Skip to main content
SpringerLink
Log in

Fully Developed Flow of Third-Grade Fluid in the Plane Duct with Convection on the Walls

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

In this paper, an appropriate analysis has been performed to study the incompressible fully developed flow of a non-Newtonian third-grade fluid in a plane duct with convection on the walls. The governing equations, continuity, momentum and energy for this problem are reduced to a nonlinear ordinary form. The momentum nonlinear equation and the resulting energy equation with Robin mixed boundary condition are solved with the collocation method. The effects of non-Newtonian parameter on dimensionless velocity profiles, dimensionless temperature profiles and also effect of Biot number on dimensionless temperature profiles have been shown. Finally, the results show that the collocation method was successfully used for solving nonlinear differential equations with Robin mixed condition.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Abbreviations

A :

Pressure gradient (Pa/m)

A n :

Rivlin–Erickson tensors

Bi :

Biot number

b i :

Constant of trial function

c i :

Constant of trial function

D :

Distant of parallel plates (m)

d :

Half distant of parallel plates (m)

h e :

External coefficient convective heat transfer (W/m2 k)

h eff :

The effective heat transfer coefficient (W/m2 k)

I :

Identity tensor

k :

Coefficient thermal conductivity (W/m k)

P :

Pressure (Pa)

p(x):

A function

R(x):

Residual function

t :

Time (s)

T *(η):

Dimensionless temperature

U(η):

Dimensionless velocity

u :

Velocity (m/s)

\(\tilde{u}\) :

Approximate function of u (m/s)

V :

Velocity field (m/s)

W i :

Weighted function

α 1, α 2 :

Material constants

β 1, β 2, β 3 :

Material constants

β :

Non-Newtonian parameter

δ :

Delta function

δ w :

Thickness of plate (m)

k w :

Thermal conductivity of plate (W/m k)

η :

Dimensionless parameter of duct width

μ :

Constant shear adhesion material (Pa s)

ρ :

Density (kg/m3)

τ :

Stress tensor (Pa)

References

  • Abbasi M, Khaki M (2015) MHD Fully developed flow of a third grade fluid in a plane duct. Int J Tech Res Appl 28:77–83

    Google Scholar 

  • Abbasi M, Ahmadian Chashmi A, Petroudi IR, Hoseinzadeh Kh (2014a) Analysis of a fourth grade fluid flow in a channel by application of VIM and HAM. Indian J Sci Res 1(2):389–395

    Google Scholar 

  • Abbasi M, Ganji DD, Petroudi IR, Khaki M (2014b) Comparative analysis of MHD boundary-layer flow of viscoelastic fluid in permeable channel with slip boundaries by using HAM, VIM, HPM. Walailak J Sci Tech 11(7):551–567

    Google Scholar 

  • Aïboud S, Saouli S (2010) Second law analysis of viscoelastic fluid over a stretching sheet subject to a transverse magnetic field with heat and mass transfer. Entropy 12:1867–1884

    Article  Google Scholar 

  • Ali I, Ali Shah R, Islam S, Khan A, Siddiqui AM (2010) Homotopy perturbation solution of second grade fluid through channels with porous walls of different permeability. World Appl Sci J 8(5):536–542

    Google Scholar 

  • Aziz A, Aziz T (2012) MHD flow of a third grade fluid in a porous half space with plate suction or injection: an analytical approach. Appl Math Comput 218:10443–10453

    MathSciNet  MATH  Google Scholar 

  • Eskin D (2015) Modeling non-Newtonian slurry flow in a flat channel with permeable walls. Chem Eng Sci 123:116–124

    Article  Google Scholar 

  • Ganj DD, Sheikholeslami Z, Hosseini M (2013) Non-Newtonian fluid flow in an axisymmetric channel with porous wall. Propuls Power Res 2(4):254–262

    Article  Google Scholar 

  • Ghasemi Moakher P, Abbasi M, Khaki M (2015) New analytical solution of MHD fluid flow of fourth grade fluid through the channel with slip condition via collocation method. Int J Adv Appl Math Mech 2(3):87–94

    MathSciNet  Google Scholar 

  • Gul T, Rehman I, Islam, S Muhammad Altafkhan, Ullah W, Shah Z (2015) Unsteady third order fluid flow with heat transfer between two vertical oscillating plates. J Appl Environ Biol Sci 5(4):72–79

    Google Scholar 

  • Hatami M, Nouri R, Ganji DD (2013) Forced convection analysis for MHD Al2O3–water nano fluid flow over a horizontal plate. J Mol Liq 187:294–301

    Article  Google Scholar 

  • Hayat T, Abbas Z (2008) Heat transfer analysis on the MHD flow of a second grade fluid in channel with porous medium. Chaos Solitons Fractals 38:556–567

    Article  MathSciNet  MATH  Google Scholar 

  • Hayat T, Ahmed N, Sajid M (2006) Analytic solution for MHD flow of a third order fluid in a porous channel. J Comput Appl Math 214:572–582

    Article  MathSciNet  MATH  Google Scholar 

  • Hayat R, Naz R, Sajid M (2010a) On the homotopy solution for Poiseuille flow of a fourth grade fluid. Commun Nonlinear Sci Numer Simulat 15:581–589

    Article  MathSciNet  MATH  Google Scholar 

  • Hayat T, Asghar Z, Asghar S, Mesloub S (2010b) Influence of inclined magnetic field on peristaltic transport of fourth grade fluid in an inclined asymmetric channel. J Taiwan Inst Chem Eng 41:553–563

    Article  Google Scholar 

  • Hayat T, Naz R, Alsaedi A, Rashidi MM (2013) Hydromagnetic rotating flow of third grade fluid. Appl Math Mech 34(12):1481–1494 (English edition)

    Article  MathSciNet  MATH  Google Scholar 

  • Hayata T, Ahmeda N, Sajidb M (2008) Analytic solution for MHD flow of a third order fluid in a porous channel. J Comput Appl Math 214:572–582

    Article  MathSciNet  Google Scholar 

  • Islam S, Bano Z, Siddique I, Siddiqui AM (2011) The optimal solution for the flow of a fourth-grade fluid with partial slip. Comput Math Appl 61:1507–1516

    Article  MathSciNet  MATH  Google Scholar 

  • Kays WM, Crawford ME (1987) Heat and mass transfer. McGraw-Hill, New York

    Google Scholar 

  • Keimanesh M, Rashidi MM, Chamkha Ali J, Jafari R (2011) Study of a third grade non-Newtonian fluid flow between two parallel plates using the multi-step differential transform method. Comput Math Appl 62:2871–2891

    Article  MathSciNet  MATH  Google Scholar 

  • Lopez de Haro M, Cuevas S, Beltran A (2014) Heat transfer and entropy generation in the parallel plate flow of a power-law fluid with asymmetric convective cooling. Energy 66:750–756

    Article  Google Scholar 

  • Mohyuddin MR (2005) On solutions of non-linear equations arising in Rivlin-Ericksen fluids. PhD Thesis. Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

  • Oosthuizen PH, Naylor D (1999) An introduction to convective heat transfer analysis. McGraw-Hill, New York

    MATH  Google Scholar 

  • Rashidi MM, Siddiqui AM, Asadi M (2010) Application of homotopy analysis method to the unsteady squeezing flow of a second-grade fluid between circular plates. Math Probl Eng 2010:1–18

    MathSciNet  MATH  Google Scholar 

  • Rashidi MM, Hayat T, Keimanesh M, Yousefian H (2013) A study on heat transfer in a second-grade fluid through a porous medium with the modified differential transform method, heat transfer—Asian. Research 42(1):31–45

    Google Scholar 

  • Reddy KR, Raju GSS (2014) Fully developed free convection flow of a third grade fluid through a porous medium in a vertical channel. Int J Concept Comput Inf Technol 2:2345–9808

    Google Scholar 

  • Siddiqui AM, Zeb A, Ghori QK, Benharbit AM (2008) Homotopy perturbation method for heat transfer flow of a third grade fluid between parallel plates. Chaos Solitons Fractals 36:182–192

    Article  MathSciNet  MATH  Google Scholar 

  • Siddiqui AM, Hameed M, Siddiqui BM, Ghori QK (2009) Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid. Commun Nonlinear Sci Numer Simulat 15:2388–2399

    Article  MathSciNet  MATH  Google Scholar 

  • Tan W, Masuoka T (2005) Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary. Int J Non-Linear Mech 40:515–522

    Article  MATH  Google Scholar 

  • Vajravelu K, Rollings D (2004) Hydromagnetic flow of a second grade fluid over a stretching sheet. Appl Math Comp 148:783–791

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran

    S. M. Ebrahimi, M. Abbasi & M. Khaki

Authors
  1. S. M. Ebrahimi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Abbasi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Khaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Abbasi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ebrahimi, S.M., Abbasi, M. & Khaki, M. Fully Developed Flow of Third-Grade Fluid in the Plane Duct with Convection on the Walls. Iran J Sci Technol Trans Mech Eng 40, 315–324 (2016). https://doi.org/10.1007/s40997-016-0031-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40997-016-0031-7

Keywords

Search

Navigation