Page 384 - Advanced engineering mathematics

364 CHAPTER 11 Vector Differential Calculus

11.5.1 A Physical Interpretation of Divergence
Suppose F(x, y, z,t) is the velocity of a fluid at point (x, y, z) and time t. Time plays no role in
computing divergence, but is included here because a flow may depend on time. We will show
that the divergence of F measures the outward flow of the fluid from any point.
Imagine a small rectangular box in the fluid, as in Figure 11.10. First look at the front and
back faces II and I, respectively. The flux of the flow out of this box across II is the normal
component of the velocity (dot product of F with i) multiplied by the area of this face:
flux outward across face II = F(x + x, y, z,t) · i y z
= f (x + x, y, z,t) y z.
On face I the unit outer normal is −i, so the flux outward across this face is
− f (x, y, z,t) y z. The total outward flux across faces II and I is therefore
[ f (x + x, y, z,t) − f (x, y, z,t)] y z.
A similar calculation holds for the pairs of other opposite sides. The total flux of fluid flowing
out of the box across its faces is
total flux =[ f (x + x, y, z,t) − f (x, y, z,t)] y z
+[g(x, y + y, z,t) − g(x, y, z,t)] x z
+[h(x, y, z + z,t) − h(x, y, z,t)] x y.

The total flux per unit volume out of the box is obtained by dividing this quantity by x y z,
obtaining
f (x + x, y, z,t) − f (x, y, z,t)
flux per unit volume =
x
g(x, y + y, z,t) − g(x, y, z,t)
+
y
h(x, y, z + z,t) − h(x, y, z,t)
+ .
z
In the limit as ( x, y, t) → (0,0,0), this sum approaches the divergence of F(x, y, z,t).

Back face I

(x, y, z) Δz

z Δx

Δy
Front face II
y

x
FIGURE 11.10 Interpretation of divergence.

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October 15, 2010 16:11 THM/NEIL Page-364 27410_11_ch11_p343-366