OHAM and FEM solutions of concentric n-layer flows of incompressible third-grade fluids in a horizontal cylindrical pipe

  • S. Iqbal
  • I. SiddiqueEmail author
  • A. M. Siddiqui


This paper examines the concentric n-layer flows for incompressible third-grade fluids through a horizontal cylindrical pipe. Such flows of multilayer fluids have a wide variety of applications in petroleum and chemical industries. The approximate solutions for velocity fields of multilayer flows are presented by the application of optimal homotopy asymptotic method and Galerkin’s finite element method. Further, it is shown that a unique maximum velocity always exists in the core of the pipe for any number of fluid layers. The effects of suitable parameters on the velocity profiles are presented graphically for multilayer flows.


n-layer flows Concentric flow Third-grade fluids OHAM and FEM Approximate solutions 



The authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper. Also the authors are highly thankful to the University of Management and Technology, Lahore, Pakistan, for supporting and facilitating.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.


  1. 1.
    Packham BA, Shail R (1971) Stratified laminar flow of two immiscible fluids. Math Proc Camb Philos Soc 69:443–448CrossRefGoogle Scholar
  2. 2.
    Shail R (1973) On laminar tow-phase flow in magnetohydrodynamic. Int J Eng Sci 11:1103–1108CrossRefGoogle Scholar
  3. 3.
    Lohrasbi J, Sahai V (1988) Magnetohydrodynamic heat transfer in two phase flow between parallel plates. Appl Sci Res 45:53–66CrossRefGoogle Scholar
  4. 4.
    Setayesh A, Sahai V (1990) Heat transfer in developing MHD Poiseuille flow and variable transport properties. Int J Heat Mass Transf 33(8):1711–1720CrossRefGoogle Scholar
  5. 5.
    Malashetty MS, Umavathi JC (1997) Two-phase magnetohydrodynamic flow and heat transfer in an inclined channel. Int J Multiph Flow 23:545–560CrossRefGoogle Scholar
  6. 6.
    Malashetty MS, Leela V (1992) Magnetohydrodynamic heat transfer in two phase flow. Int J Eng Sci 30:371–377CrossRefGoogle Scholar
  7. 7.
    Stamenković MZ (2012) MHD flow and heat transfer of two immiscible fluids with induced magnetic fields effects. Therm Sci 16(2):33–336MathSciNetGoogle Scholar
  8. 8.
    Power H, Vllegas M (1991) Weakly nonlinear instability of the flow of two immiscible liquids with different viscosities in a pipe. Fluid Dyn Res 7:215–228CrossRefGoogle Scholar
  9. 9.
    Dong L, Johnson D (2005) Experimental and theoretical study of the interfacial instability between two shear fluids in a channel Couette flow. Int J Heat Fluid Flow 26:133–140CrossRefGoogle Scholar
  10. 10.
    Pinarbasi A (2002) Interface stabilization in two-layer channel flow by heating or cooling. Eur J Mech B/Fluids 21:225–236MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shankar V (2004) Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer. J Non-Newton Fluid Mech 117:163–182CrossRefGoogle Scholar
  12. 12.
    Wong WT, Jeng CC (1987) The Stability of two concentric non-Newtonian fluids in circular pipe flow. J Non-Newton Fluid Mech 22:359–380CrossRefGoogle Scholar
  13. 13.
    Waters ND (1983) The stability of two stratified “power-law” liquids in Couette flow. J Non-Newton Fluid Mech 12:85–94CrossRefGoogle Scholar
  14. 14.
    Usha S, Ramchandra Rao A (1997) Peristaltic transport of two-layered power law fluids. J Biomech Eng 119:483–488CrossRefGoogle Scholar
  15. 15.
    Gul T, Shah RA, Islam S, Ullah M, Khan MA, Zaman A, Haq Z (2014) Exact solution of the two thin film non-Newtonian immiscible fluids on a vertical belt. J Basic Appl Sci Res 4(6):283–288Google Scholar
  16. 16.
    Brauner N, Maron DM (1992) Flow pattern transitions in two-phase liquid–liquid flow in horizontal tubes. Int J Multiph Flow 18:123–140CrossRefGoogle Scholar
  17. 17.
    Leib M, Fink M, Hasson D (1977) Heat transfer in vertical annular laminar flow of two immiscible liquids. Int J Multiph Flow 3:533–549CrossRefGoogle Scholar
  18. 18.
    Siddiqui AM, Azim QA, Rana MA (2010) On exact solutions of concentric n-layer flows of viscous fluids in a pipe. Nonlinear Sci Lett A 1(1):93–102Google Scholar
  19. 19.
    He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys 20(10):1141–1199MathSciNetCrossRefGoogle Scholar
  20. 20.
    He JH (1999) Homotopy perturbation techniques. Comput Methods Appl Mech Eng 178(4):257–262MathSciNetCrossRefGoogle Scholar
  21. 21.
    He JH (2000) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int J Nonlinear Mech 35(1):37–43MathSciNetCrossRefGoogle Scholar
  22. 22.
    He JH (2004) Comparison of homotopy perturbation and homotopy analysis method. Appl Math Comput 156(2):527–539MathSciNetzbMATHGoogle Scholar
  23. 23.
    He JH (2008) Recent development of the homotopy perturbation method. Topol Methods Nonlinear Anal 31(2):205–209MathSciNetzbMATHGoogle Scholar
  24. 24.
    Marinca V, Herisanu N (2008) Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int Commun Heat Mass Transf 35(6):710–715CrossRefGoogle Scholar
  25. 25.
    Marinca V, Herisanu N, Bota C, Marinca BV (2009) An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Appl Math Lett 22(2):245–251MathSciNetCrossRefGoogle Scholar
  26. 26.
    Marinca V, Herisanu N (2014) The optimal homotopy asymptotic method for solving Blasius equation. Appl Math Comput 231:134–139MathSciNetzbMATHGoogle Scholar
  27. 27.
    Marinca V, Herisanu N (2010) Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method. J Sound Vib 329(9):1450–1459CrossRefGoogle Scholar
  28. 28.
    Marinca V, Herisanu N (2014) On the flow of a Walters-type B’ viscoelastic fluid in a vertical channel with porous wall. Int J Heat Mass Transf 79:146–165CrossRefGoogle Scholar
  29. 29.
    Iqbal S, Idrees M, Siddiqui AM, Ansari AR (2010) Some solutions of the linear and nonlinear Klein–Gordon equations using the optimal homotopy asymptotic method. Appl Math Comput 216(10):2898–2909MathSciNetzbMATHGoogle Scholar
  30. 30.
    Iqbal S, Javid A (2011) Application of optimal homotopy asymptotic method for the analytic solution of singular Lane–Emden type equation. Appl Math Comput 217:7753–7761MathSciNetzbMATHGoogle Scholar
  31. 31.
    Iqbal S, Hashmi MS, Khan N, Javid A (2012) Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind. Appl Math Comput 218:10982–10989MathSciNetzbMATHGoogle Scholar
  32. 32.
    Zienkiewicz OC (1977) The finite element method. McGraw Hill, LondonzbMATHGoogle Scholar
  33. 33.
    Iqbal S, Mirza AM, Tirmizi IA (2010) Galerkin’s finite element formulation of the second-order boundary-value problems. Int J Comput Math 87(9):2032–2042MathSciNetCrossRefGoogle Scholar
  34. 34.
    Iqbal S, Siddique I (2012) Galerkin’s finite element method for solving special forth-order boundary-value problem. Sci Int 24(4):333–336Google Scholar
  35. 35.
    Iqbal S, Abualnaja KM (2014) Galerkin’s finite element formulation for thin film flow of a third grade fluid down an inclined plane. Sci Int 26(4):1403–1405Google Scholar
  36. 36.
    Iqbal S (2011) Galerkin’s finite element formulation of the system of fourth-order boundary-value problems. Numer Methods Partial Differ Equ 27(6):1551–1560MathSciNetCrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Informatics and Systems, School of Systems and TechnologyUniversity of Management and TechnologyLahorePakistan
  2. 2.Department of MathematicsUniversity of Management and TechnologyLahorePakistan
  3. 3.Department of MathematicsPennsylvania State UniversityYorkUSA

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