Consider a circular disc of radius a and thickness $2t.$ Both the top and bottom surfaces of the disc are exposed to a fluid at temperature $T_{}$ with a convective heat transfer coefficient $h$ while a constant heat flux $q\frac{W}{m^2}$ is applied all around the sides. Show that at steady state the temperature distribution $T(r)$ within the disc is given by:

$T(r)=T_{}+\frac{q}{k}\frac{I_0(r)}{I_1(a)}$

where $=\sqrt[\phantom{2}]{\frac{h}{kt}},k$ is the thermal conductivity of disc material, $I_0$ and $I_1$ are modified Bessel functions of the first kind of order $0$ and $1,$ respectively. Further show that at steady state the rate of heat entered from the sides is balanced by the rate of heat dissipation through convection from the top and bottom surfaces.