Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: GENERAL AND PARTICULAR SOLUTIONS

An infrared laser delivers a pulse of (heat) energyEto a pointPon a large insulated

sheet of thicknessb, thermal conductivityk, specific heatsand densityρ. The sheet is

initially at a uniform temperature. Ifu(r, t)is the excess temperature a timetlater, at a

point that is a distancer(b)fromP, then show that a suitable expression foruis

u(r, t)=

α

t

exp

(

−

r^2

2 βt

)

, (20.37)

whereαandβare constants. (Note that we userinstead ofρto denote the radial coordinate

in plane polars so as to avoid confusion with the density.)

Further,(i)show thatβ=2k/(sρ);(ii)demonstrate that the excess heat energy in the

sheet is independent oft, and hence evaluateα; and(iii)prove that the total heat flow past

any circle of radiusrisE.

The equation to be solved is the heat diffusion equation

k∇^2 u(r,t)=sρ

∂u(r,t)

∂t

.

Since we only require the solution forrbwe can treat the problem as two-dimensional
with obvious circular symmetry. Thus only ther-derivative term in the expression for∇^2 u
is non-zero, giving

k

r

∂

∂r

(

r

∂u

∂r

)

=sρ

∂u

∂t

, (20.38)

where nowu(r,t)=u(r, t).

(i) Substituting the givenexpression (20.37) into (20.38) we obtain

2 kα

βt^2

(

r^2

2 βt

− 1

)

exp

(

−

r^2

2 βt

)

=

sρα

t^2

(

r^2

2 βt

− 1

)

exp

(

−

r^2

2 βt

)

,

from which we find that (20.37) is a solution, providedβ=2k/(sρ).

(ii) The excess heat in the system at any timetis

bρs

∫∞

0

u(r, t)2πr dr=2πbρsα

∫∞

0

r

t

exp

(

−

r^2

2 βt

)

dr

=2πbρsαβ.

The excess heat is therefore independent oftand so must be equal to the total heat

inputE, implying that

α=

E

2 πbρsβ

=

E

4 πbk

.

(iii) The total heat flow past a circle of radiusris

− 2 πrbk

∫∞

0

∂u(r, t)

∂r

dt=− 2 πrbk

∫∞

0

E

4 πbkt

(

−r

βt

)

exp

(

−

r^2

2 βt

)

dt

=E

[

exp

(

−

r^2

2 βt

)]∞

0

=E for allr.

As we would expect, all the heat energyEdeposited by the laser will eventually flow past
a circle of any given radiusr.