Study of chaos in the Darcy–Bénard convection problem with Robin boundary condition on the upper surface

Abstract

Possibility of chaos is studied in Darcy–Bénard 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–Bénard convection, CDBC) and Neumann boundary condition (Barletta–Darcy–Bénard convection, BDBC). It is found that the cell size at onset is bigger in the case of BDBC compared to the generalized-Darcy–Bénard 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.