B. CONSERVATION OF MASS – THE CONTINUITY EQUATION

P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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

19



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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.

20



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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),

21



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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)



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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.

23



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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

24



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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

25



10:7



P1: JzG

0521849101c02



CUFX064/Leal



Printer: cupusbw



0 521 84910 1



April 23, 2007



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)



10:7