A radiating straight fin with constant thermal properties by modified perturbation method

Abstract

A modified perturbation method (Kato and Matsuda, 2021) is used to obtain the solution of the heat transfer problem of a radiating straight fin with constant thermal properties. The main procedure of the modified perturbation method (MPM) is: (1) A perturbation parameter ε is assumed to be included in the nonlinear term of the differential equation. The solution θ is expressed by θ = φ + θf, where θf is an initial approximation of the solution. (In this paper, θf is assumed to be a constant) (2) θ = φ + θf is substituted into the differential equation and the nonlinear term is split into linear and nonlinear terms. (3) ε which is not in the nonlinear term is replaced by a newly introduced parameter ε´. (4) An asymptotic expansion of φ in powers of ε is assumed for the solution of the differential equation, from which we obtain the perturbation solution of φ including ε and ε´. (5) ε´ in the perturbation solution of φ is replaced by ε. Then we obtain the perturbation solution of θ. The obtained solutions by MPM are found to be in good agreement with the numerical results by the finite difference method. The solutions are also compared with those by the conventional perturbation method (CPM). It is found that MPM can extend the applicable range of the small parameter ε (radiation-conduction parameter) drastically compared with that by CPM. The modifications of the perturbation method by splitting the nonlinear term help reduce the contribution of the nonlinear term, which drastically improve the convergence characteristics of the solution.

Keywords

• perturbation method
• modified perturbation method
• splitting perturbation method
• conventional perturbation method
• radiating straight fin
• small parameter
• perturbation parameter
• radiation-conduction parameter