Page 95 - Bird R.B. Transport phenomena

80 Chapter 3 The Equations of Change for Isothermal Systems

Here we have made use of the definitions of the partial derivatives. Similar equations
can be developed for the y- and z-components of the momentum balance:

(3.2-5)

(3.2-6)

By using vector-tensor notation, these three equations can be written as follows:

j pv = -[V • ф], + pgi i = x,y,z (3.2-7)
t t
That is, by letting i be successively x, y, and z, Eqs. 3.2-4, 5, and 6 can be reproduced. The
quantities pv are the Cartesian components of the vector pv, which is the momentum per
{
unit volume at a point in the fluid. Similarly, the quantities pg are the components of the
x
vector pg, which is the external force per unit volume. The term —[V • ф], is the ith com-
ponent of the vector — [V • ф].
When the ith component of Eq. 3.2-7 is multiplied by the unit vector in the ith direc-
tion and the three components are added together vectorially, we get

ф] + pg (3.2-8)

which is the differential statement of the law of conservation of momentum. It is the
translation of Eq. 3.2-1 into mathematical symbols.
In Eq. 1.7-1 it was shown that the combined momentum flux tensor ф is the sum of
the convective momentum flux tensor p w and the molecular momentum flux tensor IT,
and that the latter can be written as the sum of /?5 and т. When we insert ф = w + p8 +
p
:
т into Eq. 3.2-8, we get the following equation of motion: 2
i" - -[V -pvv] - Vp - [V • T] + pg (3.2-9)

rate of rate of momentum rate of momentum addition external force
increase of addition by by molecular transport on fluid
momentum convection per unit volume per unit
per unit per unit volume
volume volume

In this equation Vp is a vector called the "gradient of (the scalar) p" sometimes written as
"grad p." The symbol [V • т] is a vector called the "divergence of (the tensor) т" and
[V • pvv] is a vector called the "divergence of (the dyadic product) pw."
In the next two sections we give some formal results that are based on the equation
of motion. The equations of change for mechanical energy and angular momentum are
not used for problem solving in this chapter, but will be referred to in Chapter 7. Begin-
ners are advised to skim these sections on first reading and to refer to them later as the
need arises.

2
This equation is attributed to A.-L. Cauchy, Ex. de math., 2,108-111 (1827). (Baron) Augustin-Louis
Cauchy (1789-1857) (pronounced "Koh-shee" with the accent on the second syllable), originally trained
as an engineer, made great contributions to theoretical physics and mathematics, including the calculus
of complex variables.