Thermal instability in a horizontal elliptical porous duct with a conducting wall

In the present paper, a horizontal elliptical duct filled by a fluid saturated porous medium is considered. The boundary wall of the duct is assumed to be impermeable and subjected to an environment temperature thermally stratified in the vertical direction. Darcy's law and the Oberbeck-Boussinesq approximation are assumed. The stationary basic flow given by a zero velocity distribution and a linear vertical temperature distribution is determined analytically and its linear stability is studied numerically. The elliptic equations for the linear disturbances are written in terms of an eigenvalue problem and solved by employing a Galerkin finite element procedure, in order to determine the critical conditions for the onset of the convective instability. Comparisons are performed both for the special case of a circular duct and for the longitudinal rolls, with reference to a particular elliptical duct. The limiting case of a plane-parallel layer is discussed as well. The neutral stability curves, and in particular the first two branches of these curves, are obtained for different values of the aspect ratio of the duct.