Hexagonal Cell Formation in Darcy–Bénard Convection with Viscous Dissipation and Form Drag

Abstract

Recent interest in the effects of viscous dissipation on convective flows in porous media has centred almost exclusively on forced convection flows. In this paper, we investigate the manner in which it affects the onset and early stages of convection in Darcy-Bénard convection. A weakly nonlinear theory is described briefly, and it is shown that hexagonal cells are preferred over rolls when the Rayleigh number is sufficiently close to 4π 2 . At higher Rayleigh numbers, two-dimensional rolls are preferred. When weak form drag is included, then subcritical convection eventually disappears as the Forchheimer parameter increases, yielding a highly novel situation wherein hexagonal convection arises supercritically. The range of stability of hexagons is found to increase.

The energy method analysis of the Darcy–Bénard problem with viscous dissipation

“…This study is focussed on the weakly nonlinear approach to the investigation of the convection patterns and, in particular, to the formation of hexagonal cells. We believe that the present paper provides both a validation and a consolidation of the conclusions drawn in [8], as well as in [9]. In fact, the analysis carried out here is not based on numerical results relative to the special two-dimensional case, but on an entirely analytical procedure relative to general three-dimensional disturbances.…”

“…These authors performed a two-dimensional numerical study where the existence of subcritical instability caused by the viscous dissipation effect was ruled out. Another recent numerical study regarding the nonlinear instability of the Darcy-Bénard system including the effect of viscous dissipation was carried out in [9]. This study is focussed on the weakly nonlinear approach to the investigation of the convection patterns and, in particular, to the formation of hexagonal cells.…”

Nonlinear Stability for Thermal Convection in a Brinkman Porous Material with Viscous Dissipation

“…2.2.2, Nield et al (2004), Nield and Simmons (2019), Sect. 3.1, and Rees and Magyari (2017). Since the viscous dissipation effects add some strongly nonlinear terms to the governing partial differential equations, an understanding of this is vital to fully explore thermal convection from a nonlinear point of view.…”