Computer Algebra Recipes

210 CHAPTER 5. LINEAR PDE MODELS. PART 1
35
30
25
20
0 0.1 0.2 0.3 0.4 0.5
x
Figure 5.1: Initial temperature pro¯le inside the adobe wall.
On running the animation, the temperature curve will oscillate up and down
as the externally imposed temperature on the surface varies during the day.
However, the temperature variation drops to zero within a penetration depth
of less than 0:4 m or 40 cm (16 in). Well within the surface, the temperature
is in thermal equilibrium with the mean exterior temperature. Since the adobe
walls of many of the historic buildings of Arizona and New Mexico are substantially
thicker than 16 inches, the interior temperature of these buildings is quite
insensitive to external temperature variations.
PROBLEMS:
Problem 5-1: A house of wood
In the text recipe, make the following modi¯cations:
(a) The region x>0 is composed of solid wood for which K =0:15 W/(m¢K),
½ = 700 kg/m3 ,andC = 1800 J/(kg¢K).
(b) The mean daily temperature is 30 ± Candtheamplitudeofthetemperature
variation is 20 ± .
Run the animation and determine the approximate depth in centimeters at
which the temperature variation is essentially zero.
Problem 5-2: Pulsating sphere
The surface of a sphere of radius r = a, surrounded by an ideal compressible
°uid, pulsates radially with frequency !. The radial velocity of the surface is
given by V = U cos(!t). It is stated in an advanced calculus text that the
steady-state °uid velocity at an arbitrary point r>ais of the form
V =
Ua2 (c2 + a2 ! 2 ) r2 £ ¤ 2 2
(c +ar! )cos(μ)+c!(r ¡ a)sin(μ) ;