, 43:106 | Cite as

Effect of the aspect ratio on the flow characteristics of magnetohydrodynamic (MHD) third grade fluid flow through a rectangular channel



The magnetohydrodynamic (MHD) flow of a third grade fluid through a rectangular channel, considering the effect of aspect ratio, has been investigated. The flow considered is steady, laminar, incompressible and hydro-dynamically fully developed. The equation, describing the flow, is a highly non-linear partial differential equation (PDE) with remote possibility of having an exact solution and even numerical solution also is very difficult to obtain. A combination of the homotopy perturbation method (HPM) and integral method (IM) has been employed to solve the non-linear PDE which is scarce in open literature. The results of the present study are compared with the results obtained by the least square method (LSM) of the MHD third grade fluid flow through a rectangular channel, without the effect of aspect ratio and are found to be in close agreement. The results indicate that the flow field is significantly affected by the aspect ratio which should be considered for practical applications. In all the available literatures of the third grade fluid flow, the aspect ratio effect is neglected and this simplifying assumption reduces the highly complicated non-linear PDE to a non-linear ordinary differential equation (ODE). The novelty of the subject work lies in the inclusion of the effects of aspect ratio in the governing equation describing the flow of a third grade fluid through a channel and solving this by a combined analytical method (HPM and IM). Further, the effects of the Hartmann number and non-Newtonian third grade fluid parameter on the flow filed are discussed.


Third grade fluid aspect ratio homotopy perturbation method (HPM) integral method (IM) Hartmann number MHD flow 



third grade fluid parameter,

A1, A2, A3

kinematic tensors

a0,a1, b0,b1



aspect ratio of the channel,


applied magnetic field






Hartmann number,


current density respectively

L1, L2

half depth and half width of the channel respectively


non-dimensional pressure gradient,




average velocity through the channel


velocity vector

c1, c2



body force per unit volume


dimensional static pressure,


dimensional axial velocity


dimensionless coordinate in the axial direction


dimensionless coordinate in vertical direction


dimensionless coordinate in the lateral direction


an embedding parameter


dimensional pressure


dimensional axial velocity


dimensional coordinate in the vertical direction


dimensional coordinate in the lateral direction

q0, q1

embedding parameter


0th order solution for the velocity


1st order solution for the velocity

Greek Symbols


density of the fluid,


stress tensor


electrical conductivity


dynamic viscosity the fluid,

α1, α2, β1, β2, β3

material constants of the third grade fluid



The authors are thankful to the reviewers for their valuable comments and suggestions which have improved the paper.


  1. 1.
    Kundu B, Simlandi S and Das P K 2011 Analytical techniques for analysis of fully developed laminar flow through rectangular channels. Heat Mass Transf. 47(10): 1289–1299CrossRefGoogle Scholar
  2. 2.
    Tso C P, Sheeela-Francisca J and Hung Y M 2010 Viscous dissipation effects of power-law fluid flow within parallel plates with constant heat fluxes. J. Non-Newtonian Fluid Mech. 165(6): 625–630CrossRefzbMATHGoogle Scholar
  3. 3.
    Siddiqui A M, Zeb A, Ghori, Q K and Benharbit A M 2008 Homotopy perturbation method for heat transfer flow of a third grade fluid between parallel plates. Chaos Solitons Fract. 36(1): 182–192MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Danish M, Kumar S and Kumar S 2012 Exact analytical solutions for the Poiseuille and Couette-Poiseuille flow of third grade fluid between parallel plates. Commun.Nonlinear Sci. Numer. Simul. 17(3): 1089–1097MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hsiao K L 2011 MHD mixed convection for visco-elastic fluid past a porous wedge. Int. J. Non-Linear Mech. 46(1): 1–8MathSciNetCrossRefGoogle Scholar
  6. 6.
    Siddheshwar P G and Mahabaleswar U S 2005 Effects of radiation and heat source on MHD flow of a visco-elastic liquid and heat transfer over a stretching sheet. Int. J. Non-linear Mech. 40 (6): 807–820CrossRefzbMATHGoogle Scholar
  7. 7.
    Chen C H 2008 Effetcs of magnetic field and suction/injection on convection heat transfer on non-Newtonian power-law fluids past a power-law stretched sheet with surface heat flux. Int. J. Therm. Sci. 47(7):954–961CrossRefGoogle Scholar
  8. 8.
    Wang L, Jian Y, Liu Q, Li F and Chang L 2016 Electromagnetohydrodynamic flow and heat transfer of third grade fluids between two micro-parallel plates. Colloids Surf. A: Physicochem. Eng. Asp. 494(5): 87–94Google Scholar
  9. 9.
    He J H 2006 Homotopy perturbation method for solving boundary value problems. Phys. Lett. A. 350(1–2): 87–88MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    He J H 2005 Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fract. 26(3): 695–700CrossRefzbMATHGoogle Scholar
  11. 11.
    Bera P K and Sil T 2012 Homotopy perturbation method in quantum mechanical problems, Appl. Math. Comput. 219(6): 3272–3278MathSciNetzbMATHGoogle Scholar
  12. 12.
    Elsayed A F 2013 Comparison between variational method and Homotopy perturbation method for thermal diffusion and diffusion thermo effects of thixotropic fluid through biological tissues with laser radiation existence. Appl. Math. Model. 37(6): 3660–3673MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yun Y and Temuer C 2015 Application of the homotopy perturbation method for the large deglection problem of a circular plate. Appl. Math. Model. 39(3–4):1308–1316MathSciNetCrossRefGoogle Scholar
  14. 14.
    Golbabai A and Javidi A 2007 Application of homotopy perturbation method for solving eight-order boundary value problems. Appl. Math.Comput. 191(2): 334–346MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hatami M and Ganji D D 2013 Heat transfer and flow analysis for SA-TiO2 non-Newtonian nano fluid passing through the porous media between two coaxial cylinders. J. Mol. Liq. 188(12): 155–161CrossRefGoogle Scholar
  16. 16.
    Hatami M, Sheikholeslami M and Ganji D D 2014 Laminar flow and heat transfer of nanofluid between contracting and rotating disks by least square method. Powder Technol. 253(2):769–779CrossRefGoogle Scholar
  17. 17.
    Khan M, Munawar S, and Abbasbandy S 2010 Steady flow and heat transfer of a Sisko fluid in annular pipe. Int. J. Heat Mass Trans. 53(7–8):1290–1297CrossRefzbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.School of Mechanical EngineeringKIIT Deemed to be UniversityBhubaneswarIndia
  2. 2.Department of Mechanical EngineeringIndian Institute of TechnologyDhanbadIndia

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