Two-dimensional bifurcation phenomena in thermal convection in horizontal, concentric annuli containing saturated porous media

Abstract

A saturated porous medium confined between two horizontal cylinders is considered. As a result of a temperature difference between the cylinders, thermal convection is induced in the medium. The flow structure is investigated in a parameter space (R, Ra) whereRis the radii ratio andRais the Darcy-Rayleigh number. In particular, the cases ofR= 2, 2½, 21/4and 2½are considered. The fluid motion is described by the two-dimensional Darcy-Oberbeck-Boussinesq's (DOB) equations, which we solve using regular perturbation expansion. Terms up toO(Ra60) are calculated to obtain a series presentation for the Nusselt number Nu in the form\[ Nu(Ra^2) = \sum_{s=0}^{30} N_sRa^{2s}. \]This series has a limited range of utility due to singularities of the functionNu(Ra). The singularities lie both on and off the real axis in the complexRaplane. ForR= 2, the nearest singularity lies off the real axis, has no physical significance, and unnecessarily limits the range of utility of the aforementioned series. ForR= 2½, 2¼and 21/8, the singularity nearest to the origin is real and indicates that the functionNu(Ra) is no longer unique beyond the singular point.Depending on the radii ratio, the loss of uniqueness may occur as a result of either (perfect) bifurcations or the appearance of isolated solutions (imperfect bifurcations). The structure of the multiple solutions is investigated by solving the DOB equations numerically. The nonlinear partial differential equations are converted into a truncated set of ordinary differential equations via projection. The steady-state problem is solved using Newton's technique. At each step the determinant of the Jacobian is evaluated. Bifurcation points are identified with singularities of the Jacobian. Linear stability analysis is used to determine the stability of various solution branches. The results we obtained from solving the DOB equations using perturbation expansion are compared with those we obtained from solving the nonlinear partial differential equations numerically and are found to agree well.