Onset of Convection in a Porous Box with Partly Conducting and Partly Penetrative Sidewalls

Abstract

The onset of thermal convection in a 2D porous box is investigated analytically. The lateral walls are partly heat conducting and partly penetrative. The top and bottom are impermeable and perfect heat conductors. The linear stability problem is solved only for the symmetric configuration of equal conditions at each sidewall. The problem is degenerate when the parameters of semi-conduction and semi-penetration coincide. The degenerate problem has one symmetric and one antisymmetric eigenfunctions, and the cell width varies with minimum cell width in the middle. Our primary model for the partly penetrative wall is a thin and highly permeable layer near a closed wall. We also study a secondary model of a partly penetrative wall, with a thin layer of small permeability near a hydrostatic reservoir.

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

“…Moreover, a similar thermal model of the sidewalls was formulated by Nygård and Tyvand [17] and Nygård and Tyvand [18].…”

“…The proof of this result, reported also by Sutton [11], is a straightforward consequence of Eq. (17). Table 4.…”

“…Further results relative to the onset of oscillatory convection can be found in Holzbecher [14,16]. Recently, Nygård and Tyvand [17] proposed an extension of the analysis by Nilsen and Storesletten [13] on modelling the sidewalls as partially permeable and partially conducting.…”

“…A common feature of the papers by Nilsen and Storesletten [13], Rees and Tyvand [15] and Nygård and Tyvand [17] is that they are based on a two-dimensional analysis, while three-dimensional modes may be dominant at the onset of convection, as proved by Beck [12], as well as by Storesletten and Tveitereid [8] in the case of a horizontal circular channel.…”

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

“…The Biot number is, in fact, the dimensionless parameter involved when third-kind temperature conditions are used to model the boundary heat transfer (see, for instance, Incropera et al 2011). We mention that Tyvand (2010, 2011), on studying the thermal instability in a porous box (Nygård and Tyvand 2010), and in a vertical porous cylinder (Nygård and Tyvand 2011), considered third-kind boundary conditions for the temperature field in the case of partially permeable boundary walls. Mojtabi and Rees (2011), as well as Rees and Mojtabi (2011), obtained third-kind boundary conditions for the temperature disturbances in the stability analysis of a saturated porous layer bounded by walls with a finite and non-negligible thermal conductance.…”