Page 426 - Advanced engineering mathematics
P. 426
406 CHAPTER 12 Vector Integral Calculus
This is the flux of K∇u across , multiplied by the length of the time interval. But, the change
in temperature at (x, y, z) over t is approximately (∂u/∂t) t so the resulting heat loss in M is
⎛ ⎞
∂u
μρ dV t.
⎝ ⎠
∂t
M
Assuming no heat sources or sinks in M (which occur, for example, during chemical reactions
or radioactive decay), the change in heat energy in M over t must equal the heat exchange
across :
⎛ ⎞
∂u
(K∇u) · ndσ t = ⎝ μρ dV ⎠ t.
∂t
M
Therefore,
∂u
(K∇u) · ndσ = μρ dV.
∂t
M
Now use the divergence theorem to convert the surface integral to a triple integral:
(K∇u) · ndσ = ∇· (K∇u)dV.
M
Substitute this into the preceding equation to obtain
∂u
μρ −∇ · (K∇u) dV = 0.
∂t
M
Now is an arbitrary closed surface within the medium. If the integrand in the last equation
were, say, positive at some point P 0 , then it would be positive throughout some (perhaps very
small) sphere about P 0 , and we could choose as this sphere. But then the triple integral over M
of a positive quantity would be positive, not zero, a contradiction. The same conclusion follows
if this integrand were negative at some P 0 .
Vanishing of the last integral for every closed surface in the medium therefore forces the
integrand to be identically zero:
∂u
μρ −∇ · (K∇u) = 0.
∂t
This is the partial differential equation
∂u
μρ =∇ · (K∇u)
∂t
for the temperature function. This equation is called the heat equation.
We can expand
∂u ∂u ∂u
∇· (K∇u) =∇ · K i + K j + K k
∂x ∂y ∂z
∂ ∂u ∂ ∂u ∂ ∂u
= K + K + K
∂x ∂x ∂y ∂y ∂z ∂z
2
2
2
∂K ∂u ∂K ∂u ∂K ∂u ∂ u ∂ u ∂ u
= + + + K + +
∂x ∂x ∂y ∂y ∂z ∂z ∂x 2 ∂y 2 ∂z 2
2
=∇K ·∇u + K∇ u.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
October 14, 2010 14:53 THM/NEIL Page-406 27410_12_ch12_p367-424