Application of the Homotopy Perturbation Method for Solving the Foam Drainage Equation

Abdolhosein Fereidoon,Hessameddin Yaghoobi,Mohammadreza Davoudabadi

doi:10.1155/2011/864023

Abdolhosein Fereidoon, Hessameddin Yaghoobi + Show 1 more

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https://doi.org/10.1155/2011/864023

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Abstract

We use homotopy perturbation method (HPM) to handle the foam drainage equation. Foaming occurs in many distillation and absorption processes. The drainage of liquid foams involves the interplay of gravity, surface tension, and viscous forces. The concept of He′s homotopy perturbation method is introduced briefly for applying this method for problem solving. The results of HPM as an analytical solution are then compared with those derived from Adomian′s decomposition method (ADM) and the variational iteration method (VIM). The results reveal that the HPM is very effective and convenient in predicting the solution of such problems, and it is predicted that HPM can find a wide application in new engineering problems.

Highlights

Most scientific problems and physical phenomena occur nonlinearly
The results which are obtained by the HPM and the exact solutions are quite similar
The results show that this perturbation scheme provides excellent approximations to the solution of this nonlinear equation with high accuracy and avoids linearization and physically unrealistic assumptions

Summary

Introduction

Most scientific problems and physical phenomena occur nonlinearly. Except in a limited number of these problems, finding the exact analytical solutions of such problems are rather difficult. Most of the basic rules that explain the stability of liquid gas foams were introduced over 100 years ago by the Belgian Joseph Plateau who was blind before he completed his important book on the subject This modern-day book by Weaire and Hutzler provides valuable summaries of plateaus work on the laws of equilibrium of soap films, and it is especially useful since the original 1873 French text does not appear to be in a fully translated English version. Recent theoretical studies by Verbist and Weaire describe the main features of both free drainage 44, , where liquid drains out of a foam due to gravity, and forced drainage , where liquid is introduced to the top of a column of foam In the latter case, a solitary wave of constant velocity is generated when liquid is added at a constant rate. The results of HPM as an analytical solution are compared with those derived from Adomian’s decomposition method and the variational iteration method

Basic Idea of Homotopy Perturbation Method

Implementation of HPM

Results and Discussion

Conclusion