Analytic solution for MHD flow of a third order fluid in a porous channel. (English) Zbl 1144.76059
Summary: The present study investigates the channel flow of a third order fluid. The fluid is electrically conducting in the presence of a magnetic field applied transversely to the porous walls of a channel. Expression for velocity is developed by an analytic method, namely the homotopy analysis method (HAM). Convergence of the obtained solution is properly checked. The feature of the analytic solution as function of the physical parameters of the problem are discussed with the help of graphs. It is observed that unlike the flow of second grade fluid, the obtained solution for a third order fluid is non-similar. Also, the behavior of Hartmann number on the velocity is different to that of the Reynold’s number.
MSC:
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
References:
| [1] | Abbas, Z.; Sajid, M.; Hayat, T., MHD boundary layer flow of an upper-convected Maxwell fluid in a porous channel, Theoret. Comput. Fluid Dynamics, 20, 229-238 (2006) · Zbl 1109.76065 |
| [2] | Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A, 360, 109-113 (2006) · Zbl 1236.80010 |
| [3] | Asghar, S.; Masood Khan; Hayat, T., Magnetohydrodynamic transient flows of a non-Newtonian fluid, Internat. J. Non-Linear Mech., 40, 589-601 (2005) · Zbl 1343.76071 |
| [4] | Chen, C. I.; Chen, C. K.; Yang, Y. T., Unsteady unidirectional flow of an Oldroyd-B fluid in a circular duct with different given volume flow rate conditions, Heat Mass Transfer, 40, 203-209 (2004) |
| [5] | Choi, J. J.; Rusak, Z.; Tichy, J. A., Maxwell fluid suction flow in a channel, J. Non-Newtonian Fluid Mech., 85, 165-187 (1999) · Zbl 0941.76079 |
| [6] | Fetecau, C.; Fetecau, C., Unsteady flow of Oldroyd-B fluids in a channel of rectangular cross-section, Internat. J. Non-Linear Mech., 40, 1214-1219 (2005) · Zbl 1287.76045 |
| [7] | Fetecau, C.; Fetecau, C., Decay of a potential vortex in an Oldroyed-B fluid, Internat. J. Eng. Sci., 43, 340-351 (2005) · Zbl 1211.76008 |
| [8] | Fetecau, C.; Fetecau, C., On some axial Couette flows of non-Newtonian fluids, Z. Angew. Math. Phys., 56, 1098-1106 (2005) · Zbl 1096.76003 |
| [9] | Fetecau, C.; Fetecau, C., Starting solutions for some unsteady unidirectional flows of a second grade fluid, Internat. J. Eng. Sci., 43, 781-789 (2005) · Zbl 1211.76032 |
| [10] | Gorla, R. S.R.; Shammugam, K.; Kumari, M., Nonsimilar solutions for mixed convection in non-Newtonoian fluids along horizontal surfaces in porous media, Trans. Porous Media, 28, 319-334 (1997) |
| [11] | Hayat, T.; Ali, N., Peristaltically induced motion of a MHD third grade fluid in a deformable tube, Physica A, 370, 225-239 (2006) |
| [12] | Hayat, T.; Khan, M., Homotopy solution for a generalized second grade fluid past a porous plate, Non-Linear Dynamics, 42, 395-405 (2005) · Zbl 1094.76005 |
| [13] | Hayat, T.; Khan, M.; Asghar, S., Magnetohydrodynamic flow of an Oldroyd 6-constant fluid, Appl. Math. Comput., 155, 417-425 (2004) · Zbl 1126.76388 |
| [14] | Hayat, T.; Khan, M.; Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta Mech., 168, 213-232 (2004) · Zbl 1063.76108 |
| [15] | Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Internat. J. Eng. Sci., 42, 123-135 (2004) · Zbl 1211.76009 |
| [16] | Hayat, T.; Khan, M.; Ayub, M., Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field, J. Math. Anal. Appl., 298, 225-244 (2004) · Zbl 1067.35074 |
| [17] | Hayat, T.; Khan, M.; Siddiqui, A. M.; Asghar, S., Transient flows of a second grade fluid, Internat. J. Non-Linear Mech., 39, 1621-1633 (2004) · Zbl 1348.76017 |
| [18] | Hayat, T.; Siddiqui, A. M.; Asghar, S., Some simple flows of an Oldroyd-B fluid, Internat. J. Eng. Sci., 39, 135-147 (2001) |
| [19] | Hayat, T.; Wang, Y.; Hutter, K., Hall effects on the unsteady hydromagnetic oscillatory flow of a second grade fluid, Internat. J. Non-Linear Mech., 39, 1027-1037 (2004) · Zbl 1348.76187 |
| [20] | Khan, M.; Hayat, T.; Asghar, S., Exact solution for MHD flow of a generalized Oldroyed-B fluid with modified Darcy’s law, Internat. J. Eng. Sci., 44, 333-339 (2006) · Zbl 1213.76024 |
| [21] | Liao, S. J., A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J. Fluid Mech., 385, 101-128 (1999) · Zbl 0931.76017 |
| [22] | Liao, S. J., Beyond Perturbation: Introduction to Homotopy Analysis Method (2003), London: London Chapman & Hall/CRC Press, Boca Raton |
| [23] | Liao, S. J., An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate, Comm. Non-Linear Sci. Numer. Simmulation, 11, 326-339 (2006) · Zbl 1078.76022 |
| [24] | Liao, S. J.; Campo, A., Analytic solutions of the temperature distribution in Blasius viscous flow problems, J. Fluid Mech., 453, 411-425 (2002) · Zbl 1007.76014 |
| [25] | Liao, S. J.; Cheung, K. F., Homotopy analysis of nonlinear progressive waves in deep water, J. Eng. Math., 45, 105-116 (2003) · Zbl 1112.76316 |
| [26] | Rajagopal, K. R., A note on unsteady unidirectional flows of non-Newtonian fluid, Internat. J. Non-Linear Mech., 17, 369-373 (1982) · Zbl 0527.76003 |
| [27] | Rajagopal, K. R., On the creeping flow of the second order fluid, J. Non-Newtonian Fluid Mech., 15, 239-246 (1984) · Zbl 0568.76015 |
| [28] | Rajagopal, K. R., On boundary conditions for fluid of differential type, (Sequiera, A., Navier-Stokes Equations and Related Non-linear Problems (1995), Plenum Press: Plenum Press New York) · Zbl 0846.35107 |
| [29] | Reddy, R. S.; Kumari, M., Nonsimilar solution for a mixed convection non-Newtonian fluids along a vertical plate in a porous medium, Trans. Porous Media, 33, 295-307 (1998) |
| [30] | Sadeghy, K.; Najafi, A. H.; Saffaripour, M., Sakiadis flow of an upper-convected Maxwell fluid, Internat. J. Non-Linear Mech., 40, 1220-1228 (2005) · Zbl 1349.76081 |
| [31] | Sajid, M.; Hayat, T.; Asghar, S., On the analytic solution of the steady flow of a fourth grade fluid, Phys. Lett. A., 355, 18-26 (2006) |
| [32] | Sarpakaya, T., Flow of non-Newtonian fluids in a magnetic field, AIChE J., 7, 324-328 (1961) |
| [33] | Tan, W. C.; Masuoka, T., Stokes first problem for a second grade fluid in a porous half space with heated boundary, Internat. J. Non-Linear Mech., 40, 515-522 (2005) · Zbl 1349.76830 |
| [34] | Tan, W. C.; Masuoka, T., Stokes first problem for an Oldroyd-B fluid in a porous half space, Phys. Fluids, 17, 23101-23107 (2005) · Zbl 1187.76517 |
| [35] | Tan, Y.; Xu, H.; Liao, S. J., Explicit series solution of travelling waves with a front of Fisher equation, Chaos Solitons & Fractals, 31, 462-472 (2007) · Zbl 1143.35313 |
| [36] | Wang, Y.; Hayat, T., Hydromagnetic rotating flow of fourth order fluid past a porous plate, Math. Methods Appl. Sci., 27, 477-496 (2004) · Zbl 1128.76378 |
| [37] | Wu, W.; Liao, S. J., Solving solitary waves with discontinuity by means of homotopy analysis method, Chaos Solitons & Fractals, 26, 177-185 (2005) · Zbl 1071.76009 |
| [38] | Zhu, S. P., An exact and explicit solution for the valuation of American put options, Quantitative Finance, 6, 229-242 (2006) · Zbl 1136.91468 |
| [39] | Zhu, S. P., A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield, Anziam J., 47, 477-494 (2006) · Zbl 1147.91336 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
