Heat transfer in a porous radial fin

Abstract

In this thesis, a time dependent nonlinear partial differential equation modelling heat transfer in a porous radial fin is derived. The equation considered is nonlinear in nature due to the presence of two nonlinear terms i.e. the bouyancy/natural convection parameter and the radiation parameter, making obtaining the solution of the equation via analytical methods challenging. As such, we consider semi-analytical and numerical methods for the solution of this problem. A semi-analytical method namely, the Differential Transformation method, is employed to study the behaviour of the heat transfer described by the partial differential equation. By using the Differential Transformation method the nonlinear equations are reduced to recurrence relations and the boundary conditions are transformed into a set of algebraic equations. The solution obtained from the Differential Transformation method is compared to the work of Darvishi et al. [1]. Computational methods such as the Crank-Nicolson scheme with and without the Newton-Raphson and Usmani method as well as the Numerical Well-balancing scheme are used to obtain new numerical solutions which are useful indicators for the exactness of the semi-analytical solution obtained. Lie Symmetry analysis is conducted to derive an appropriate initial condition by means of infinitesimal generators, since the equation under consideration is sensitive to the initial condition when employing the Numerical Well-balancing scheme. These results are then compared to the solution obtained as per the work of Darvishi et al. [1] and the semi-analytical solution. New physical and mathematical insights are revealed through the numerical methods and steady state solutions found and the comparisons made between them and other already existing solutions. The findings reveal that the results obtained via the Crank-Nicolson scheme and the Well-balancing scheme match those in the work by Darvishi et al. [1]; the Crank-Nicolson method, incorporating the Newton-Raphson and Usmani method, and the Differential Transformation method do not match these results even though the behaviour observed is appropriate and comparable. In order to engage with the dynamics of this scheme we conduct a dynamical systems analysis. In doing so, we obtain an assessment of the impact of the nonlinear sink terms on the stability of the numerical scheme employed and on the dynamics of the solutions