Comparative analysis of the withdrawal of thin-film flow of a third-grade fluid under the influence of MHD using two homotopy based techniques

The purpose of this work is to examine the thin film flow of a third grade fluid having appropriate boundary conditions using two homotopy based techniques, that is, the optimal homotopy asymptotic method with DJ polynomial (OHAM-DJ) and the asymptotic homotopy perturbation method (AHPM), where the belt is considered to be moving vertically. The novelty of this research lies in the comparative application of two perturbation methods to the same flow problem, which not only validates the results but also highlights the efficiency and accuracy of each technique. The governing equations are reduced to ordinary differential equations (ODEs). The influences of important dimensionless parameters such as the magnetic parameter the non-Newtonian parameter , and the Stokes number on the velocity profile and flow behavior are carefully investigated. Solutions are given in the form of graphs and tables, illuminating a strong agreement between both methods. This work finds that increasing the magnetic parameter slows down the fluid motion because of Lorentz force, whereas higher non-Newtonian parameter and the Stokes number significantly alter the velocity. This dual-technique methodology ensures solution robustness and gives deeper understanding into the complex rheology of third-grade fluid flows in practical applications. The proposed methods are compared with the optimal auxiliary functions method (OAFM), and it is found that both these techniques yield better results than OAFM.