Abstract

The linear and weakly nonlinear stability analyses of Darcy-Benard convection of a Newtonian fluid experiencing a uniform vertical cross-flow is investigated in the paper for various aspect ratios. Making use of the Maclaurin series representation, an expression for axial eigenfunctions is obtained with the radial eigenfunction being a Bessel function of first kind. These eigenfunctions are influenced by the Peclet number, Pe, the non-dimensional number that signifies the rate of vertical cross-flow. The modified-Vadasz-Lorenz model obtained in this paper has newly defined non-dimensional parameters that capture the influence of vertical cross-flow. From the linear stability analysis, it is found that the effect of introducing vertical cross-flow is to stabilize the system. Using a weakly nonlinear stability analysis, the closed-form expression of the Hopf-Rayleigh number as a function of Pe is obtained. Furthermore, the behavior of the modified-Vadasz-Lorenz model is analyzed using the largest Lyapunov exponent and the bifurcation diagram. This gives information about the intensity of chaos and occurrence of the periodic motion. We observe that the influence of vertical cross-flow is to increase the value of the Hopf-Rayleigh number and thereby to delay the onset of chaos. Furthermore, the appearance of the first periodic point is preponed due to the vertical cross-flow. As the rate of vertical cross-flow increases, the intensity of chaos decreases, thereby indicating that the effect of introducing vertical cross-flow is to suppress chaos.