Pattern selection for Darcy-Bénard convection with local thermal nonequilibrium

•This short paper presents a weakly nonlinear analysis of the onset of Darcy-Bénard convection in the presence of local thermal nonequilibrium. Attention is focussed on the cross-roll instability.•The aim is to determine whether the ensuing convection pattern immediately after onset is two-dimensional or three-dimensional. We find that it is two-dimensional in all cases.•We also find that the cross-roll instability does not always arise as a potential instability mechanism for those cases where the convecting roll has a detuned wavenumber. A weakly nonlinear analysis is performed on the classical Darcy-Bénard problem to determine the effects of local thermal nonequilibrium on the planform of convection immediately post-onset. It is found that two-dimensional rolls are always favoured. Although disturbances which are perpendicular to the roll whose stability is being assessed usually form the most dangerous mode of instability, it is also found that there are regions of parameter space where the cross-roll instability becomes inoperative as an instability mechanism.