# American Institute of Mathematical Sciences

December  2016, 9(6): 2031-2046. doi: 10.3934/dcdss.2016083

## Second-order slip flow of a generalized Oldroyd-B fluid through porous medium

 1 Gengdan Institute of Beijing University of Technology, Beijing 101301, China 2 School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received  August 2015 Revised  September 2016 Published  November 2016

This work is concerned the flow of a generalized Oldroyd-B fluid in a porous half-space with second-order slip effect. The fractional calculus approach is used to establish the constitutive relationship of the non-Newtonian fluid model. A new motion model is firstly proposed by modifying the boundary condition with second-order slip effect. Exact solutions for velocity and shear stress are obtained in terms of Fox H-function by using the discrete inverse Laplace transform of the sequential fractional derivatives. The similar solutions for the generalized Oldroyd-B fluid with first-order slip or no slip, and the solutions for a generalized Oldroyd-B fluid in nonporous medium, are obtained as the limiting cases of our solutions. Furthermore, the behavior of various parameters on the corresponding flow characteristics is shown graphical through different diagrams.
Citation: Yaqing Liu, Liancun Zheng. Second-order slip flow of a generalized Oldroyd-B fluid through porous medium. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2031-2046. doi: 10.3934/dcdss.2016083
##### References:
 [1] R. L. Bagley and P. T. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27(3) (1983), 201-210. [2] R. L. Bagley and P. T. Torvik, On the fractional calculus model of viscoelastic behavior, J. Rheol., 30 (1986), 133-155. [3] A. Beskok and G. E. Karniadakis, A model for flows in channels pipes, and ducts at micro and nano scales, Microscale Therm. Eng., 3 (1999), 43-77. [4] C. Fetecau, T. Hayat, C. Fetecau and N. Alia, Unsteady flow of a second grade fluid between two side walls perpendicular to a plate, Nonlinear Anal. RWA, 9 (2008), 1236-1252. doi: 10.1016/j.nonrwa.2007.02.014. [5] C. Fetecau, M. Nazar and C. Fetecau, Unsteady flow of an Oldroyd-B fluid generated by a constantly accelerating plate between two side walls perpendicular to the plate, Int. J. Non-Linear Mech., 44 (2009), 1039-1047. doi: 10.1016/j.ijnonlinmec.2009.08.008. [6] C. Fetecau, C. Fetecau, M. Kamranc and D. Vieru, Exact solutions for the flow of a generalized Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate, J. Non-Newtonian Fluid Mech., 156 (2009), 189-201. doi: 10.1016/j.jnnfm.2008.06.005. [7] Chr. Friedrich, Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheol. Acta, 30 (1991), 151-158. [8] S. H. Han, L. C. Zheng and X. X. Zhang, Slip effects on a generalized Burgers' fluid flow between two side walls with fractional derivative, J. Egypt. Math. Soc., 24 (2016), 130-137. doi: 10.1016/j.joems.2014.10.004. [9] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4612-0873-0. [10] J. C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature, Proc. R. Soc. Lond., 27 (1879), 304-308. doi: 10.1098/rspl.1878.0052. [11] M. Navier, Memoire sur les lois du movement des fluids, Mem. L'Acad. Sci. L'Inst. France, 6 (1823), 389-440. [12] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. doi: 10.1007/978-1-4612-0873-0. [13] H. T. Qi and M. Y. Xu, Stokes' first problem for a viscoelastic fluid with the generalized Oldroyd-B model, Acta Mech. Sin., 23 (2007), 463-469. doi: 10.1007/s10409-007-0093-2. [14] H. T. Qi and M. Y. Xu, Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative, Appl. Math. Model., 33 (2009), 4184-4191. doi: 10.1016/j.apm.2009.03.002. [15] I. N. Sneddon, Fourier Transforms, McGraw-Hill Book Company, Inc., New York, Toronto, London, 1951. doi: 10.1007/978-1-4612-0873-0. [16] D. Y. Song and T. Q. Jiang, Study on the constitutive equation with fractional derivative for the viscoelastic fluids Modified Jeffreys model and its application, Rheol Acta, 27 (1998), 512-517. [17] W. C. Tan and T. Masuoka, Stokes' first problem for a second grade fluid in a porous half-space with heated boundary, Int. J. Non-Linear Mech., 40 (2005), 515-522. [18] W. C. Tan and T. Masuoka, Stokes' first problem for an Oldroyd-B fluid in a porous half-space, Phys. Fluid, 17 (2005), 023101, 7pp. doi: 10.1063/1.1850409. [19] W. C. Tan, Velocity over shoot of start-up flow for a Maxwell fluid in a porous half-space, Chin. Phys., 15 (2006), 2644-2650. [20] W. C. Tan and T. Masuoka, Exact solutions of the Rayleigh-Stokes problem for a heated generalized second grade fluid in a porous half-space, Appl. Math. Model, 33 (2009), 524-531. doi: 10.1016/j.apm.2007.11.015. [21] C. F. Xue and J. X. Nie, An exact solution of start-up flow for the fractional generalized Burgers' fluid in a porous half-space, Rheol Acta, 30 (1991), 151-158. [22] C. F. Xue, J. X. Nie and W. C. Tan, An exact solution of start-up flow for the fractional generalized Burgers' fluid in a porous half-space, Nonlinear Anal. RWA, 9 (2008), 1628-1637. doi: 10.1016/j.nonrwa.2007.04.007. [23] T. T. Zhang, L. Jia and Z. C. Wang, Validation of Navier-Stokes equations for slip flow analysis within transition region, Int. J. Heat Mass Transfer, 51 (2008), 6323-6327. doi: 10.1016/j.ijheatmasstransfer.2008.04.049. [24] T. T. Zhang, L. Jia, Z. C. Wang and X. Li, The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels, Phys. Lett. A , 372 (2008), 3223-3227. doi: 10.1016/j.physleta.2008.01.077. [25] L. C. Zheng, X. X. Zhang and C. Q. Lu, Heat transfer of power law non-Newtonian, Chin. Phys. Lett., 23 (2006), 3301-3304. [26] L. C. Zheng, Y. Q. Liu and X. X. Zhang, Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative, Nonlinear Anal. RWA, 13 (2012), 513-523. doi: 10.1016/j.nonrwa.2011.02.016. [27] J. Zhu, L. C. Zheng and Z. G. Zhang, The effect of the slip condition on the MHD stagnation-point over a power-law stretching sheet, Appl. Math. Mech., 31 (2010), 439-448. doi: 10.1007/s10483-010-0404-z.

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##### References:
 [1] R. L. Bagley and P. T. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27(3) (1983), 201-210. [2] R. L. Bagley and P. T. Torvik, On the fractional calculus model of viscoelastic behavior, J. Rheol., 30 (1986), 133-155. [3] A. Beskok and G. E. Karniadakis, A model for flows in channels pipes, and ducts at micro and nano scales, Microscale Therm. Eng., 3 (1999), 43-77. [4] C. Fetecau, T. Hayat, C. Fetecau and N. Alia, Unsteady flow of a second grade fluid between two side walls perpendicular to a plate, Nonlinear Anal. RWA, 9 (2008), 1236-1252. doi: 10.1016/j.nonrwa.2007.02.014. [5] C. Fetecau, M. Nazar and C. Fetecau, Unsteady flow of an Oldroyd-B fluid generated by a constantly accelerating plate between two side walls perpendicular to the plate, Int. J. Non-Linear Mech., 44 (2009), 1039-1047. doi: 10.1016/j.ijnonlinmec.2009.08.008. [6] C. Fetecau, C. Fetecau, M. Kamranc and D. Vieru, Exact solutions for the flow of a generalized Oldroyd-B fluid induced by a constantly accelerating plate between two side walls perpendicular to the plate, J. Non-Newtonian Fluid Mech., 156 (2009), 189-201. doi: 10.1016/j.jnnfm.2008.06.005. [7] Chr. Friedrich, Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheol. Acta, 30 (1991), 151-158. [8] S. H. Han, L. C. Zheng and X. X. Zhang, Slip effects on a generalized Burgers' fluid flow between two side walls with fractional derivative, J. Egypt. Math. Soc., 24 (2016), 130-137. doi: 10.1016/j.joems.2014.10.004. [9] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function: Theory and Applications, Springer, New York, 2010. doi: 10.1007/978-1-4612-0873-0. [10] J. C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature, Proc. R. Soc. Lond., 27 (1879), 304-308. doi: 10.1098/rspl.1878.0052. [11] M. Navier, Memoire sur les lois du movement des fluids, Mem. L'Acad. Sci. L'Inst. France, 6 (1823), 389-440. [12] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. doi: 10.1007/978-1-4612-0873-0. [13] H. T. Qi and M. Y. Xu, Stokes' first problem for a viscoelastic fluid with the generalized Oldroyd-B model, Acta Mech. Sin., 23 (2007), 463-469. doi: 10.1007/s10409-007-0093-2. [14] H. T. Qi and M. Y. Xu, Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative, Appl. Math. Model., 33 (2009), 4184-4191. doi: 10.1016/j.apm.2009.03.002. [15] I. N. Sneddon, Fourier Transforms, McGraw-Hill Book Company, Inc., New York, Toronto, London, 1951. doi: 10.1007/978-1-4612-0873-0. [16] D. Y. Song and T. Q. Jiang, Study on the constitutive equation with fractional derivative for the viscoelastic fluids Modified Jeffreys model and its application, Rheol Acta, 27 (1998), 512-517. [17] W. C. Tan and T. Masuoka, Stokes' first problem for a second grade fluid in a porous half-space with heated boundary, Int. J. Non-Linear Mech., 40 (2005), 515-522. [18] W. C. Tan and T. Masuoka, Stokes' first problem for an Oldroyd-B fluid in a porous half-space, Phys. Fluid, 17 (2005), 023101, 7pp. doi: 10.1063/1.1850409. [19] W. C. Tan, Velocity over shoot of start-up flow for a Maxwell fluid in a porous half-space, Chin. Phys., 15 (2006), 2644-2650. [20] W. C. Tan and T. Masuoka, Exact solutions of the Rayleigh-Stokes problem for a heated generalized second grade fluid in a porous half-space, Appl. Math. Model, 33 (2009), 524-531. doi: 10.1016/j.apm.2007.11.015. [21] C. F. Xue and J. X. Nie, An exact solution of start-up flow for the fractional generalized Burgers' fluid in a porous half-space, Rheol Acta, 30 (1991), 151-158. [22] C. F. Xue, J. X. Nie and W. C. Tan, An exact solution of start-up flow for the fractional generalized Burgers' fluid in a porous half-space, Nonlinear Anal. RWA, 9 (2008), 1628-1637. doi: 10.1016/j.nonrwa.2007.04.007. [23] T. T. Zhang, L. Jia and Z. C. Wang, Validation of Navier-Stokes equations for slip flow analysis within transition region, Int. J. Heat Mass Transfer, 51 (2008), 6323-6327. doi: 10.1016/j.ijheatmasstransfer.2008.04.049. [24] T. T. Zhang, L. Jia, Z. C. Wang and X. Li, The application of homotopy analysis method for 2-dimensional steady slip flow in microchannels, Phys. Lett. A , 372 (2008), 3223-3227. doi: 10.1016/j.physleta.2008.01.077. [25] L. C. Zheng, X. X. Zhang and C. Q. Lu, Heat transfer of power law non-Newtonian, Chin. Phys. Lett., 23 (2006), 3301-3304. [26] L. C. Zheng, Y. Q. Liu and X. X. Zhang, Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative, Nonlinear Anal. RWA, 13 (2012), 513-523. doi: 10.1016/j.nonrwa.2011.02.016. [27] J. Zhu, L. C. Zheng and Z. G. Zhang, The effect of the slip condition on the MHD stagnation-point over a power-law stretching sheet, Appl. Math. Mech., 31 (2010), 439-448. doi: 10.1007/s10483-010-0404-z.
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