Abstract

Possibility of chaos is studied in Darcy-B & eacute;nard convection using the Dirichlet and the Robin boundary condition at the lower and upper boundaries, respectively. Comparison is made with the results of Dirichlet (classical-Darcy-B & eacute;nard convection, CDBC) and Neumann boundary condition (Barletta-Darcy-B & eacute;nard convection, BDBC). It is found that the cell size at onset is bigger in the case of BDBC compared to the generalized-Darcy-B & eacute;nard convection (GDBC) and much bigger compared to CDBC. The critical-Darcy-Rayleigh number of BDBC is found to be the least and that of CDBC is the largest. Nonlinear-stability-analysis is performed leading to the scaled-generalized-Vadasz-Lorenz model (SGVLM). In deriving this model, help is sought from a local-nonlinear-stability-analysis that yields the form of the convective-mode. The SGVLM is shown to be dissipative and conservative, with its bounded solution trapped within an ellipsoid. Onset of chaos and its characteristics are studied using the Hopf-Rayleigh-number, the Lorenz-butterfly-diagram, and the plot of the amplitude of the convective-mode vs the control-parameter, R, which is the eigenvalue. Chaos sets in earlier in CDBC and much later in BDBC when compared to that in GDBC. Beyond the onset of chaos is seen a sequence of chaotic and periodic motions, with the latter sometimes being present for an extended period.