Onset of convective rolls in a circular porous duct with external heat transfer to a thermally stratified environment

A horizontal circular duct filled with a fluid saturated porous medium is studied. The external wall is assumed to exchange heat with an external environment thermally stratified in the vertical direction. The external heat transfer is modeled through a third kind boundary condition, and a Biot number associated with the external heat transfer coefficient is defined. The linear stability of the basic state where the velocity field is zero is studied numerically. The condition of neutral stability is determined, by solving the system of elliptic governing equations for the disturbances through a Galerkin finite-element method. The neutral stability curves, together with the critical values of the wave number and of the Rayleigh number, are obtained for different values of the Biot number. The case of a duct with a finite axial length, having impermeable and thermally insulated axial boundaries, is considered. On increasing the length-to-radius aspect ratio, the transition from a two-dimensional to a three-dimensional pattern of instability at the onset of convection is described.