Let’s solve a heat conduction problem. A hemisphere 0 ≤ r ≤ b, 0 ≤ μ ≤ 1 is initially at temperature T = F (r, μ). For times t > 0, the boundary at the spherical surface r = b is maintained at zero temperature, and at the base μ = 0 is perfectly insulated. Obtain an expression for the temperature distribution T(r,μ,t) for times t > 0.

## Heat Conduction problem solving

I will solve the below problem:

A hemisphere 0 ≤ r ≤ b, 0 ≤ μ ≤ 1 is initially at temperature T = F (r, μ). For times t > 0, the boundary at the spherical surface r = b is maintained at zero temperature, and at the base μ = 0 is perfectly insulated. Obtain an expression for the temperature distribution T(r,μ,t) for times t > 0.

(sol)

$$\frac{\partial^2 T}{\partial r^2} + \frac{2}{r} \frac{\partial T}{\partial r} + \frac{1}{r^2} \frac{\partial}{\partial \mu} \left[ (1-\mu^2) \frac{\partial T}{\partial \mu} \right]= \frac{1}{\alpha} \frac{ \partial T}{\partial t}$$

BC1: $T(r \to 0 )$ is finite, BC2: $T(r=b)=0$, BC3: $T(\mu \to +1) $ is finite, BC4: $\frac{\partial T}{\partial \mu} = 0$ at $\mu =0$, IC:$T(t=0) = F(r,\mu)$

Let $V = r^{1/2} T$.

Then,

$$ \frac{\partial^2 V}{\partial r^2} + \frac{1}{r} \frac{\partial V}{\partial r} – \frac{1}{4} \frac{ V}{r^2} + \frac{1}{r^2} \frac{\partial } {\partial \mu} \left[ (1-\mu^2) \frac{\partial V}{\partial \mu} \right] = \frac{1}{\alpha} \frac{\partial V}{\partial t}$$

BC1: $V(r\to 0)$ is finite, BC2: $V(r=b)=0$, BC3: $V(\mu \to +1 )$ is finite, BC4: $\frac{\partial V}{\partial \mu} =0$ at $\mu=0$, IC $V(t=0) = r^{1/2} F(r,\mu)$.

Let $V(r,\mu,t) = R(r) M ( \mu) \Gamma(t)$.

$$ \frac{1}{R} \left(\frac{\partial^2 R}{\partial r^2} + \frac{1}{r} \frac{\partial R}{\partial r} – \frac{1}{4} \frac{ R}{r^2} \right)+ \frac{1}{r^2 M} \frac{\partial } {\partial \mu} \left[ (1-\mu^2) \frac{\partial M}{\partial \mu} \right] = \frac{1}{\alpha} \frac{\partial \Gamma}{\partial t} = – \lambda^2$$

$$\Gamma(t) = C_1 e^{-\alpha \lambda^2 t}$$

$$ \frac{r^2}{R} \left(\frac{\partial^2 R}{\partial r^2} + \frac{1}{r} \frac{\partial R}{\partial r} – \frac{1}{4} \frac{ R}{r^2} \right)+\lambda^2 r^2= – \frac{1}{ M} \frac{\partial } {\partial \mu} \left[ (1-\mu^2) \frac{\partial M}{\partial \mu} \right] =n(n+1)$$

$$M(\mu) = C_2 P_n(\mu) + C_3 Q_n(\mu)$$

From BC3, $C_3=0$. From BC4, $n=0,2,4,…$.

$$R(r) = C_4 J_{n+1/2}(\lambda r) + C_5 Y_{n+1/2} (\lambda r)$$

From BC1, $C_5=0$. From BC2 $J_{n+1/2}(\lambda b) = 0$, $\lambda_{nm}$ for $m=1,2,3…$.

Since $J_{n+1/2}(0)=0$, $\lambda_{nm} \neq 0$.

$$V(r,\mu,t) = \sum_{n=0, even}^\infty \sum_{m=1}^\infty C_{nm} J_{n+1/2} (\lambda_{nm} r) P_n (\mu) e^{-\alpha \lambda_{nm}^2 t}$$

From IC, $$C_{nm} = \frac{\int_0^1\int_0^b r^{3/2} F(r,\mu) J_{n+1/2} (\lambda_{nm}r) P_n (\mu) dr d\mu}{\int_0^b J_{n+1/2}^2(\lambda_{nm}r)dr \int_0^1 [P_n(\mu)]^2 d\mu}$$

$$T(r,\mu,t) = V(r,\mu,t) / r^{1/2}$$

Reference:

Problem 5.14 in Hahn, D. W., & Özisik, M. N. (2012). Heat conduction. John Wiley & Sons.

HW problem in Advanced heat transfer lecture by Prof. Seol (설재훈 교수님) in GIST, Fall semester 2023

[Heat Conduction Problem Solving 6.3] In a one-dimensional infinite medium, −∞