PHYSICS 312
Problem set 3:

A rod of length a is initially (t=0) at temperature T=0. It is put into convective contact with a fluid with temperature T₀. The temperatures at the ends (x=0) and (x=a) are kept at T=0, but towards the middle the temperature rises towards T₀. Assume that the temperature distribution satisfies the differential equation

$$\frac{\partial^2T}{\partial x^2}-\gamma^2(T-T_0)=\frac{1}{k}\frac{\partial T}{\partial t}$$

for t>0, with k and $\gamma$ constants.

a:
Find the steady state temperature distribution T_S(x).
b:
Find the differential equation satisfied by the transient.

u(x,t)=T(x,t)-T_S(x)

c:
What is the initial temperature distribution for the transient?
d:
Solve the transient problem formally, by assuming that the initial temperature u(x,0) has a Fourier sine series:

$$u(x,0)=\sum_{n=0}^\infty c_n\sin(\frac{n\pi x}{a})$$

(for this part of the problem you don't need to calculate the Fourier coefficients).
e:
Could you find the time dependent temperature distribution directly without subtracting the steady state?
f:
Plot the temperature distribution at time t=1 for $a=k=\gamma=T_0=1$. (For this part of the problem you need to compute the first few Fourier coefficients either with or without the steady state subtracted.) Return to title page.