The core principle governing unsteady heat conduction in cylindrical coordinates is that variable surface area prevents similarity solutions for finite geometries due to a radial dependence term ($1/r$) in the differential operator, necessitating separation of variables instead. Consequently, spatial eigenfunctions are identified as Bessel functions rather than sinusoidal functions, and physically admissible solutions at the central axis require derivative symmetry conditions ($\partial T/\partial r = 0$). This framework establishes that discrete eigenvalues correspond to zeros of Bessel functions $J_0(\beta_n)$, defining a generalized Sturm-Liouville problem distinct from Cartesian systems where boundary surfaces are planar.
The core principle governing unsteady heat conduction in cylindrical coordinates is that variable surface area prevents similarity solutions for finite geometries due to a radial dependence term ($1/…