A three-dimensional study of the onset of convection in a horizontal, rectangular porous channel heated from below

The onset of convection is studied in a rectangular channel filled with a fluid saturated porous medium, bounded above and below by impermeable isothermal walls at unequal temperatures and laterally by partially conducting walls. A three-dimensional linear stability analysis is carried out under the assumption of an infinite longitudinal channel length. Then, this assumption is relaxed in order to determine the threshold length for the three-dimensional convection to be the preferred mode at onset. Sensible parameters influencing the conditions for the instability are the aspect ratio of the transverse cross-section and the Biot number associated with the sidewall heat transfer to the external environment. The neutral stability is investigated by expressing the Darcy-Rayleigh number as a function of the longitudinal wave number, for assigned values of the transverse aspect ratio and of the Biot number