On the extinction of hexagonal cells in Darcy–Bénard convection

A weakly nonlinear analysis is used to investigate the combined effects of viscous dissipation and inclination on Darcy–Bénard convection. For a horizontal layer and in the absence of viscous dissipation then, as the Darcy–Rayleigh number is increased, the first pattern to arise takes the form of two-dimensional rolls. If a weak viscous dissipation effect is included for a horizontal layer, this causes the preferred pattern near onset to change to a hexagonal planform. Here, we consider the additional effect of introducing an inclination to the layer. This causes an increase in the critical Darcy–Rayleigh number for each of the constituent rolls of the hexagon; this size of this increase depends on the orientation of the roll. In consequence, the hexagons no longer consist of three rolls of equal amplitudes and this modifies the stability criteria substantially. A detailed analysis is given for the range of existence and stability of both longitudinal rolls and what are termed pseudo-hexagons. It is shown that there is a maximum inclination for which a stable three-dimensional convection pattern is possible.