Onset of Convection in a Porous Rectangular Channel with External Heat Transfer to Upper and Lower Fluid Environments

Abstract

The conditions for the onset of convection in a horizontal rectangular channel filled with a fluid saturated porous medium are studied. The vertical sidewalls are assumed to be impermeable and adiabatic. The horizontal upper and lower boundary walls are considered as impermeable and subject to external heat transfer, modelled through a third-kind boundary condition on the temperature field. The external fluid environments above and below the channel, kept at different temperatures, provide the heating-from-below mechanism which may lead to the onset of the thermal instability in the porous medium. The linear response of the fluid saturated porous channel, in a basic motionless state, is tested with respect to threedimensional normal mode disturbances of the temperature field and of the pressure field. The linearised disturbance equations are solved analytically leading to an implicit-form expression of the neutral stability condition, formulated as a functional relationship between the Darcy-Rayleigh number and the continuous longitudinal wave number of the normal modes, for any assigned aspect ratio of the cross-section and for any given Biot number. The analysis of the neutral stability is carried out. The analysis is extended to the case of a channel with a finite length in the longitudinal direction, and with adiabatic and impermeable capped ends.

Effect of a finite external heat transfer coefficient on the Darcy-Bénard instability in a vertical porous cylinder

“…(16) and (26), defines implicitly the functional relationship 21 so that we refer the reader to this previous study for any further detail. On the other hand, the geometry of the vertical sidewall plays an essential role when the allowed values of λ have to be determined.…”

“…The values of R c have been computed and reported in a table contained in the paper by Barletta and Storesletten. 21 Similarly, in Fig. 3, the horizontal dotted lines correspond to the critical values of R for each assigned Biot number.…”

“…The possibility that the heating from below, or the cooling from above, is caused by a forced convection flow over the horizontal bounding wall has been regarded by Kubitschek and Weidman 19,20 as well as by Barletta and Storesletten. 21 These authors modelled the heating/cooling processes over the horizontal boundaries by means of third-kind boundary conditions, depending on the Biot number, for the temperature field. In particular, the study carried out by Kubitschek and Weidman 20 refers to a vertical porous cylinder, where the lower boundary is subject to a third-kind temperature condition, while the upper boundary is considered as isothermal (Dirichlet temperature condition).…”

“…The same choice was made by Kubitschek and Weidman 19 as well as by Barletta and Storesletten. 21 Thus, the approach followed in our present paper will be to consider the most general time-dependent and three-dimensional normal modes, and then proving that only the stationary modes are allowed at neutral stability. The normal modes of perturbation that are most dangerous at the onset of convection will be determined on varying the aspect ratio of the vertical cylinder, viz., the radius-to-height ratio.…”

The impact of imperfect heat transfer on the convective instability of a thermal boundary layer in a porous media

“…Nygård & Tyvand 2010) or horizontal walls (e.g. Barletta & Storesletten 2012). Barletta, Tyvand & Nygård (2015) took this further and applied a Robin boundary condition to both the thermal and vertical velocity perturbations on the upper and lower boundaries.…”