Separation of variables in the steady heat transfer and fluid dynamics in a cylindrical sector domain

In the present paper, an analytical solution of Poisson’s equation is presented. Reference is made to a two–dimensional r , φ domain and the separation of variables method is employed. Usually, the eigenvalue problems display a finite spectrum whenever the considered domain is finite, while a continuous spectrum arises when the domain is infinite. In the present paper an exception to the rule is presented. In fact, a finite domain is considered displaying a continuum eigenvalue spectrum. In detail, the conduction equation, as well as the momentum balance equation for steady parallel flows, is written as a Poisson equation. Then a change of variable is proposed in order to map the finite cylindrical sector domain into an infinite rectangular strip domain. Then, Poisson’s equation is solved for different boundary conditions. Finally, in the fluid dynamic case the analytical solution is compared with a numerical solution obtained by employing the finite element software package Comsol Multiphysics (© Comsol, AB). The comparison reveals an excellent agreement.