- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in Mathematical Physics

Volume 2012 (2012), Article ID 634925, 11 pages

http://dx.doi.org/10.1155/2012/634925

## A Study on the Convergence of Series Solution of Non-Newtonian Third Grade Fluid with Variable Viscosity: By Means of Homotopy Analysis Method

^{1}Department of Mechanical Engineering, University of California Riverside, Bourns Hall, A373, Riverside, CA 92521, USA^{2}Department of Mathematics & Statistics, FBAS, IIU, H-10, Islamabad 44000, Pakistan

Received 14 December 2011; Accepted 27 January 2012

Academic Editor: Teoman Özer

Copyright © 2012 R. Ellahi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work is concerned with the series solutions for the flow of third-grade non-Newtonian fluid with variable viscosity. Due to the nonlinear, coupled, and highly complicated nature of partial differential equations, finding an analytical solution is not an easy task. The homotopy analysis method (HAM) is employed for the presentation of series solutions. The HAM is accepted as an elegant tool for effective solutions for complicated nonlinear problems. The solutions of (Hayat et al., 2007) are developed, and their convergence has been discussed explicitly for two different models, namely, constant and variable viscosity. An error analysis is also described. In addition, the obtained results are illustrated graphically to depict the convergence region. The physical features of the pertinent parameters are presented in the form of numerical tables.

#### 1. Introduction

During the last few years, there has been substantial progress in the steady and unsteady flows of non-Newtonian fluids. A huge amount of literature is now available on the topic (see some studies [1–6]). All real fluids are diverse in nature. Hence in view of rheological characteristics, all non-Newtonian fluids cannot be explained by employing one constitutive equation. This is the striking difference between viscous and the non-Newtonian fluids. The rheological parameters appearing in the constitutive equations lead to a higher-order and complicated governing equations than the Navier-Stokes equations. The simplest subclass of differential-type fluids is called the second grade. In steady flow such fluids can predict the normal stress and does not show shear thinning and shear thickening behaviors. The third-grade fluid puts forward the explanation of shear thinning and shear thickening properties. Therefore, the present paper aims to study the pipe flow of a third-grade fluid. Some progress on the topic is mentioned in the studies [7, 8] and many references therein. In all these studies, variable viscosity is used. Massoudi and Christie [9] numerically examined the pipe flow of a third-grade fluid when viscosity depends upon temperature. Hayat et al. [10] presented the homotopy solution of the problem considered in [10] up to second-order deformation.

In this paper, the motivation comes from a desire to understand the convergence of the problem discussed in [10]. The relevant equations for flow and temperature have been solved analytically by using homotopy analysis method [11–15]. Here the convergence of the obtained solutions is explicitly shown,and that was not previously given in [10].

#### 2. Problem

From [10], we have the equations (2.1) to (3.4) in nondimensional and nonlinear coupled partial differential equations of the form subject to boundary conditions

#### 3. Solution of the Problem

Our interest is to carry out the analysis for the homotopy solutions for two cases of viscosity, namely, constant and space-dependent viscous dissipation.

*Case I. *For constant viscosity model, we choose
For HAM solution, we select
as initial approximations of and , respectively, which satisfy the linear operator and corresponding boundary conditions. We use the method of higher-order differential mapping [16] to choose the linear operator which is defined by
such that
where and are the arbitrary constants.

If the convergence parameter is and is an embedding parameter, then the th-order problems become
where the nonlinear parameters and are defined by
For and , we have
When increases from 0 to 1, vary from to , respectively. By Taylor’s theorem and (3.7), one can get
where
The convergence of the series (3.8) depends upon . We choose in such a way that the series (3.8) is convergent at ; then, due to (3.7), we get
The th-order deformation problems are
where the recurrence formulae and are given by
in which
For constant viscosity, the velocity and temperature expressions up to second-order deformation are

*Case II. *For space-dependent viscosity, we take
For HAM solution, we select
As the initial approximation of and . We select
such that
where and are arbitrary constants. The - and -order deformation problems are
where
For variable viscosity, the velocity and temperature expressions up to second-order deformation are
where the constant coefficients – can be easily obtained through the routine calculation.

*th-order solutions*

In both cases, for and , we have
When increases from 0 to 1, , varies from to and , respectively. By Taylor’s theorem and (3.24) the general solutions can be written as
where
The convergence of (3.25) depends upon ; therefore, we choose in such a way that it should be convergent at . In view of (3.24), finally the general form of th-order solutions is

#### 4. Discussion

It is noticed that the explicit, analytical expressions (3.11), (19), (3.19), and (3.20) contain the auxiliary parameter . As pointed out by Liao [17], the convergence region and rate of approximations given by the HAM are strongly dependent upon . Figures 1 and 2 show the -curves of velocity and temperature profiles, respectively, just to find the range of for the case of constant viscosity. The range for admissible values of for velocity is and for temperature is . Figures 4 and 5 represent the -curves for variable viscosity. The admissible ranges for both velocity and temperature profiles are and , respectively. In Figures 3 and 6, the graphs of residual error are plotted for constant and variable viscosity, respectively. The error of norm 2 of two successive approximations over with HAM by -order approximations is calculated by It is seen that the error is minimum at . These values of also lie in the admissible range of .

We use the widely applied symbolic computation software MATHEMATICA to see the effects of sundry parameters by Tables 1, 2, and 3.

#### 5. Conclusion

In this paper, the convergence of series solution for constant and variable viscosity in a third-grade fluid is presented. The steady pipe flow is considered. Convergence values and residual error are also examined in Figures 1 to 6. To see the effects of emerging parameters for constant and variable viscosity, Tables 1 to 3 have been displayed. In Tables 1 and 2, it is found that the velocity and temperature increase with the decrease in pressure gradient and third-grade parameter, respectively, whereas Table 3 explains the variation of viscous dissipation parameter on velocity and temperature distributions. Here, it is revealed that the velocity and temperature decrease by increasing the viscous dissipation. It is observed that the results and figures [10] for important parameters and are correct and remain unchanged.

#### Acknowledgments

R. Ellahi thanks the United State Education Foundation Pakistan and CIES USA for honoring him by the Fulbright Scholar Award for the year 2011-2012. R. Ellahi is also grateful to the Higher Education Commission and PCST of Pakistan to award him the awards of NRPU and Productive Scientist, respectively.

#### References

- W. Tan and T. Masuoka, “Stability analysis of a Maxwell fluid in a porous medium heated from below,”
*Physics Letters A*, vol. 360, no. 3, pp. 454–460, 2007. View at Publisher · View at Google Scholar · View at Scopus - T. Hayat and F. M. Mahomed, “Note on an exact solution for the pipe flow of a third-grade fluid,”
*Acta Mechanica*, vol. 190, no. 1–4, pp. 233–236, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - W. Tan and T. Masuoka, “Stokes' first problem for a second grade fluid in a porous half-space with heated boundary,”
*International Journal of Non-Linear Mechanics*, vol. 40, no. 4, pp. 515–522, 2005. View at Publisher · View at Google Scholar · View at Scopus - M. Y. Malik, A. Hussain, and S. Nadeem, “Flow of a Jeffery-six constant fluid between coaxial cylinders with heat transfer analysis,”
*Communications in Theoretical Physics*, vol. 56, no. 2, pp. 345–351, 2011. View at Publisher · View at Google Scholar - M. Y. Malik, A. Hussain, S. Nadeem, and T. Hayat, “Flow of a third grade fluid between coaxial cylinders with variable viscosity,”
*Zeitschrift fur Naturforschung A*, vol. 64, no. 9-10, pp. 588–596, 2009. View at Google Scholar · View at Scopus - M. Hameed and S. Nadeem, “Unsteady MHD flow of a non-Newtonian fluid on a porous plate,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 1, pp. 724–733, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Ellahi,
*Steady and Unsteady Flow for Newtonian and Non-Newtonian Fluids: Basics, Concepts and Methods*, VDM, Saarbrücken, Germany, 2009. - R. Ellahi and S. Afzal, “Effects of variable viscosity in a third grade fluid with porous medium: an analytic solution,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2056–2072, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Massoudi and I. Christie, “Effects of variable viscosity and viscous dissipation on the flow of a third grade fluid in a pipe,”
*International Journal of Non-Linear Mechanics*, vol. 30, no. 5, pp. 687–699, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - T. Hayat, R. Ellahi, and S. Asghar, “The influence of variable viscosity and viscous dissipation on the non-Newtonian flow: an analytical solution,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 12, no. 3, pp. 300–313, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. J. Liao,
*The proposed homotopy analysis technique for the solution of nonlinear problems*, Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China, 1992. - S. Liao,
*Beyond Perturbation: Introduction to Homotopy Analysis Method*, vol. 2, Chapman & Hall, Boca Raton, Fla, USA, 2004. - S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,”
*Physics Letters A*, vol. 360, no. 1, pp. 109–113, 2006. View at Publisher · View at Google Scholar - R. Ellahi, “Effects of the slip boundary condition on non-Newtonian flows in a channel,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 4, pp. 1377–1384, 2009. View at Publisher · View at Google Scholar - T. Hayat, R. Ellahi, P. D. Ariel, and S. Asghar, “Homotopy solution for the channel flow of a third grade fluid,”
*Nonlinear Dynamics*, vol. 45, no. 1-2, pp. 55–64, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. A. Van Gorder and K. Vajravelu, “On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 12, pp. 4078–4089, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S.-J. Liao, “An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude,”
*International Journal of Non-Linear Mechanics*, vol. 38, no. 8, pp. 1173–1183, 2003. View at Publisher · View at Google Scholar