Darcy-Bénard convection of Newtonian liquids and Newtonian nanoliquids in cylindrical enclosures and cylindrical annuli

An analytical study of linear and nonlinear Darcy-Bénard convection of Newtonian liquids and Newtonian nanoliquids confined in a cylindrical porous enclosure is made. The effect of concentric insertion of a solid cylinder into the hollow circular cylinder on onset and heat transport is also investigated. An axisymmetric mode is considered, and the Bessel functions are the eigenfunctions for the problem. The two-phase model is used in the case of nanoliquids. Weakly nonlinear stability analysis is performed by considering the double Fourier-Bessel series expansion for velocity, temperature, and nanoparticle concentration fields. Water well-dispersed with copper nanoparticles of very high thermal conductivity, and one of the five different shapes is chosen as the working medium. The thermophysical properties of nanoliquids are calculated using the phenomenological laws and the mixture theory. It is found that the effect of concentric insertion of a solid cylinder into the hollow cylinder is to enhance the heat transport. The results of rectangular enclosures are obtained as limiting cases of the present study. In general, curvature enhances the heat transport and hence the heat transport is maximum in the case of a cylindrical annulus followed by that in cylindrical and rectangular enclosures. Among the five different shapes of nanoparticles, blade-shaped nanoparticles help transport maximum heat. An analytical expression is obtained for the Hopf bifurcation point in the cases of the fifth-order and the third-order Lorenz models. Regular, chaotic, mildly chaotic, and periodic behaviors of the Lorenz system are discussed using plots of the maximum Lyapunov exponent and the bifurcation diagram.