Onset of Thermal Convection in a Vertical Porous Cylinder with a Partly Conducting and Partly Penetrative Cylinder Wall

The onset of thermal convection in a vertical porous cylinder in three dimensions is investigated analytically. Top and bottom of the cylinder are set to be perfectly heat conducting and impermeable, and is uniformly heated from below. The convection problem is solved for a cylinder wall that is partly conducting and partly penetrative. The expressions for semi-conduction and semi-penetration are based on a porous medium separated from its surroundings by a thin wall. The eigenvalue problem is split into two Helmholtz equations, and the results are expressed by Bessel functions in the radial direction. Comparisons are made with existing solutions for the limit cases of a closed cylinder wall that is either conducting or insulating. Two different models are compared for the kinematic limit condition of an open boundary.