Chinese Journal of Engineering

Volume 2013 (2013), Article ID 808342, 7 pages

http://dx.doi.org/10.1155/2013/808342

## Flow of an Eyring-Powell Model Fluid between Coaxial Cylinders with Variable Viscosity

Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan

Received 21 July 2013; Accepted 18 August 2013

Academic Editors: G. Chen and S. Wei-dong

Copyright © 2013 Azad Hussain et al. 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

We consider the flow of Eyring-Powell model fluid in the annulus between two cylinders whose viscosity depends upon the temperature. We consider the steady flow in the annulus due to the motion of inner cylinder and constant pressure gradient. In the problem considered the flow is found to be remarkedly different from that for the incompressible Navier-Stokes fluid with constant viscosity. An analytical solution of the nonlinear problem is obtained using homotopy analysis method. The behavior of pertinent parameters is analyzed and depicted through graphs.

#### 1. Introduction

The analysis of the behaviour of the fluid motion of the non-Newtonian fluids becomes much complicated and subtle as compared to Newtonian fluids due to the fact that non-Newtonian fluids do not exhibit the linear relationship between stress and strain. Rivlin and Ericksen [1] and Truesdell and Noll [2] classified viscoelastic fluids with the help of constitutive relations for the stress tensor as a function of the symmetric part of the velocity gradient and its higher (total) derivatives. In recent years, there have been several studies [3–12] on flows of non-Newtonian fluids. It is a well-known fact that it is not possible to obtain a single constitutive equation exhibiting all properties of all non-Newtonian fluids from the available literature. That is why several models of non-Newtonian fluids have been proposed in the literature. Eyring-Powell model fluid is one of these models. Eyring-Powell model was first introduced by Powell and Eyring in 1944. However, the literature survey indicates that very low energy has been devoted to the flows of Eyring-Powell model fluid with variable viscosity. Massoudi and Christie [13] have considered the effects of variable viscosity and viscous dissipation on the flow of a third grade fluid in a uniform pipe. Massoudi and Christie [13] found the numerical solutions with the help of straight forward finite difference method. They also discussed that the flow of a fluid-solid mixture is very complicated and may depend on many variables such as physical properties of each phase and size and shape of solid particles. Later on, the influence of constant and space dependent viscosity on the flow of a third grade fluid in a pipe has been discussed analytically by Hayat et al. [14]. The approximate and analytical solution of non-Newtonian fluid with variable viscosity has been analyzed by Yürüsoy and Pakdermirli [15] and Pakdemirli and Yilbas [16]. The pipe flow of non-Newtonian fluid with variable viscosity keeping no slip and partial slip has been discussed analytically by Nadeem and Ali [17] and Nadeem et al. [18]. More recently, Nadeem and Akbar [19] studied the effects of temperature dependent viscosity on peristaltic flow of a Jeffrey-six constant fluid in a uniform vertical tube. The main aim of the present study is to venture further in the regime of Eyring-Powell model fluid with variable viscosity. To the best of the authors knowledge no attempt has been made to investigate Eyring-Powell model fluid in the annulus between two cylinders whose viscosity depends upon the temperature. The governing equations for Eyring-Powell model fluid are formulated considering cylindrical coordinates system. The equations are simplified using the assumptions of long wave length and low Reynolds number approximation. The obtained non-linear problem is solved using homotopy analysis method [20–28]. The effects of the emerging parameters are analyzed and depicted through graphs.

#### 2. Mathematical Model

The constitutive equation for a Cauchy stress in an Eyring-Powell model fluid is given by where is the velocity, is the Cauchy stress tensor, is the coefficient of shear viscosity, and and are the material constants. We take the velocity and stress as

#### 3. Physical Model

Consider the steady flow of an Eyring-Powell model fluid with variable temperature dependent viscosity between coaxial cylinders. The motion is caused due to a constant pressure gradient and by the motion of the inner cylinder parallel to its length, whereas the outer cylinder is kept stationary. The heat transfer analysis is also taken into account. The dimensionless problem which can describe the flow is whence where , , and are, respectively, the reference viscosity, a reference temperature (the bulk mean fluid temperature), and reference velocity is related to the Prandtl number and Eckert number.

#### 4. Series Solutions for Reynolds’ Model

Here the viscosity is expressed in the form which by Maclaurin’s series can be written as Note that corresponds to the case of constant viscosity. Invoking the above equation into (3) one has For HAM solution, we choose the following initial guesses: The auxiliary linear operators are in the form which satisfy where , , , and are the constants.

If is an embedding parameter and and are auxiliary parameters, then the problems at the zero and th order are, respectively, given by The boundary conditions at the th order are In (11)–(13) By Mathematica the solutions of (21) can be written as where and are constants which can be determined on substituting (22) into (15) and (16).

#### 5. Series Solutions for Vogel’s Model

Here which by Maclaurin’s series reduces to Invoking the above expressions, (1) become With the following initial guesses and auxiliary linear operators the th-order deformation problems are The expressions of and are finally given by

#### 6. Graphical Results and Discussion

In order to report the convergence of the obtained series solutions and the effects of sundry parameters in the present investigation we plotted Figures 1–13. Figures 1–4 are prepared to see the convergence region. Figures 1 and 2 correspond to Reynolds’ model whereas Figures 3 and 4 relate to Vogel’s model. Figure 5 shows the temperature variation for different values of for Reynolds’ model. It can be seen that temperature decreases as increases. Figure 6 depicts the velocity variation for Reynolds’ model for different values of . Velocity also decreases as increases. Figure 7 shows the velocity variation for different values of for Reynolds’ model. It can be seen that velocity increases as increases. Figure 8 is plotted in order to see the temperature variation for Reynolds’ model for different values of ; it is depicted that temperature increases as increases. Figures 9–13 correspond to Vogel’s model. Figure 9 depicts temperature variation for Vogel’s model for different values of . It is seen that temperature increases as increases. Figure 10 shows the velocity variation for Vogel’s model for different values of . It is observed that velocity decreases as increases. Figure 11 is prepared to observe the temperature variation for Vogel’s model for different values of . It is observed that temperature decreases as increases. Figure 12 is plotted to see the the velocity variation for Vogel’s model for different values of . It is observed that velocity decreases as increases. Figure 13 depicts the velocity variation for Vogel’s model for different values of . It is observed that velocity decreases as increases.

#### 7. Conclusions

In this paper, we consider the flow of Eyring-Powell model fluid in the annulus between two cylinders whose viscosity depends upon the temperature. We discussed the steady flow in the annulus due to the motion of inner cylinder and constant pressure gradient. In the problem considered the flow is found to be remarkedly different from that for the incompressible Navier-Stokes fluid with constant viscosity. The behavior of pertinent parameters is analyzed and depicted through graphs. Using usual similarity transformations the governing equations have been transformed into non-linear ordinary differential equations. The highly non-linear problem is then solved by homotopy analysis method. Effects of the various parameters on velocity and temperature profiles are examined.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

- R. S. Rivlin and J. L. Ericksen, “Stress deformation relations for isotropic materials,”
*Rational Mechanics and Analysis Journal*, vol. 4, pp. 323–425, 1955. View at Google Scholar - C. Truesdell and W. Noll,
*The Non-Linear Field Theories of Mechanics*, Springer, New York, NY, USA, 2nd edition, 1992. - K. R. Rajagopal, “A note on unsteady unidirectional flows of a non-Newtonian fluid,”
*International Journal of Non-Linear Mechanics*, vol. 17, no. 5-6, pp. 369–373, 1982. View at Google Scholar · View at Scopus - K. R. Rajagopal and A. S. Gupta, “An exact solution for the flow of a non-newtonian fluid past an infinite porous plate,”
*Meccanica*, vol. 19, no. 2, pp. 158–160, 1984. View at Publisher · View at Google Scholar · View at Scopus - K. R. Rajagopal and R. K. Bhatnagar, “Exact solutions for some simple flows of an Oldroyd-B fluid,”
*Acta Mechanica*, vol. 113, no. 1–4, pp. 233–239, 1995. View at Publisher · View at Google Scholar · View at Scopus - K. R. Rajagopal, “On the creeping flow of the second-order fluid,”
*Journal of Non-Newtonian Fluid Mechanics*, vol. 15, no. 2, pp. 239–246, 1984. View at Google Scholar · View at Scopus - K. R. Rajagopal, “Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid,”
*Acta Mechanica*, vol. 49, no. 3-4, pp. 281–285, 1983. View at Publisher · View at Google Scholar · View at Scopus - A. M. Benharbit and A. M. Siddiqui, “Certain solutions of the equations of the planar motion of a second grade fluid for steady and unsteady cases,”
*Acta Mechanica*, vol. 94, no. 1-2, pp. 85–96, 1992. View at Publisher · View at Google Scholar · View at Scopus - T. Hayat, S. Asghar, and A. M. Siddiqui, “Periodic unsteady flows of a non-Newtonian fluid,”
*Acta Mechanica*, vol. 131, no. 3-4, pp. 169–175, 1998. View at Google Scholar · View at Scopus - A. M. Siddiqui, T. Hayat, and S. Asghar, “Periodic flows of a non-Newtonian fluid between two parallel plates,”
*International Journal of Non-Linear Mechanics*, vol. 34, no. 5, pp. 895–899, 1999. View at Google Scholar · View at Scopus - T. Hayat, S. Asghar, and A. M. Siddiqui, “On the moment of a plane disk in a non-Newtonian fluid,”
*Acta Mechanica*, vol. 136, no. 3, pp. 125–131, 1999. View at Google Scholar · View at Scopus - T. Hayat, S. Asghar, and A. M. Siddiqui, “Some unsteady unidirectional flows of a non-Newtonian fluid,”
*International Journal of Engineering Science*, vol. 38, no. 3, pp. 337–346, 2000. View at Publisher · View at Google Scholar · View at Scopus - 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 Google Scholar · 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 Scopus - M. Yürüsoy and M. Pakdermirli, “Approximate analytical solutions for flow of a third grade fluid in a pipe,”
*International Journal of Non-Linear Mechanics*, vol. 37, no. 2, pp. 187–195, 2002. View at Google Scholar - M. Pakdemirli and B. S. Yilbas, “Entropy generation for pipe flow of a third grade fluid with Vogel model viscosity,”
*International Journal of Non-Linear Mechanics*, vol. 41, no. 3, pp. 432–437, 2006. View at Publisher · View at Google Scholar · View at Scopus - S. Nadeem and M. Ali, “Analytical solutions for pipe flow of a fourth grade fluid with Reynold and Vogel's models of viscosities,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2073–2090, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. Nadeem, T. Hayat, S. Abbasbandy, and M. Ali, “Effects of partial slip on a fourth-grade fluid with variable viscosity: an analytic solution,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 2, pp. 856–868, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Nadeem and N. S. Akbar, “Effects of temperature dependent viscosity on peristaltic flow of a Jeffrey-six constant fluid in a non-uniform vertical tube,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 12, pp. 3950–3964, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Y. Malik, A. Hussain, and S. Nadeem, “Analytical treatment of an oldroyd 8-constant fluid between coaxial cylinders with variable viscosity,”
*Communications in Theoretical Physics*, vol. 56, no. 5, pp. 933–938, 2011. View at Publisher · View at Google Scholar - S. Nadeem, T. Hayat, S. Abbasbandy, and M. Ali, “Effects of partial slip on a fourth-grade fluid with variable viscosity: an analytic solution,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 2, pp. 856–868, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Y. Malik, A. Hussain, S. Nadeem, and T. Hayat, “Flow of a third grade fluid between coaxial cylinders with variable viscosity,”
*Zeitschrift für Naturforschung A*, vol. 64, no. 9-10, pp. 588–596, 2009. View at Google Scholar · View at Scopus - S. J. Liao,
*Beyond Perturbation: Introduction to Homotopy Analysis Method*, CRC Press, Boca Raton, Fla, USA, 2003. - S. J. Liao, “On the homotopy analysis method for nonlinear problems,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 499–513, 2004. View at Publisher · View at Google Scholar · View at Scopus - S. J. Liao, “An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 11, no. 3, pp. 326–339, 2006. View at Publisher · View at Google Scholar · View at Scopus - 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 · View at Scopus - S. Abbasbandy, “Homotopy analysis method for heat radiation equations,”
*International Communications in Heat and Mass Transfer*, vol. 34, no. 3, pp. 380–387, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Abbasbandy, Y. Tan, and S. J. Liao, “Newton-homotopy analysis method for nonlinear equations,”
*Applied Mathematics and Computation*, vol. 188, no. 2, pp. 1794–1800, 2007. View at Publisher · View at Google Scholar · View at Scopus