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Exact and approximate analytic solutions of the thin film flow of fourth-grade fluids by the modified Adomian decomposition method

Lazhar Bougoffa (Department of Mathematics, Al-Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia)
Jun-Sheng Duan (School of Sciences, Shanghai Institute of Technology, Shanghai, China)
Randolph Rach (The George Adomian Center for Applied Mathematics, Hartford, Michigan, USA)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 7 November 2016

Abstract

Purpose

The purpose of this paper is to first deduce a new form of the exact analytic solution of the well-known nonlinear second-order differential equation subject to a set of mixed nonlinear Robin and Neumann boundary conditions that model the thin film flows of fourth-grade fluids, and second to compare the approximate analytic solutions by the Adomian decomposition method (ADM) with the new exact analytic solution to validate its accuracy for parametric simulations of the thin film fluid flows, even for more complex models of non-Newtonian fluids in industrial applications.

Design/methodology/approach

The approach to calculating a new form of the exact analytic solution of thin film fluid flows rests upon a sequence of transformations including the modification of the classic technique due to Scipione del Ferro and Niccolò Fontana Tartaglia. Next the authors establish a lemma that justifies the new expression of the exact analytic solution for thin film fluid flows of fourth-grade fluids. Second, the authors apply a modification of the systematic ADM to quickly and easily calculate the sequence of analytic approximate solutions for this strongly nonlinear model of thin film flow of fourth-grade fluids. The ADM has been previously demonstrated to be eminently practical with widespread applicability to frontier problems arising in scientific and engineering applications. Herein, the authors seek to establish the relative merits of the ADM in the context of the thin film flows of fourth-grade fluids.

Findings

The ADM is shown to closely agree with the new expression of the exact analytic solution. The authors have calculated the error remainder functions and the maximal error remainder parameters in the error analysis to corroborate the solutions. The error analysis demonstrates the rapid rate of convergence and that we can approximate the exact solution as closely as we please; furthermore the rate of convergence is shown to be approximately exponential, and thus only a low-stage approximation will be adequate for engineering simulations as previously documented in the literature.

Originality/value

This paper presents an accurate work for solving thin film flows of fourth-grade fluids. The authors have compared the approximate analytic solutions by the ADM with the new expression of the exact analytic solution for this strongly nonlinear model. The authors commend this technique for more complex thin film fluid flow models.

Keywords

Citation

Bougoffa, L., Duan, J.-S. and Rach, R. (2016), "Exact and approximate analytic solutions of the thin film flow of fourth-grade fluids by the modified Adomian decomposition method", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 26 No. 8, pp. 2432-2440. https://doi.org/10.1108/HFF-07-2015-0278

Publisher

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Emerald Group Publishing Limited

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