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
References
[1]
R. S. Rivlin and J. L. Ericksen, “Stress deformation relations for isotropic materials,” Rational Mechanics and Analysis Journal, vol. 4, pp. 323–425, 1955.
[2]
C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer, New York, NY, USA, 2nd edition, 1992.
[3]
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.
[4]
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.
[5]
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.
[6]
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.
[7]
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.
[8]
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.
[9]
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.
[10]
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.
[11]
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.
[12]
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.
[13]
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.
[14]
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.
[15]
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.
[16]
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.
[17]
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.
[18]
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.
[19]
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.
[20]
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.
[21]
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.
[22]
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.
[23]
S. J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, CRC Press, Boca Raton, Fla, USA, 2003.
[24]
S. J. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004.
[25]
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.
[26]
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.
[27]
S. Abbasbandy, “Homotopy analysis method for heat radiation equations,” International Communications in Heat and Mass Transfer, vol. 34, no. 3, pp. 380–387, 2007.
[28]
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.
