zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

76W05Magnetohydrodynamics and electrohydrodynamics
80A20Heat and mass transfer, heat flow
76Z05Physiological flows
76M25Other numerical methods (fluid mechanics)
Full Text: DOI
[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.
[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)
[8] Chang, C. C.; Lundgren, T. S.: Duct flow in magnetohydrodynamics, Zamp 12, 100-114 (1961) · Zbl 0115.21904 · doi:10.1007/BF01601011
[9] Gold, R. R.: Magnetohydrodynamic pipe flow. Part 1, J fluid mech 13, 505-512 (1962) · Zbl 0117.43301 · doi:10.1017/S0022112062000889
[10] Hunt, J. C. R.: Magnetohydrodynamic flow in rectangular ducts, J fluid mech 21, 577-590 (1965) · Zbl 0125.18401 · doi:10.1017/S0022112065000344
[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 · doi:10.1016/j.apnum.2008.05.001
[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)
[13] Shakeri, F.; Dehghan, M.: A finite volume spectral element method for solving magnetohydrodynamic (MHD) equations, Appl numer math 61, 1-23 (2011) · Zbl 05825496
[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 · doi:10.1007/BF01178406
[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 05261399
[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 · doi:10.1016/j.jcp.2006.06.045
[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] Verardi, S. L. Lopes; 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 · doi:10.1137/S003614299732615X
[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 · doi:10.1016/j.cma.2004.06.035
[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 · doi:10.1002/fld.1650110614
[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 · doi:10.1016/j.jcp.2006.03.022
[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 · doi:10.1016/S0045-7825(01)00196-7
[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 · doi:10.1002/fld.1650040307
[28] Tezer-Sezgin, M.; Aydýn, S. Han: Dual reciprocity boundary element method for magnetohydrodynamic flow using radial basis functions, Int J comput fluid dyn 16, No. 1, 88-92 (2002) · Zbl 1082.76580 · doi:10.1080/10618560290004026
[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 05191108
[30] Zhang, M.; Yu, S. T. John; Lin, S. C. Henry; 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 · doi:10.1016/j.jcp.2005.10.006
[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 · doi:10.1016/j.camwa.2010.03.018
[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 · doi:10.1016/j.camwa.2011.01.037
[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, No. 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, No. 2, 342-345 (2011)
[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. · 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 · doi:10.1002/num.20460
[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) · Zbl 1185.65187
[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 · doi:10.1016/j.cpc.2010.03.014
[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 · doi:10.1002/num.20482
[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 · doi:10.1002/mma.1329
[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 · doi:10.1016/j.mcm.2007.09.016
[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 · doi:10.1016/j.cnsns.2008.03.001
[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 · doi:10.1016/j.ijengsci.2007.04.009
[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 · doi:10.1016/j.nonrwa.2008.01.019
[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 · doi:10.1016/j.physleta.2007.10.036
[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 · doi:10.1016/j.physleta.2008.05.006
[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 · doi:10.1016/j.ijheatmasstransfer.2006.08.014
[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 · doi:10.1016/j.ijheatmasstransfer.2007.05.031
[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 · doi:10.1016/j.cnsns.2009.01.030
[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 · doi:10.1016/j.cnsns.2006.07.009
[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 · doi:10.1016/j.camwa.2006.12.036
[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 · doi:10.1016/j.cnsns.2009.09.033
[57] Hayat, T.; Khan, M.; Asghar, S.: Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta mech 168, No. 3-4, 213-232 (2004) · Zbl 1063.76108 · doi:10.1007/s00707-004-0085-2
[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, No. 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, No. 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, No. 5, 548-563 (2010) · Zbl 05706886
[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 · doi:10.1016/j.cnsns.2008.11.011
[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 · doi:10.1016/j.euromechflu.2005.12.003
[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.
[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 · doi:10.1016/j.compfluid.2006.09.005
[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.; B., B. Pal; 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 · doi:10.1142/S0218202598000627
[69] 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)
[71] Walters, K.: Second-order effects in elasticity, plasticity and fluid dynamics, (1964) · Zbl 0136.22901