The application of homotopy analysis method to thin film flows of a third order fluid

The aim of the current article is to provide the analytic solutions to two thin film flows of a third order fluid. These are: (i) when the fluid moves on a belt and (ii) when the fluid moves down an inclined plane. Both problems have been solved using homotopy analysis method (HAM). These problems were already solved by Siddiqui et al. [Siddiqui AM, Mahmood R, Ghori QK. Thin film flow of a third grade fluid on a moving belt by He’s homotopy perturbation method. Int J Non-Linear Sci Numer Simul 2006;7:1–8, Siddiqui AM, Mahmood R, Ghori QK. Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. Chaos, Solitons & Fractals in press] using homotopy perturbation method (HPM) and traditional perturbation technique. With the help of two examples, it is shown that HPM is a special case of HAM. It has been noted that the solution up to second order is not enough in the case of flow on a moving belt. It is explicitly proved that the solutions of the flow down an inclined plane given in reference [Siddiqui AM, Mahmood R, Ghori QK. Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. Chaos, Solitons & Fractals in press] are divergent and hence have no meanings. The variation of velocity field corresponding to pertinent flow parameters is graphically presented and discussed.