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Unsteady solutions in a third-grade fluid filling the porous space. (English) Zbl 1151.76595

Summary: An analysis is made of the unsteady flow of a third-grade fluid in a porous medium. A modified Darcy’s law is considered in the flow modelling. Reduction and solutions are obtained by employing similarity and numerical methods. The effects of pertinent parameters on the flow velocity are studied through graphs.

MSC:

76S05 Flows in porous media; filtration; seepage
76A05 Non-Newtonian fluids
76M55 Dimensional analysis and similarity applied to problems in fluid mechanics
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References:

[1] K. R. Rajagopal, “Boundedness and uniqueness of fluids of the differential type,” Acta Ciencia Indica, vol. 8, no. 1-4, pp. 28-38, 1982. · Zbl 0555.76011
[2] K. R. Rajagopal, “On boundary conditions for fluids of the differential type,” in Navier-Stokes Equations and Related Nonlinear Problems, A. Sequira, Ed., pp. 273-278, Plenum Press, New York, NY, USA, 1995. · Zbl 0846.35107
[3] K. R. Rajagopal, A. Z. Szeri, and W. Troy, “An existence theorem for the flow of a non-Newtonian fluid past an infinite porous plate,” International Journal of Non-Linear Mechanics, vol. 21, no. 4, pp. 279-289, 1986. · Zbl 0599.76013 · doi:10.1016/0020-7462(86)90035-1
[4] K. R. Rajagopal and P. N. Kaloni, “Some remarks on boundary conditions for flows of fluids of the differential type,” in Continuum Mechanics and Its Applications, G. A. C. Graham and S. K. Malik, Eds., pp. 935-942, Hemisphere, New York, NY, USA, 1989. · Zbl 0706.76008
[5] K. R. Rajagopal and T. Y. Na, “On Stokes/ problem for a non-Newtonian fluid,” Acta Mechanica, vol. 48, no. 3-4, pp. 233-239, 1983. · Zbl 0528.76003 · doi:10.1007/BF01170422
[6] K. R. Rajagopal and T. Y. Na, “Natural convection flow of a non-Newtonian fluid between two vertical flat plates,” Acta Mechanica, vol. 54, no. 3-4, pp. 239-246, 1985. · Zbl 0558.76008 · doi:10.1007/BF01184849
[7] 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. · Zbl 0552.76008 · doi:10.1007/BF01560464
[8] T. Hayat, M. Khan, and M. Ayub, “Some analytical solutions for second grade fluid flows for cylindrical geometries,” Mathematical and Computer Modelling, vol. 43, no. 1-2, pp. 16-29, 2006. · Zbl 1090.76006 · doi:10.1016/j.mcm.2005.04.009
[9] T. Hayat, Z. Abbas, and S. Asghar, “Heat transfer analysis on rotating flow of a second-grade fluid past a porous plate with variable suction,” Mathematical Problems in Engineering, vol. 2005, no. 5, pp. 555-582, 2005. · Zbl 1200.76213 · doi:10.1155/MPE.2005.555
[10] T. Hayat, S. Nadeem, S. Asghar, and A. M. Siddiqui, “Effects of Hall current on unsteady flow of a second grade fluid in a rotating system,” Chemical Engineering Communications, vol. 192, no. 10, pp. 1272-1284, 2005. · doi:10.1080/009864490515810
[11] T. Hayat, R. Ellahi, P. D. Ariel, and S. Asghar, “Homotopy solution for the channel flow of a third grade fluid,” Nonlinear Dynamics, vol. 45, no. 1-2, pp. 55-64, 2006. · Zbl 1100.76005 · doi:10.1007/s11071-005-9015-7
[12] T. Hayat and N. Ali, “Peristaltically induced motion of a MHD third grade fluid in a deformable tube,” Physica A, vol. 370, no. 2, pp. 225-239, 2006. · doi:10.1016/j.physa.2006.02.029
[13] P. D. Ariel, T. Hayat, and S. Asghar, “The flow of an elastico-viscous fluid past a stretching sheet with partial slip,” Acta Mechanica, vol. 187, no. 1-4, pp. 29-35, 2006. · Zbl 1103.76010 · doi:10.1007/s00707-006-0370-3
[14] T. Hayat and A. H. Kara, “Couette flow of a third-grade fluid with variable magnetic field,” Mathematical and Computer Modelling, vol. 43, no. 1-2, pp. 132-137, 2006. · Zbl 1105.76007 · doi:10.1016/j.mcm.2004.12.009
[15] M. B. Abd-el-Malek, N. A. Badran, and H. S. Hassan, “Solution of the Rayleigh problem for a power law non-Newtonian conducting fluid via group method,” International Journal of Engineering Science, vol. 40, no. 14, pp. 1599-1609, 2002. · Zbl 1211.76003 · doi:10.1016/S0020-7225(02)00037-X
[16] C. Wafo Soh, “Invariant solutions of the unidirectional flow of an electrically charged power-law non-Newtonian fluid over a flat plate in presence of a transverse magnetic field,” Communications in Nonlinear Science and Numerical Simulation, vol. 10, no. 5, pp. 537-548, 2005. · Zbl 1303.76116 · doi:10.1016/j.cnsns.2003.12.008
[17] C.-I. Chen, T. Hayat, and J.-L. Chen, “Exact solutions for the unsteady flow of a Burger/s fluid in a duct induced by time-dependent prescribed volume flow rate,” Heat and Mass Transfer, vol. 43, no. 1, pp. 85-90, 2006. · doi:10.1007/s00231-006-0092-z
[18] C. Fetecau and C. Fetecau, “Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder,” International Journal of Engineering Science, vol. 44, no. 11-12, pp. 788-796, 2006. · Zbl 1213.76014 · doi:10.1016/j.ijengsci.2006.04.010
[19] C. Fetecau and C. Fetecau, “Decay of a potential vortex in an Oldroyd-B fluid,” International Journal of Engineering Science, vol. 43, no. 3-4, pp. 340-351, 2005. · Zbl 1211.76008 · doi:10.1016/j.ijengsci.2004.08.013
[20] C. Fetecau and C. Fetecau, “Starting solutions for some unsteady unidirectional flows of a second grade fluid,” International Journal of Engineering Science, vol. 43, no. 10, pp. 781-789, 2005. · Zbl 1211.76032 · doi:10.1016/j.ijengsci.2004.12.009
[21] W. Tan and T. Masuoka, “Stokes/ first problem for a second grade fluid in a porous half-space with heated boundary,” International Journal of Non-Linear Mechanics, vol. 40, no. 4, pp. 515-522, 2005. · Zbl 1349.76830 · doi:10.1016/j.ijnonlinmec.2004.07.016
[22] W. Tan and T. Masuoka, “Stokes/ first problem for an Oldroyd-B fluid in a porous half space,” Physics of Fluids, vol. 17, no. 2, Article ID 023101, 7 pages, 2005. · Zbl 1187.76517 · doi:10.1063/1.1850409
[23] C. Truesdell and W. Noll, “The non-linear field theories of mechanics,” in Handbuch der Physik, Band III/3, pp. 1-602, Springer, Berlin, Germany, 1965. · Zbl 1068.74002
[24] R. S. Rivlin and J. L. Ericksen, “Stress-deformation relations for isotropic materials,” Journal of Rational Mechanics and Analysis, vol. 4, pp. 323-425, 1955. · Zbl 0064.42004
[25] R. L. Fosdick and K. R. Rajagopal, “Thermodynamics and stability of fluids of third grade,” Proceedings of the Royal Society of London, vol. 369, no. 1738, pp. 351-377, 1980. · Zbl 0441.76002 · doi:10.1098/rspa.1980.0005
[26] M. G. Alishayev, “Proceedings of Moscow Pedagogy Institute,” Hydrodynamics, vol. 3, pp. 166-174, 1974 (Russian).
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