Nonlinear Rayleigh-Bénard magnetoconvection of a weakly electrically conducting Newtonian liquid in shallow cylindrical enclosures

A study of nonlinear axisymmetric Rayleigh-Bénard magnetoconvection in a cylindrical enclosure filled with a dilute concentration of carbon-based nanotubes in a weakly electrically conducting Newtonian liquid heated from below for various aspect ratios is carried out. Cylindrical geometry is the prototype for heat storage devices and thermal coolant systems with a controlled environment. There is an analogy between porous media and magnetohydrodynamic problems and hence Rayleigh-Bénard magnetoconvection problem is practically important. The solution of the velocity and the temperature is in terms of the Bessel functions of the first kind and hyperbolic functions that are further used to study the marginal stability curves, heat transport, and the dynamical system. Symmetric and asymmetric boundaries of the realistic-type are considered on the horizontal and vertical bounding surfaces. The results of these boundaries are compared with those of the idealistic-type which are symmetric. A unified analysis approach is adopted for all boundary combinations in deriving the Lorenz model and studying the nonlinear dynamics. The time-dependent Nusselt numbers incorporating the effect of the curvature of the cylinder accurately captures the enhanced heat transport situation in the regular convective regime. Further, the influence of various parameters on the indicators of chaos such as the rH-plots, Lorenz attractor, bifurcation diagram, and the time series plot is investigated. The rH-plots clearly point to the appearance of chaos and also assist in determining its intensity and periodicity. The trapping region of the solution of the Lorenz model having the shape like that of a rugby-ball is highlighted in the paper. The size of the ellipsoid shrinks with increase in the strength of the magnetic field and also depends on the boundary conditions.