Project
Problem 1
Consider an outdoor swimming pool (one that is heated by the sun). The temperature of
the water near the surface fluctuates with the daily variation of the temperature in the air.
For the beginning assume a rectangular pool (3m × 10m) whose lateral and bottom sides
are insulated. At the top temperature changes according with the surrounding temperature
Ts = 28 + 7 cos(π(t − 12)/12).
Temperature is measured in C. Thermal diffusivity of the water is 1.4e − 3cm2 /sec (assume
independent of the temperature).
R
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1) Calculate and plot the average temperature of the water ( |D|
T (x, t)dx where D is
D
the whole domain) as a function of time. When does the pool have the highest temperature?
2) Discuss how the shape of the pool (assuming it is still rectangular) affects the average
temperature (assume the total volume of the water is the same).
Problem 2
Consider a unit square conductor with conductivity 1 and an inclusion with volume
fraction 0.1 that is not good conductor (take conductivity to be 0.02). Mathematically this
problem can be formulated using Ohm’s law (J = σ∇φ, where J is the flux and φ is the
potential) and the fact that ∇ · J = 0.
∇ · σ(x)∇φ = 0,
(1)
where σ(x) = 0.02 if x belongs to the inclusion, and 1 otherwise. The important quantity of
this conductor is its effective (overall) conductivity, σ ∗ , defined as
J∗ = σ ∗ ∇φ∗
where both J∗ and φ∗ are volume averages of J and φ respectively,
Z
Z
1
1
∗
∗
J(x, y)dxdy, φ =
φ(x, y)dxdy.
J =
|D| D
|D| D
Here φ is the solution (1) with boundary conditions φ = 1 at x = 0 and φ = 0 at x = 1, and
no flow on the rest of the boundary.
a) As a first step assume that the inclusion has a circular form and place it in the middle
of the square. Calculate the effective conductivity of the material both in x and y directions.
b) Move the circular inclusion to a couple of a different location and calculate the effective
conductivity. Compare the results with part a).
c) How do you think volume fraction affects the effective conductivity?
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