Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall. (English) Zbl 1316.76111
Summary: We analyze the flow and heat transfer characteristics of a magnetohydrodynamic (MHD) viscoelastic fluid in a parallel plate channel with a stretching wall. Homotopy analysis method (HAM) is used to obtain analytical solutions of the governing nonlinear differential equations. The analytical solutions are obtained in the form of infinite series and the convergence of the series solution is discussed explicitly. The obtained results are presented through graphs for several sets of values of the parameters, and the salient features of the solutions are analyzed. A comparison of our HAM results (for a special case of the study) with the available results in the literature (obtained by other methods) shows that our results are accurate for a wide range of parameters. Further, we point that our analysis finds application to the study of the haemodynamic flow of blood in the cardiovascular system subject to external magnetic field.
MSC:
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 76Z05 | Physiological flows |
| 76M25 | Other numerical methods (fluid mechanics) (MSC2010) |
References:
| [1] | Alfvén, H., Existence of electromagnetic-hydrodynamic waves, Nature, 150, 405-406 (1942) |
| [2] | Hartmann, J.; Hg-Dynamics, I., Theory of the laminar flow of an electrically conducting liquid in a homogeneous magnetic field, K. Dan. Vidensk. Selsk. Mat.-Fys. Medd., 15, 1-27 (1937) |
| [3] | Smith, P., Some asymptotic extremum principles for magnetohydrodynamic pipe flow, Appl Sci Res, 24, 452-466 (1971) · Zbl 0235.76053 |
| [4] | Branover H, Gershon P. MHD turbulence study, Ben-Gurion University, Rept BGUN-RDA-100-76; 1976.; Branover H, Gershon P. MHD turbulence study, Ben-Gurion University, Rept BGUN-RDA-100-76; 1976. |
| [5] | Holroyd, R. J., An experimental study of the effects of wall conductivity non-uniform magnetic field variable-area ducts on liquid metal flow at high Hartmann number. Part 1: Ducts with non-conducting walls, J Fluid Mech, 93, 609-630 (1979) |
| [6] | Holroyd, R. J., MHD flow in a rectangular duct with pairs of conducting and non-conducting walls in the presence of a non-uniform magnetic field, J Fluid Mech, 96, 335-353 (1980) |
| [7] | Davidson, J.; Thess, A.; Davidson, P. A., Magnetohydrodynamics (2002), Springer: Springer Vienna · Zbl 1030.00041 |
| [8] | Chang, C. C.; Lundgren, T. S., Duct flow in magnetohydrodynamics, ZAMP, 12, 100-114 (1961) · Zbl 0115.21904 |
| [9] | Gold, R. R., Magnetohydrodynamic pipe flow. Part 1, J Fluid Mech, 13, 505-512 (1962) · Zbl 0117.43301 |
| [10] | Hunt, J. C.R., Magnetohydrodynamic flow in rectangular ducts, J Fluid Mech, 21, 577-590 (1965) · Zbl 0125.18401 |
| [11] | Dehghan, M.; Mirzaei, D., Meshless local Petrov-Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity, Appl Numer Math, 59, 1043-1058 (2009) · Zbl 1159.76034 |
| [12] | Dehghan, M.; Mirzaei, D., Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes, Comput Phys Commun, 180, 1458-1466 (2009) · Zbl 07872387 |
| [13] | Shakeri, F.; Dehghan, M., A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations, Appl Numer Math, 61, 1-23 (2011) · Zbl 1427.76276 |
| [14] | Demendy, Z.; Nagy, T.; Hungary, M.-E., A new algorithm for solution of equations of MHD channel flows at moderate Hartmann numbers, Acta Mech, 123, 135-149 (1997) · Zbl 0902.76058 |
| [15] | Bozkaya, N.; Tezer-Sezgin, M., Time-domain BEM solution of convection-diffusion-type MHD equations, Int J Numer Methods Fluids, 56, 1969-1991 (2008) · Zbl 1388.76191 |
| [16] | Fuchs, F. G.; Mishra, S.; Risebro, N. H., Splitting based finite volume schemes for ideal MHD equations, J Comput Phys, 228, 641-660 (2009) · Zbl 1259.76021 |
| [17] | Gardner, L. R.T.; Gardner, G. A., A two-dimensional bi-cubic B-spline finite element used in a study of MHD duct flow, Comput Methods Appl Mech Eng, 124, 365-375 (1995) |
| [18] | Guermond, J. L.; Laguerre, R.; Léorat, J.; Nore, C., An interior penalty Galerkin method for the MHD equations in heterogeneous domains, J Comput Phys, 221, 349-369 (2007) · Zbl 1108.76040 |
| [19] | Han, J.; Tang, H., An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics, J Comput Phys, 220, 791-812 (2007) · Zbl 1235.76084 |
| [20] | Lopes Verardi, S. L.; Machado, J. M.; Shiyou, Y., The application of interpolating MLS approximations to the analysis of MHD flows, Source Finite Elem Anal Des, 39, 1173-1187 (2003) |
| [21] | Meir, A. J.; Schmidt, P. G., Analysis and numerical approximation of a stationary MHD flow problem with nonideal boundary, SIAM J Numer Anal, 36, 1304-1332 (1999) · Zbl 0948.76091 |
| [22] | Neslitürk, A. I.; Tezer-Sezgin, M., The finite element method for MHD flow at high Hartmann numbers, Comput Methods Appl Mech Eng, 194, 1201-1224 (2005) · Zbl 1091.76036 |
| [23] | Ramos, J. I.; Winowich, N. S., Finite difference and finite element methods for MHD channel flows, Int J Numer Methods Fluids, 11, 907-934 (1990) · Zbl 0704.76066 |
| [24] | Reynolds, D. R.; Samtaney, R.; Woodward, C. S., A fully implicit numerical method for single-fluid resistive magnetohydrodynamics, J Comput Phys, 219, 144-162 (2006) · Zbl 1103.76036 |
| [25] | Salah, N. B.; Soulaimani, A.; Habashi, W. G., A finite element method for magnetohydrodynamics, Comput Methods Appl Mech Eng, 190, 5867-5892 (2001) · Zbl 1044.76030 |
| [26] | Singh, B.; Lal, J., FEM in MHD channel flow problems, Int J Numer Methods Eng, 18, 1104-1111 (1982) · Zbl 0489.76119 |
| [27] | Singh, B.; Lal, J., FEM for unsteady MHD flow through pipes with arbitrary wall conductivity, Int J Numer Methods Fluids, 4, 291-302 (1984) · Zbl 0547.76119 |
| [28] | Tezer-Sezgin, M.; Han Aydýn, S., Dual reciprocity boundary element method for magnetohydrodynamic flow using radial basis functions, Int J Comput Fluid Dyn, 16, 1, 88-92 (2002) |
| [29] | Yee, H. C.; Sjögreen, B., Development of low dissipative high order filter schemes for multiscale Navier-Stokes/MHD systems, J Comput Phys, 225, 910-934 (2007) · Zbl 1343.76053 |
| [30] | Zhang, M.; John Yu, S. T.; Henry Lin, S. C.; Chang, S. C.; Blankson, I., Solving the MHD equations by the space-time conservation element and solution element method, J Comput Phys, 214, 5 99-617 (2006) · Zbl 1136.76399 |
| [31] | Raftari, B.; Yildirim, A., The application of homotopy perturbation method for MHD flows of UCM fluids above porous stretching sheets, Comput Math Appl, 59, 3328-3337 (2010) · Zbl 1198.65148 |
| [32] | Raftari, B.; Yildirim, A., Series solution of a nonlinear ODE arising in magnetohydrodynamic by HPM-Padé technique, Comput Math Appl, 61, 1676-1681 (2011) · Zbl 1217.76055 |
| [33] | Raftari, B.; Yildirim, A., A new modified homotopy perturbation method with two free auxiliary parameters for solving MHD viscous flow due to a shrinking sheet, Eng Comput, 28, 5, 528-539 (2011) · Zbl 1284.76320 |
| [34] | Raftari, B.; Mohyud-Din, S. T.; Yildirim, A., Solution to the MHD flow over a non-linear stretching sheet by homotopy perturbation method, Sci China Phys Mech Astron, 54, 2, 342-345 (2011), February |
| [35] | Raftari B, Yildirim A, Application of Homotopy perturbation method for heat and mass transfer in MHD flow. J Thermophy Heat Transfer 2011, in press.; Raftari B, Yildirim A, Application of Homotopy perturbation method for heat and mass transfer in MHD flow. J Thermophy Heat Transfer 2011, in press. · Zbl 1284.76320 |
| [36] | Dehghan, M.; Manafian, J.; Saadatmandi, A., Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer Methods Partial Differ Equat, 26, 448-479 (2010) · Zbl 1185.65187 |
| [37] | Dehghan, M.; Shakeri, F., Use of He’s homotopy perturbation method for solving a partial differential equation arising in modeling of flow in porous media, J Porous Media, 11, 765-778 (2008) |
| [38] | Dehghan, M.; Manafian, J.; Saadatmandi, A., The solution of the linear fractional partial differential equations using the homotopy analysis method, Z Naturforsch, 65a, 935-949 (2010) |
| [39] | Dehghan, M.; Salehi, R., Solution of a nonlinear time-delay model in biology via semi-analytical approaches, Comput Phys Commun, 181, 1255-1265 (2010) · Zbl 1219.65062 |
| [40] | Dehghan, M.; Salehi, R., A seminumeric approach for solution of the Eikonal partial differential equation and its applications, Numer Methods Partial Differ Equat, 26, 702-722 (2010) · Zbl 1189.65237 |
| [41] | Dehghan, M.; Manafian, J.; Saadatmandi, A., Application of semi-analytic methods for the Fitzhugh-Nagumo equation which models the transmission of nerve impulses, Math Methods Appl Sci, 33, 1384-1398 (2010) · Zbl 1196.35025 |
| [42] | Shakeri, F.; Dehghan, M., Solution of delay differential equations via a homotopy perturbation method, Math Comput Model, 48, 486-498 (2008) · Zbl 1145.34353 |
| [43] | Salehi, R.; Dehghan, M., The use of homotopy analysis method to solve the time-dependent nonlinear Eikonal partial differential equation, Z Naturforsch, 66a, 259-271 (2011) |
| [44] | Alizadeh-Pahlavan, A.; Sadeghy, K., On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets, Commun Nonlinear Sci Numer Simul, 14, 1355-1365 (2009) · Zbl 1221.76213 |
| [45] | Alizadeh-Pahlavan, A.; Aliakbar, V.; Vakili-Farahani, F.; Sadeghy, K., MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method, Commun Nonlinear Sci Numer Simul, 14, 473-488 (2009) |
| [46] | Aliakbar, V.; Alizadeh-Pahlavan, A.; Sadeghy, K., The influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets, Commun Nonlinear Sci Numer Simul, 14, 779-794 (2009) |
| [47] | Hayat, T.; Sajid, M., Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid, Int J Eng Sci, 45, 393-401 (2007) · Zbl 1213.76137 |
| [48] | Hayat, T.; Javed, T.; Abbas, Z., MHD flow of a micropolar fluid near a stagnation-point towards a non-linear stretching surface, Nonlinear Anal Real World Appl, 10, 1514-1526 (2009) · Zbl 1160.76055 |
| [49] | Hayat, T.; Fetecau, C.; Sajid, M., On MHD transient flow of a Maxwell fluid in a porous medium and rotating frame, Phys Lett A, 372, 1639-1644 (2008) · Zbl 1217.76086 |
| [50] | Hayat, T.; Abbas, Z.; Ali, N., MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species, Phys Lett A, 372, 4698-4704 (2008) · Zbl 1221.76031 |
| [51] | Hayat, T.; Abbas, Z.; Sajid, M.; Asghar, S., The influence of thermal radiation on MHD flow of a second grade fluid, Int J Heat Mass Transfer, 50, 931-941 (2007) · Zbl 1124.80325 |
| [52] | Abbas, Z.; Hayat, T., Radiation effects on MHD flow in a porous space, Int J Heat Mass Transfer, 51, 1024-1033 (2008) · Zbl 1141.76069 |
| [53] | Abbasbandy, S.; Hayat, T., Solution of the MHD Falkner-Skan flow by homotopy analysis method, Commun Nonlinear Sci Numer Simul, 14, 3591-3598 (2009) · Zbl 1221.76133 |
| [54] | Hayat, T.; Sajid, M.; Ayub, M., On explicit analytic solution for MHD pipe flow of a fourth grade fluid, Commun Nonlinear Sci Numer Simul, 13, 745-751 (2008) · Zbl 1221.76221 |
| [55] | Hayat, T.; Ahmed, Naveed; Sajid, M.; Asghar, S., On the MHD flow of a second grade fluid in a porous channel, Comput Math Appl, 54, 407-414 (2007) · Zbl 1123.76072 |
| [56] | Hayat, T.; Sajjad, R.; Asghar, S., Series solution for MHD channel flow of a Jeffery fluid, Commun Nonlinear Sci Numer Simulat, 15, 2400-2406 (2010) · Zbl 1222.76076 |
| [57] | Hayat, T.; Khan, M.; Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta Mech, 168, 3-4, 213-232 (2004) · Zbl 1063.76108 |
| [58] | Hayat, T.; Qasim, M.; Mesloub, S., MHD flow and heat transfer over permeable stretching sheet with slip conditions, Int J Numer Methods Fluids, 66, 8, 963-975 (2011) · Zbl 1285.76044 |
| [59] | Sweet, E.; Vajravelu, K.; Van Gorder, R. A., Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator, Central Euro J Phys, 9, 1, 167-175 (2011) |
| [60] | Joneidi, A. A.; Domairry, G.; Babaelahi, M.; Mozaffari, M., Analytical treatment on magnetohydrodynamic (MHD) flow and heat transfer due to a stretching hollow cylinder, Int J Numer Methods Fluids, 63, 5, 548-563 (2010) · Zbl 1423.76339 |
| [61] | Kumari, M.; Nath, G., Analytical solution of unsteady three-dimensional MHD boundary layer flow and heat transfer due to impulsively stretched plane surface, Commun Nonlinear Sci Numer Simulat, 14, 3339-3350 (2009) · Zbl 1221.76226 |
| [62] | Xu, H.; Liao, S. J.; Pop, I., Series solutions of unsteady three-dimensional MHD flow and heat transfer in the boundary layer over an impulsively stretching plate, Euro J Mech B Fluids, 26, 15-27 (2007) · Zbl 1105.76061 |
| [63] | Misra, JC, Pal B, Gupta AS. Hall effects on magnetohydrodynamic flow and heat transfer over a stretching surface. In: Malik SK (ed.), Mathematics and its application to industry and new engineering area. New Delhi: INSA; 2001. p. 49-62.; Misra, JC, Pal B, Gupta AS. Hall effects on magnetohydrodynamic flow and heat transfer over a stretching surface. In: Malik SK (ed.), Mathematics and its application to industry and new engineering area. New Delhi: INSA; 2001. p. 49-62. |
| [64] | Misra, J. C.; Chakravarty, S., Flow in arteries in the presence of stenosis, J Biomech, 19, 907-918 (1986) |
| [65] | Misra, J. C.; Patra, M. K.; Misra, S. C., A non-Newtonian fluid model for blood flow through arteries under stenotic conditions, J Biomech, 26, 1129-1141 (1993) |
| [66] | Misra, J. C.; Shit, G. C.; Rath, H. J., Flow and heat transfer of a MHD viscoelastic fluid in a channel with stretching walls: some applications to haemodynamics, Comput Fluids, 37, 1-11 (2008) · Zbl 1194.76287 |
| [67] | Misra, J. C.; Pal, B., Hydromagnetic flow of a viscoelastic fluid in a parallel plate channel with stretching walls, Ind J Math, 41, 231-247 (1999) · Zbl 1137.76830 |
| [68] | Misra, J. C.; Pal B, B.; Gupta, A. S., Hydromagnetic flow of a second-grade fluid in a channel-some applications to physiological systems, Math Model Methods Appl Sci, 8, 1323-1342 (1998) · Zbl 0963.76610 |
| [69] | Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD Thesis, Shanghai Jiao Tong University; 1992.; Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD Thesis, Shanghai Jiao Tong University; 1992. |
| [70] | Liao, S. J., Beyond perturbation: introduction to homotopy analysis method (2003), Chapman & Hall/CRC Press: Chapman & Hall/CRC Press Boca Raton |
| [71] | Walters, K., Second-order effects in elasticity, plasticity and fluid dynamics (1964), Pergamon |
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.
