Question

(b) Write the BVP that solves for the steady state temperature of an object in the shape of a quarter disk of radius 5. The flat sides of the disk are insulated and the round edge is kept at temperature 7 degrees Celsius. Use polar coordinates ρ for the radius and φ for the polar angle.

Answer 1

Diffusion equation is

K ot

where

$\kappa=\frac{K}{C}$ K is thermal conductivity and C is specific heat.

In cylindrical polar coordinates

$\bigtriangledown ^2T(\rho,\varphi ,t)=\frac{\partial^2 T}{\partial \rho^2}+\frac{1}{r}\frac{\partial T}{\partial \rho}+\frac{1}{r^2}\frac{\partial^2 T}{\partial \varphi^2}=\frac{1}{\kappa}\frac{\partial T}{\partial t}$

Boundary conditions are

flat sides insulated so rate of change of temperature with time is 0.

$\frac{\partial T(\rho,0,t)}{\partial t}=\frac{\partial T(\rho,\frac{\pi}{2}.t)}{\partial t}=0$

curved side is maintained at 7⁰

$T(5,\varphi,t)=7^o$