Consider a short cylinder of radius \(r_{o}\) and height \(H\) in which heat is generated at a constant rate of \(\dot{e}_{\text {gen. }}\). Heat is lost from the cylindrical surface at \(r=r_{o}\) by convection to the surrounding medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). The bottom surface of the cylinder at \(z=0\) is insulated, while the top surface at \(z=H\) is subjected to uniform heat flux \(\dot{q}_{H}\). Assuming constant thermal conductivity and steady two-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary conditions) of this heat conduction problem. Do not solve.

Based on the given information, the mathematical formulation for the heat conduction problem in a short cylinder can be derived as the following: 1. Governing equation in cylindrical coordinates (steady state, 2D, constant thermal conductivity, and heat generation): \(\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right) + \frac{\partial^2 T}{\partial z^2} = \frac{\dot{e}_{\text{gen}}}{k}\) 2. Boundary conditions: a. Heat loss from the cylindrical surface: \(\left.-k\frac{\partial T}{\partial r}\right|_{r=r_o} = h(T(r_o, z) - T_\infty)\) b. Insulated bottom surface: \(\left.\frac{\partial T}{\partial z}\right|_{z=0} = 0\) c. Uniform heat flux from the top surface: \(\left.-k\frac{\partial T}{\partial z}\right|_{z=H} = \dot{q}_H\) d. Symmetry condition at the centerline: \(\left.\frac{\partial T}{\partial r}\right|_{r=0} = 0\) These equations and boundary conditions together define the mathematical model for the heat transfer problem in the short cylinder.