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MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel. (English) Zbl 1109.76065

Summary: Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell fluid is investigated in a channel. The walls of the channel are taken as porous. Using the similarity transformations and boundary layer approximations, the nonlinear partial differential equations are reduced to an ordinary differential equation. The developed nonlinear equation is solved analytically using the homotopy analysis method. An expression for the analytic solution is derived in the form of a series. The convergence of the obtained series is shown. The effects of the Reynolds number \(Re\), Deborah number \(De\) and Hartman number \(M\) are shown through graphs and discussed for both the suction and injection cases.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
76S05 Flows in porous media; filtration; seepage
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[1] Hayat T., Khan M., Siddiqui A.M., Asghar S. (2004) Transient flows of a second grade fluid. Int. J. Non-Lin. Mech. 39, 1621–163 · Zbl 1348.76017 · doi:10.1016/j.ijnonlinmec.2002.12.001
[2] Hayat T., Wang Y., Hutter K. (2004) Hall effects on the unsteady hydromagnetic oscillatory flow of a second grade fluid. Int. J. Non-Lin. Mech. 39, 1027–1037 · Zbl 1348.76187 · doi:10.1016/S0020-7462(03)00094-5
[3] Rajagopal K.R. (1982) A note on unsteady unidirectional flows of a non-Newtonian fluid. Int. J. Non-Lin. Mech. 17, 369–373 · Zbl 0527.76003 · doi:10.1016/0020-7462(82)90006-3
[4] Rajagopal K.R. (1984) On the creeping flow of the second grade fluid. J. Non-Newton. Fluid Mech. 15, 239–246 · Zbl 0568.76015 · doi:10.1016/0377-0257(84)80008-7
[5] Bandelli R. (1995) Unsteady unidirectional flows of second grade fluid in domains with heated boundaries. Int. J. Non-Lin. Mech. 30, 263–269 · Zbl 0837.76004 · doi:10.1016/0020-7462(94)00051-B
[6] Fetecau C., Fetecau C. (2003) A new exact solution for the flow of a Maxwell fluid past an infinite plate. Int. J. Non-Lin. Mech. 38, 423–427 · Zbl 1287.76041 · doi:10.1016/S0020-7462(01)00062-2
[7] Fetecau C., Fetecau C. (2003) The Rayleigh–Stokes problem for a fluid of Maxwellian type. Int. J. Non-Lin. Mech. 38, 603–607 · Zbl 1287.76042 · doi:10.1016/S0020-7462(01)00078-6
[8] Fetecau C., Fetecau C. (2003) Decay of a potential vortex in a Maxwell fluid. Int. J. Non-Lin. Mech. 38, 985–990 · Zbl 1287.76043 · doi:10.1016/S0020-7462(02)00042-2
[9] Fetecau C., Fetecau C. (2005) Starting solutions for some unsteady unidirectional flows of a second grade fluid. Int. J. Eng. Sci. 43, 781–789 · Zbl 1211.76032 · doi:10.1016/j.ijengsci.2004.12.009
[10] Asghar S., Hayat T., Siddiqui A.M. (2002) Moving boundary in a non-Newtonian fluid. Int. J. Non-Lin. Mech. 37, 75–80 · Zbl 1116.76310 · doi:10.1016/S0020-7462(00)00096-2
[11] Hayat T. (2005) Oscillatory solution in rotating flow of a Johnson–Segalman fluid. Z. Angew. Math. Mech. 85, 449–456 · Zbl 1071.76063 · doi:10.1002/zamm.200310173
[12] Hayat T., Nadeem S., Asghar S. (2004) Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model. Appl. Math. Comput. 151, 153–161 · Zbl 1161.76436 · doi:10.1016/S0096-3003(03)00329-1
[13] Chen C.I., Chen C.K., Yang Y.T (2003) Unsteady unidirectional flow of an Oldroyd-B fluid in a circular duct with different given volume flow rate conditions. Heat Mass Transf. 40, 203–209 · doi:10.1007/s00231-002-0350-7
[14] Tan W.C., Masuoka T. (2005) Stokes first poroblem for second grade fluid in a porous half space. Int. J. Non-Lin. Mech. 40, 515–522 · Zbl 1349.76830 · doi:10.1016/j.ijnonlinmec.2004.07.016
[15] Tan W.C., Masuoka T. (2005) Stokes first problem for an Oldroyd-B fluid in a porous half space. Phys. Fluid. 17: 023101–7 · Zbl 1187.76517 · doi:10.1063/1.1850409
[16] Fosdick R.L., Rajagopal K.R. (1979) Anomalous features in the model of second grade fluids. Arch. Rat. Mech. Anal. 70, 145–152 · Zbl 0427.76006 · doi:10.1007/BF00250351
[17] Dunn J.E., Rajagopal K.R. (1995) Fluids of differential type: critical review and thermodynamic analysis. Int. J. Eng. Sci. 33, 689–729 · Zbl 0899.76062 · doi:10.1016/0020-7225(94)00078-X
[18] Sadeghy K., Sharifi M. (2004) Local similarity solution for the flow of a ”second-grade” viscoelastic fluid above a moving plate. Int. J. Non-Lin. Mech. 39, 1265–1273 · Zbl 1348.76064 · doi:10.1016/j.ijnonlinmec.2003.08.005
[19] Bird R.B., Armstrong R.C., Hassager O. (1987) Dynamics of Polymeric Liquids, vols. I and II. Wiley, New York
[20] Choi J.J., Rusak Z., Tichy J.A. (1999) Maxwell fluid suction flow in a channel. J. Non-Newton. Fluid Mech. 85, 165–187 · Zbl 0941.76079 · doi:10.1016/S0377-0257(98)00197-9
[21] Sadeghy K., Najafi A.H., Saffaripour M. (2005) Sakiadis flow of an upper-convected Maxwell fluid. Int. J. Non-Lin. Mech. 40, 1220–1228 · Zbl 1349.76081 · doi:10.1016/j.ijnonlinmec.2005.05.006
[22] Shercliff I. (1953) Steady motion of conducting fluids in pipes under transverse magnetic fields. Proc. Camb. Philos. Soc. 49, 136–144 · Zbl 0050.19404 · doi:10.1017/S0305004100028139
[23] Grinberg G.A. (1961) On steady flow of a conducting fluid in a rectangular tube with two non-conducting walls and two conducting ones parallel to an external magnetic field. PMM 25, 1024–1034
[24] Hunt J.C.R. (1965) Magnetohydrodynamic flow in rectangular ducts. J. Fluid Mech. 21, 577–590 · Zbl 0125.18401 · doi:10.1017/S0022112065000344
[25] Hunt J.C.R., Stewartson K. (1965) Magnetohydrodynamic flow in rectangular ducts II. J. Fluid Mech. 23, 563–581 · doi:10.1017/S0022112065001544
[26] Chiang D., Lundgren T. (1967) Magnetohydrodynamic flow in rectangular duct with perfectly conducting electrodes. Z. Angew. Math. Phys. 18, 92–105 · doi:10.1007/BF01593897
[27] Singh B., Lal J. (1984) Kantorovich method in magnetohydrodynamic flow problems through channels. Ind. J. Pure Appl. Math. 15, 1048–1063 · Zbl 0568.76109
[28] Rajagopal K.R., Gupta A.S., Na T.Y. (1983) A note on the Falkner–Skan flows of a non-Newtonian fluid. Int. J. Non-Lin. Mech. 18,313–320 · Zbl 0527.76010 · doi:10.1016/0020-7462(83)90028-8
[29] Cortell R. (1994) Similarity solutions for flow and heat transfer in a viscoelastic fluid over a stretching sheet. Int. J. Non-Lin. Mech. 29, 155–16 · Zbl 0795.76010 · doi:10.1016/0020-7462(94)90034-5
[30] Cortell R. (2005a) A note on magnetohydrodynamic flow of a power-law fluid over a stretching sheet. Appl. Math. Comp. 168, 557–5 · Zbl 1081.76059 · doi:10.1016/j.amc.2004.09.046
[31] Vajravelu K., Roper T. (1999) Flow and heat transfer in a second grade fluid over a stretching sheet. Int. J. Non-Lin. Mech. 34, 1031–1036 · Zbl 1006.76005 · doi:10.1016/S0020-7462(98)00073-0
[32] Geeindreau C., Auriault J.L. (2001) Magnetohydrodynamic flows through porous media. C. R. Acad. Sci. 329 (SerieIIb) 445
[33] Cortell R. (2006a) A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet. Int. J. Non-Lin. Mech. 41, 78–85 · Zbl 1160.80302 · doi:10.1016/j.ijnonlinmec.2005.04.008
[34] Cortell R. (2006b) Flow and heat transfer of an electrically conducting fluid of a second grade over a stretching sheet subject to suction and to a transverse magnetic field. Int. J. Heat Mass Transf. 49, 1851–1856 · Zbl 1189.76778 · doi:10.1016/j.ijheatmasstransfer.2005.11.013
[35] Khan S.K. (2006) Heat transfer in a viscoelastic fluid flow over a stretching surface with heat source/sink, suction/blowing and radiation. Int. J. Heat Mass Transf. 49, 628–639 · Zbl 1189.76055 · doi:10.1016/j.ijheatmasstransfer.2005.07.049
[36] Liao S.J. (2003) Beyond perturbation: introduction to homotopy analysis method. Chapman & Hall/CRC, Boca Raton
[37] Liao S.J. (2004) On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 · Zbl 1086.35005 · doi:10.1016/S0096-3003(02)00790-7
[38] Liao S.J. (1999) A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate. J. Fluid Mech. 385, 101–128 · Zbl 0931.76017 · doi:10.1017/S0022112099004292
[39] Liao S.J., Campo A. (2002) Analytic solutions of the temperature distribution in Blasius viscous flow problems. J. Fluid Mech. 453, 411–42 · Zbl 1007.76014 · doi:10.1017/S0022112001007169
[40] Liao S.J. (2003) On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech. 488, 189–212 · Zbl 1063.76671 · doi:10.1017/S0022112003004865
[41] Liao S.J., Cheung K.F. (2003) Homotopy analysis of nonlinear progressive waves in deep water. J. Eng. Math. 45, 105–116 · Zbl 1112.76316 · doi:10.1023/A:1022189509293
[42] Liao S.J., Pop I. (2004) Explicit analytic solution for similarity boundary layer equations. Int. J. Heat Mass Transf. 46, 1855–1860 · Zbl 1029.76050
[43] Ayub M., Rasheed A., Hayat T. (2003) Exact flow of a third grade fluid past a porous plate using homotopy analysis method. Int. J. Eng. Sci. 41, 2091–2103 · Zbl 1211.76076 · doi:10.1016/S0020-7225(03)00207-6
[44] Hayat T., Khan M., Ayub M. (2004) On the explicit analytic solutions of an Oldroyd 6-constant fluid. Int. J. Eng. Sci. 42, 123–135 · Zbl 1211.76009 · doi:10.1016/S0020-7225(03)00281-7
[45] Hayat T., Khan M., Ayub M. (2004) Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field. J. Math. Anal. Appl. 298, 225–244 · Zbl 1067.35074 · doi:10.1016/j.jmaa.2004.05.011
[46] Hayat T., Khan M., Asghar S. (2004) Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid. Acta Mech. 168, 213–232 · Zbl 1063.76108 · doi:10.1007/s00707-004-0085-2
[47] Yang C., Liao S.J. (2006) On the explicit purely analytic solution of Von Karman swirling viscous flow. Comm. Non-Lin. Sci. Numer. Simm. 11, 83–93 · Zbl 1075.35059 · doi:10.1016/j.cnsns.2004.05.006
[48] Liao S.J. (2005) A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Transf. 48, 2529–2539 · Zbl 1189.76142 · doi:10.1016/j.ijheatmasstransfer.2005.01.005
[49] Liao S.J. (2006) An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate. Comm. Non-Lin. Sci. Numer. Simm. 11, 326–339 · Zbl 1078.76022 · doi:10.1016/j.cnsns.2004.09.004
[50] Cheng J., Liao S.J., Pop I. (2005) Analytic series solution for unsteady mixed convection boundary layer flow near the stagnation point on a vertical surface in a porous medium. Transp. Porous Media 61, 365-379 · doi:10.1007/s11242-005-0546-7
[51] Xu H., Liao S.J. (2005) Series solutions of unsteady magnetohydrodynamic flows of non-Newtonian fluids caused by an impulsively stretching plate. J. Non-Newton. Fluid Mech. 129, 46–55 · Zbl 1195.76069 · doi:10.1016/j.jnnfm.2005.05.005
[52] Hayat T., Khan M. (2005) Homotopy solution for a generalized second grade fluid past a porous plate. Non-Lin. Dyn. 42, 395–405 · Zbl 1094.76005 · doi:10.1007/s11071-005-7346-z
[53] Wu W., Liao S.J. (2005) Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos Solitons Fractals 26, 177–185 · Zbl 1071.76009 · doi:10.1016/j.chaos.2004.12.016
[54] Wu Y.Y., Wang C., Liao S.J. (2005) Solving the one loop solution of the Vakhnenko equation by means of the homotopy analysis method. Chaos Solitons Fractals 23, 1733–1740 · Zbl 1069.35060
[55] Hayat T., Khan M., Asghar S. (2004) Magnetohydrodynamic flow of an Oldroyd 6-constant fluid. Appl. Math. Comput. 155, 417–425 · Zbl 1126.76388 · doi:10.1016/S0096-3003(03)00787-2
[56] Sajid M., Hayat T., Asghar S. (2006) On the analytic solution of the steady flow of a fourth grade fluid. Phys. Lett. A. 355, 18–26 · doi:10.1016/j.physleta.2006.01.092
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