Finite amplitude cellular convection under the influence of a vertical magnetic field

At the onset of stationary convection, the effect of a vertical magnetic field on the heat transfer of Rayleigh-Benard convection for an electrically conducting fluid is studied. The nonlinear governing equations describing the motion, temperature and magnetic fields are expanded as the sequence of non-homogeneous linear equations, which depend on the solutions of the linear stability problem. Infinite number of steady state with finite amplitude solutions are obtained for the stress-free boundary conditions. The perturbation method proposed by Kuo (1961) is used for the first time to highlight the heat transfer features of magnetoconvection. An explicit expression at the onset of convection in terms of parameters of the system is obtained. The dependence of heat transfer rate on Rayleigh number (R), Chandrasekhar number, thermal and magnetic Prandtl numbers is extensively examined until sixth order using an expansion of R as proposed by Kuo (1961). The results show that the magnetic field dampens the heat flow for stationary convection, i.e., the onset of convection shifts to higher values of R as the vertical magnetic field increases. Under the uniform magnetic field, heat flow gets enhanced as the thermal Prandtl number increases, whereas heat flow diminishes for the increase in magnetic Prandtl number. The results of flow field and heat transfer characteristics are depicted in the form of streamlines and isotherms, respectively. The presence of magnetic field changes the flow structure of streamlines from unicellular to multicellular patterns. This is due to the magnetic susceptibility of colder fluid flow towards the magnetic field. The flow field is analyzed with respect to the topological invariant relation. To trace the path of convective heat transport, the concept of Heatfunction has been employed. This methodology explains the comprehensive interpretation of energy distribution in terms of heatlines.