Page 94 - Bird R.B. Transport phenomena

§3.2 The Equation of Motion 79

Fig. 3.2-1. Fixed volume element Ax
Pzx 12 + Д21 ( x + Ax, у + Ay, z + Az) Ay Az, with six arrows indicating the
flux of x-momentum through the sur-
faces by all mechanisms. The shaded
faces are located at x and x + Ax.

(x, y, z)

Note that Eq. 3.2-1 is an extension of Eq. 2.1-1 to unsteady-state problems. Therefore we
proceed in much the same way as in Chapter 2. However, in addition to including the
unsteady-state term, we must allow the fluid to move through all six faces of the volume
element. Remember that Eq. 3.2-1 is a vector equation with components in each of the
three coordinate directions x, y, and z. We develop the x-component of each term in Eq.
3.2-1; the y- and z-components may be treated analogously. 1
First, we consider the rates of flow of the x-component of momentum into and out of
the volume element shown in Fig. 3.2-1. Momentum enters and leaves Ax Ay Az by two
mechanisms: convective transport (see §1.7), and molecular transport (see §1.2).
The rate at which the x-component of momentum enters across the shaded face at
x by all mechanisms—both convective and molecular—is (ф )\ Ay Az and the rate at
хх х
which it leaves the shaded face at x + Ax is (ф )\ А Х Ay Az. The rates at which
хх
х+
x-momentum enters and leaves through the faces at у and у + Ay are (ф )\ у Az Ax and
ух
(Фух)1,+А У &z A*/ respectively. Similarly, the rates at which x-momentum enters and
leaves through the faces at z and z + Az are (ф )\ г Ах Ay and ty )\ b Ax Ay. When
zx
гх
z+ Z
these contributions are added we get for the net rate of addition of x-momentum
Ay Аг(ф \ - Az Ах{ф \ - ф \ ) + Ах Ау(ф \ - (3.2-2)
хх х
ух у+Ау
гх х
ух у
across all three pairs of faces.
Next there is the external force (typically the gravitational force) acting on the fluid
in the volume element. The x-component of this force is
pg Ax Ay Az (3.2-3)
x
Equations 3.2-2 and 3.2-3 give the x-components of the three terms on the right side of
Eq. 3.2-1. The sum of these terms must then be equated to the rate of increase of
x-momentum within the volume element: Ax Ay Az d{pv )/dt. When this is done, we
x
have the x-component of the momentum balance. When this equation is divided by
Ax Ay Az and the limit is taken as Ax, Ay, and Az go to zero, the following equation
results:
Pgx (3.2-4)

1
In this book all the equations of change are derived by applying the conservation laws to a region
Ax Ay Az fixed in space. The same equations can be obtained by using an arbitrary region fixed in space
or one moving along with the fluid. These derivations are described in Problem 3D.1. Advanced students
should become familiar with these derivations.