Onset of convection in a vertical porous cylinder with a permeable and conducting side boundary

The onset of natural convection in a vertical porous cylinder saturated by a fluid is studied. The lateral confinement of the porous cylinder is due to an external porous medium having a permeability much smaller than that of the cylinder. Thus, the vertical side boundary of the cylinder is permeable and constrained by given pressure and temperature distributions. The lower and upper plane boundaries of the cylinder are impermeable walls. The lower wall is subject to a uniform heat flux, while the upper wall has a uniform temperature. The basic motionless state displays a uniform and vertical temperature gradient oriented downward. The linear stability analysis is carried out by using an analytical dispersion relation. The allowed modes of perturbation are determined as solutions of the Helmholtz-Dirichlet problem. First, the natural convection problem is formulated for a circular cylinder. Then, the investigation is generalised to an arbitrary cross-sectional shape of the cylinder. The sample case of an elliptical cylinder is studied in detail, by adopting an analytical solution based on Mathieu functions.