Shanks Transformation for the Asymptotic Relation of Heat Transfer Rate and Rayleigh Number for Convection in Mantle

The relation between heat transfer coefficient, Nusselt number and the Rayleigh number has a significant role in mantle convection. By boundary layer theory, it is found that for constant viscosity case, and for temperature-dependent viscosity case, , where is the viscous temperature parameter in the variable viscosity function defined in Arrhenius form and the coefficient depends on cell-width. The values of Nusselt number are obtained for unit aspect ratio cell for Rayleigh-B´enard convection model where the cases like constant viscosity and temperature dependent viscosity are considered with high Rayleigh Number (). The convection model is solved numerically by the finite element method based PDE solver for different Rayleigh numbers and viscosity contrast. The values of found numerically are used to obtain theoretically the proportionality constant by using the Shanks transformation which is one of the best all-purpose methods for accelerating convergence of sequences. The results are presented by several tables and figures. It is found that Shanks transformation successfully predicts the value of for constant viscosity convection case, but failed to predict a constant value for the temperature-dependent viscosity case.