Page 99 - Bird R.B. Transport phenomena

84 Chapter 3 The Equations of Change for Isothermal Systems

Table 3.5-1 The Equations of Change for Isothermal Systems in the D/Df-Form"
Note: At the left are given the equation numbers for the d/'dt forms.

(3.1-4) - -p(V •v) (A)
Dt
Dv
(3.2-9) = -V P -[V • T] + pg (B)
D
(3.3-1) P GV) = - ( v Vp) - (v • [V • T])+ p(v •g) (C)
Dt
D +
)
+
(3.4-1) p [rXv] 1 • (r X p8 ] - [v - (r X т ] + [r X pg] (D)"
nt
a Equations (A) through (C) are obtained from Eqs. 3.1-4, 3.2-9, and 3.3-1 with no
assumptions. Equation (D) is written for symmetrical т only.

The quantity in the second parentheses in the second line is zero according to the equa-
tion of continuity. Consequently Eq. 3.5-3 can be written in vector form as
/ p j | (3.5-4)

Similarly, for any vector function f{x, y, z, t),

| pg (3.5-5)

These equations can be used to rewrite the equations of change given in §§3.1 to 3.4 in
terms of the substantial derivative as shown in Table 3.5-1.
Equation A in Table 3.5-1 tells how the density is decreasing or increasing as one
moves along with the fluid, because of the compression [(V • v) < 0] or expansion of the
fluid [(V • v) > 0]. Equation В can be interpreted as (mass) X (acceleration) = the sum of
the pressure forces, viscous forces, and the external force. In other words, Eq. 3.2-9 is
equivalent to Newton's second law of motion applied to a small blob of fluid whose en-
velope moves locally with the fluid velocity v (see Problem 3D.1).
We now discuss briefly the three most common simplifications of the equation of
motion. 1
(i) For constant p and JJL, insertion of the Newtonian expression for т from Eq. 1.2-7
into the equation of motion leads to the very famous Navier-Stokes equation, first de-
veloped from molecular arguments by Navier and from continuum arguments by
Stokes: 2

2
p ^ v = - Vp + /*Vv + pg or p ^ v = - W> + /xVv (3.5-6, 7)
2
-
In the second form we have used the "modified pressure" 2P = p + pgh introduced in
Chapter 2, where h is the elevation in the gravitational field and gh is the gravitational

1 For discussions of the history of these and other famous fluid dynamics relations, see H. Rouse
and S. Ince, History of Hydraulics, Iowa Institute of Hydraulics, Iowa City (1959).
2
L. M. H. Navier, Memoires de VAcademie Royale des Sciences, 6, 389^40 (1827); G. G. Stokes, Proc.
Cambridge Phil Soc, 8, 287-319 (1845). The name Navier is pronounced "Nah-vyay."