In this article, an analytical technique has been proposed for solving the model of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability. The idea of combining the Fourier transform and the homotopy perturbation method to yield a new technique was successful. The tables and graphs of the results of new analytical approximate solutions have illustrated the importance, usefulness, and necessity of using the new method. The results obtained showed the accuracy and efficiency of the new method compared to the previous methods, which were used to find the analytical approximate solutions for the current problem.

In this study, a hybrid method combining the homotopy perturbation method (HPM) and Fourier transform (FT) is developed and denoted as FT-HPM. This novel algorithm leverages the properties of convolution theory to facilitate calculations and is applied to obtain approximate analytical solutions for the two-dimensional natural convection between two concentric horizontal circular cylinders maintained at various uniform temperatures. The effects of Rayleigh number, Prandtl number, and radius variation on the fluid flow (air) and heat transfer are investigated. Furthermore, velocity distributions are examined and discussed, while the Nusselt number is calculated to represent local and general heat transfer rates through the relevant Nusselt numbers. The convergence of the FT-HPM method is discussed theoretically, with the formulation of theorems that are applied to the results of the obtained solutions. Tables and graphs of the analytical solutions demonstrate the feasibility and potential usefulness of the proposed algorithm for addressing various nonlinear problems, particularly natural convection problems. This research contributes to the understanding of natural convection in complex geometries and provides a foundation for future studies in this field.