TY - JOUR
T1 - A bifurcation study of chemically driven convection in a porous medium
AU - Nandakumar, K.
AU - Weinitschke, H. J.
N1 - Funding Information:
Acknowledgements-Continuing support from NSERC to KN and from Deutsche Forschungsgemeinschaft to HJW as well as a partial support from the Central Research Fund of the University of Alberta are gratefully acknowledged.
PY - 1992
Y1 - 1992
N2 - We present a bifurcation study of the chemically driven convection problem in a porous medium. The effect of nonuniform heat generation on the convection is of interest in chemical reactor theory as found in the recent works of Farr and Viljoen . We focus primarily on the effects of weak convection (driven by buoyancy force) on the bifurcation structure of the basic reaction-convection equations. The parameter that governs the magnitude of the convection is the Rayleigh number, Ra. Perturbation methods for both regular and singular solutions are developed. Such methods are found to be adequate for Ra ≈ 15. The full power of the numerical methods is required to uncover the more complicated bifurcation structure in certain regions of the parameter space. Arc-length continuation is used to construct the solution branches and an extended system formulation is used to precisely locate the limit and symmetry-breaking bifurcation points. The bifurcation structure characteristic of the convection problem, studied earlier by Weinitschke., is observed even when the heat generation is nonuniform. For nonzero Ra, the equations have reflection symmetry only about y-axis. Even this symmetry is spontaneously broken at specific values of ε, The Frank-Kamenetskii parameter, and asymmetric solutions emerging from such singular points occur in pairs. In addition to the primary solution branches, which start at low values of ε and possess the expected symmetries, there are also isolated solution branches with symmetric flow structures when convection is strong relative to the reaction. In the limit of no convection (Ra → O) the classical reaction (or combustion) problem governed by the partial differential equation Δθ + εe(θ/1+μθ) = 0 is recovered. Here, μ is the Arrhenius factor. In this limit, the equation has reflection symmetry about both the x- and y-axes. For the case μ = O, previously computed solutions that conform to the expected symmetries are confirmed. In addition, certain hitherto unknown solutions are found. These additional solutions do not conform to the expected symmetries based on the properties of the governing equations. Hence, they occur with multiplicities of four. They survive grid refinement tests with up to 129 × 129 grid points on a square domain, thus suggesting that such asymmetric solutions are not artifacts of the numerical scheme.
AB - We present a bifurcation study of the chemically driven convection problem in a porous medium. The effect of nonuniform heat generation on the convection is of interest in chemical reactor theory as found in the recent works of Farr and Viljoen . We focus primarily on the effects of weak convection (driven by buoyancy force) on the bifurcation structure of the basic reaction-convection equations. The parameter that governs the magnitude of the convection is the Rayleigh number, Ra. Perturbation methods for both regular and singular solutions are developed. Such methods are found to be adequate for Ra ≈ 15. The full power of the numerical methods is required to uncover the more complicated bifurcation structure in certain regions of the parameter space. Arc-length continuation is used to construct the solution branches and an extended system formulation is used to precisely locate the limit and symmetry-breaking bifurcation points. The bifurcation structure characteristic of the convection problem, studied earlier by Weinitschke., is observed even when the heat generation is nonuniform. For nonzero Ra, the equations have reflection symmetry only about y-axis. Even this symmetry is spontaneously broken at specific values of ε, The Frank-Kamenetskii parameter, and asymmetric solutions emerging from such singular points occur in pairs. In addition to the primary solution branches, which start at low values of ε and possess the expected symmetries, there are also isolated solution branches with symmetric flow structures when convection is strong relative to the reaction. In the limit of no convection (Ra → O) the classical reaction (or combustion) problem governed by the partial differential equation Δθ + εe(θ/1+μθ) = 0 is recovered. Here, μ is the Arrhenius factor. In this limit, the equation has reflection symmetry about both the x- and y-axes. For the case μ = O, previously computed solutions that conform to the expected symmetries are confirmed. In addition, certain hitherto unknown solutions are found. These additional solutions do not conform to the expected symmetries based on the properties of the governing equations. Hence, they occur with multiplicities of four. They survive grid refinement tests with up to 129 × 129 grid points on a square domain, thus suggesting that such asymmetric solutions are not artifacts of the numerical scheme.
UR - http://www.scopus.com/inward/record.url?scp=0026933864&partnerID=8YFLogxK
U2 - 10.1016/0009-2509(92)85161-4
DO - 10.1016/0009-2509(92)85161-4
M3 - 文章
AN - SCOPUS:0026933864
SN - 0009-2509
VL - 47
SP - 4107
EP - 4120
JO - Chemical Engineering Science
JF - Chemical Engineering Science
IS - 15-16
ER -