Page 93 - Bird R.B. Transport phenomena

78 Chapter 3 The Equations of Change for Isothermal Systems

Here (V • pv) is called the "divergence of pv," sometimes written as "div pv." The vector
pv is the mass flux, and its divergence has a simple meaning: it is the net rate of mass ef-
flux per unit volume. The derivation in Problem 3D.1 uses a volume element of arbitrary
shape; it is not necessary to use a rectangular volume element as we have done here.
A very important special form of the equation of continuity is that for a fluid of con-
stant density, for which Eq. 3.1-4 assumes the particularly simple form

(incompressible fluid) (V • v) = 0 (3.1-5)
Of course, no fluid is truly incompressible, but frequently in engineering and biological
applications, the assumption of constant density results in considerable simplification
and very little error. '
1 2

EXAMPLE 3.1-1 Show that for any kind of flow pattern, the normal stresses are zero at fluid-solid boundaries,
for Newtonian fluids with constant density. This is an important result that we shall use
Normal Stresses at often.
Solid Surf aces for
Incompressible SOLUTION
Newtonian Fluids
We visualize the flow of a fluid near some solid surface, which may or may not be flat. The
flow may be quite general, with all three velocity components being functions of all three co-
ordinates and time. At some point P on the surface we erect a Cartesian coordinate system
with the origin at P. We now ask what the normal stress r is at P.
22
According to Table B.I or Eq. 1.2-6, T = —lyXdvJdz), because (V • v) = 0 for incompress-
ZZ
ible fluids. Then at point P on the surface of the solid
to,
r zz \ z=0 = - (3.1-6)
' dz
First we replaced the derivative dvjdz by using Eq. 3.1-3 with p constant. However, on the
solid surface at z = 0, the velocity v x is zero by the no-slip condition (see §2.1), and therefore
the derivative dv /dx on the surface must be zero. The same is true of dv /dy on the surface.
x
y
Therefore r, is zero. It is also true that т and r yy are zero at the surface because of the vanish-
2
хх
ing of the derivatives at z = 0. (Note: The vanishing of the normal stresses on solid surfaces
does not apply to polymeric fluids, which are viscoelastic. For compressible fluids, the nor-
mal stresses at solid surfaces are zero if the density is not changing with time, as is shown in
Problem 3C.2.)
§3.2 THE EQUATION OF MOTION

To get the equation of motion we write a momentum balance over the volume element
Ax Ay Az in Fig. 3.2-1 of the form
rate of rate of rate of external
increase = momentum - momentum + force on (3.2-1)
of momentum in out the fluid

1
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford (1987), p. 21, point out
that, for steady, isentropic flows, commonly encountered in aerodynamics, the incompressibility
assumption is valid when the fluid velocity is small compared to the velocity of sound (i.e., low Mach
number).
2
Equation 3.1-5 is the basis for Chapter 2 in G. K. Batchelor, An Introduction to Fluid Dynamics,
Cambridge University Press (1967), which is a lengthy discussion of the kinematical consequences of the
equation of continuity.