible to neglect the variations of density in the
continuity equation. We will return to this issue of how to treat the density in nonisothermal
flows later in the book.
In any case, if the fluid is isothermal and the density is approximated as a constant, the
continuity equation takes the simpler form
∇ · u ≡ div u = 0.
(2–20)
Vector fields whose divergence vanishes are sometimes referred to as solenoidal. A
more comprehensive discussion of the conditions for approximating the velocity field as
solenoidal has been given by Batchelor.8 These imply that, in cases in which the fluid is subjected to an oscillating pressure, the characteristic velocity in the Mach number condition
should be interpreted as the product of the frequency times the linear dimension of the fluid
domain, and that the difference in static pressures over the length scale of the domain must
be small compared with the absolute pressure. Because our subject matter will frequently
deal with incompressible, isothermal fluids, we shall often make use of (2–20) in lieu of the
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C. Newton’s Laws of Mechanics
more general form, (2–5). In the presence of heat transfer, however, the density will not be
constant because of temperature variations in the fluid, and the simple form (2–20) cannot
be used even for an incompressible fluid unless additional approximations are made. We
shall return to this point later in our presentation.
C. NEWTON’S LAWS OF MECHANICS
We have shown how a pointwise DE can be derived by application of the macroscopic
principle of mass conservation to a material (control) volume of fluid. In this section,
we consider the derivation of differential equations of motion by application of Newton’s
second law of motion, and its generalization from linear to angular momentum, to the same
material control volume. It may be noted that introductory chemical engineering courses in
transport phenomena often approach the derivation of these same equations of motion as an
application of the “conservation of linear and angular momentum” applied to a fixed control
volume. In my view, this obscures the fact that the equations of motion in fluid mechanics are
nothing more than the familiar laws of Newtonian mechanics that are generally introduced
in freshman physics.
We begin with Newton’s second law, which may be stated in the form
⎧
⎫
⎪ the time rate of change ⎪
⎪
⎪
⎪
⎪ of linear momentum ⎪
⎪
⎨
⎬
the sum of forces acting
of a given body,
=
.
(2–21)
on the body
⎪
⎪
⎪ relative to an inertial ⎪
⎪
⎪
⎪
⎪
⎩
⎭
reference frame
This can be applied directly to the material (control) volume of fluid, Vm (t), which was
introduced in the last section. As required for application of (2–21), this is a fixed body of
material in the continuum sense. The resulting equation is
D
Dt
ρud V =
Vm (t)
sum of forces
.
acting on Vm (t)
(2–22)
The fact that the material control volume has a time-dependent shape does not lead to any
complication of principle in applying Newton’s second law. To proceed further, we must
consider the types of forces that appear on the right-hand side of (2–22).
From the purely continuum mechanics viewpoint that we have now adopted, we recognize two kinds of forces acting on the material control volume. First are the body forces,
associated with the presence of external fields, that are capable of penetrating to the interior
of the fluid and acting equally on all elements (per unit mass). The most familiar body
force is gravity, and we will be concerned exclusively with this single type of body force
in this book. Another example is the action of an electromagnetic field when the fluid is
an electrical conductor, which leads to an extension of fluid mechanics into the topic of
magnetohydrodynamics (see Problem 2–11 at the end of the chapter). The second type of
force is a surface force, which acts from the fluid outside the material control volume on the
fluid inside and vice versa. In reality, there may exist short-range forces of molecular origin
in the fluid. With the scale of resolution inherent in the use of the continuum approximation,
these will appear as surface-force contributions in the basic balance (2–22). In addition, the
surface-force terms will always include an effective surface-force contribution to simulate
the transport of momentum across the boundaries of the material control volume that is due
to random differences between the continuum velocity u and the actual molecular velocities.
As explained earlier, the necessity for these latter effective surface-force contributions is
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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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