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L2

;

Problem a: Sketch this temperature distribution and indicate the role of the constants

T0 and L.

We will assume that the temperature pro le maintains a Gaussian shape at later times

but that the peak value and the width may change, i.e. we will consider a solution of the

following form:

T(x t) = F(t) exp H(t)x2 : (8.30)

;

At this point the function F(t) and H(t) are not yet known.

Problem b: Show that these functions satisfy the initial conditions:

H(0) = 1=L2 :

F(0) = T0 (8.31)

Problem c: Show that for the special solution (8.30) the heat equation reduces to:

@F x2 F @H = 4F H 2 x2 2F H : (8.32)

@t @t

; ;

It is possible to derive equations for the time evolution of F and H by recognizing that

(8.32) can only be satis ed for all values of x when all terms proportional to x2 balance

and when the terms independent of x balance.

Problem d: Use this to show that F(t) and H(t) satisfy the following di erential equa-

tions:

@F = FH (8.33)

@t ;2

@H = H2 : (8.34)

@t ;4

It is easiest to solve the last equation rst because it contains only H(t) whereas (8.33)

contains both F(t) and H(t).

8.4. THE HEAT EQUATION 81

Problem e: Solve (8.34) with the initial condition (8.31) and show that:

H(t) = 4 t 1 L2 : (8.35)

+

Problem f: Solve (8.33) with the initial condition (8.31) and show that:

L

F(t) = T0 : (8.36)

4 t + L2

p

Inserting these solutions in expression (8.30) gives the temperature eld at all times t 0:

!

x2

L

T(x t) = T0 4 t + L2 :

exp (8.37)

4 t + L2

p ;

Problem g: Sketch the temperature for several later times and describe using the solution

(8.37) how the temperature pro le changes as time progresses.

R1

The total heat Qtotal (t) at time t is given by Qtotal (t) = C ;1 T(x t)dx, where C is

the heat capacity.

Problem h: Show that the total heat does1not change with time for the solution (8.37).

; R 1 exp ; 2 du

R

Hint: reduce any integral of the form ;1 exp x2 dx to the integral ;1 ;

R1

; ;u

with a suitable change of variables. You don't even have to use that ;1 exp 2 du =

;u

.

p

Problem i: Show that for any solution of the heat equation (8.28) where the heat ux

t) = 0) the total heat Qtotal (t) is constant

vanishes at the endpoints ( @x T(x = 1

in time.

Problem j: What happens to the special solution (8.37) when the temperature eld

evolves backward in time? Consider in particular times earlier than t = 2 =4 .

;L

Problem k: The peak value of the temperature eld (8.37) decays as 1= 4 t + L2 with

p

time. Do you expect that in more dimensions this decay is more rapid or more slowly

with time? Don't do any calculations but use your common sense only!

Up to this point, we considered the conduction of heat in a medium without ow

(v = 0). In many applications the ow in the medium plays a crucial role in redistributing

heat. This is particular the case when heat is the source of convective motions, as for

example in the earth's mantle, the atmosphere and the central heating system in buildings.

As an example of the role of advection we will consider the cooling model of the oceanic

lithosphere proposed by Parsons and Sclater 45].

At the mid-oceanic ridges lithospheric material with thickness H is produced. At the

ridge the temperature of this material is essentially the temperature Tm of mantle material.

As shown in gure (8.2) this implies that at x = 0 and at depth z = H the temperature

is given by the mantle temperature: T(x = 0 z) = T(x z = H) = Tm . We assume that

the velocity with which the plate moves away from the ridge is constant:

v =U^ :

x (8.38)

CHAPTER 8. CONSERVATION LAWS

82

x=0

T=0

000

111

000

111

111

000

000T = T

111

000

111

000

111

U H U

m

000

111

000

111

000

111 T = Tm

111

000

000

111

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