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Advances in Mathematical Physics
Volume 2012 (2012), Article ID 634925, 11 pages
A Study on the Convergence of Series Solution of Non-Newtonian Third Grade Fluid with Variable Viscosity: By Means of Homotopy Analysis Method
1Department of Mechanical Engineering, University of California Riverside, Bourns Hall, A373, Riverside, CA 92521, USA
2Department 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.
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.
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  numerically examined the pipe flow of a third-grade fluid when viscosity depends upon temperature. Hayat et al.  presented the homotopy solution of the problem considered in  up to second-order deformation.
In this paper, the motivation comes from a desire to understand the convergence of the problem discussed in . 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 .
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  to choose the linear operator which is defined by
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.
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
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 , 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 .
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  for important parameters and are correct and remain unchanged.
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.
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