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Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure
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 |
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. |
show all references
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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