Thermophysical properties of unsteady 3D flow of magneto Carreau fluid in the presence of chemical species: a numerical approach

  • M. Khan
  • M. Irfan
  • W. A. Khan
Technical Paper


In this article, we establish a mathematical relation for unsteady 3D flow of a magneto-Carreau liquid over a bidirectional stretched surface. The impact of convective heat transfer for magneto-Carreau liquid is investigated in the presence of non-linear thermal radiation and heat absorption/generation. Additionally, in this analysis, the proposed model of heterogeneous–homogeneous processes with equivalent diffusivities for reactant and autocatalysis is considered. The modeled boundary layer equations are reduced to a system of nonlinear ordinary differential equations using the appropriate transformation. The resulting equations are then solved by utilizing two different techniques, namely the bvp4c function in Matlab and homotopy analysis method (HAM). The numerical data for the velocity, temperature and concentration fields are graphically sketched and characteristics of the relevant parameters are deliberated in detail. Moreover, the velocity gradients and the rate of heat transfer at the stretched surface for different values of the pertaining parameters are given in tabulated form. It is observed that the temperature profile enhance for higher values of magnetic parameter M and heat generation parameter \((\delta >0)\), whereas it decline for augmented values of heat absorption \((\delta <0)\) parameter. In addition, the concentration profile decline for increasing values of homogeneous reaction parameter \(k_{1}\)and unsteadiness parameter S. To see the validity of the numerical computations, we compared these results of the numerical techniques bvp4c with an efficient analytical method, namely the homotopy analysis method (HAM) and perceived an outstanding correlation between these techniques.


3D-Carreau fluid MHD Thermal radiation Heterogeneous–Homogenous reactions Heat source/sink. 


  1. 1.
    Nandeppanavar MM, Vajravelu K, Abel M, Ravi S, Jyoti H (2012) Heat transfer in a liquid film over an unsteady stretching sheet. Int J Heat Mass Transf 55:1316–1324CrossRefMATHGoogle Scholar
  2. 2.
    Haq RU, Khan ZH, Khan WA (2014) Thermophysical effects of carbon nanotubes on MHD flow over a stretching surface. Phys E 63:215–222CrossRefGoogle Scholar
  3. 3.
    Hayat T, Khan MI, Waqas M, Alsaedi A, Khan MI (2017) Radiative flow of micropolar nanofluid accounting thermophoresis and Brownian moment. Int J Hydro Energy 42:16821–16833CrossRefGoogle Scholar
  4. 4.
    Anwar MS, Rasheed A (2017) A microscopic study of MHD fractional inertial flow through Forchheimer medium. Chin J Phys.
  5. 5.
    Rehman FU, Nadeem S, Haq RU (2017) Heat transfer analysis for three-dimensional stagnation-point flow over an exponentially stretching surface. Chin J Phys 55:1552–1560CrossRefGoogle Scholar
  6. 6.
    Anwar MS, Rasheed A (2017) Heat transfer at microscopic level in a MHD fractional inertial flow confined between non-isothermal boundaries. Eur Phy J Plus 132:305. CrossRefGoogle Scholar
  7. 7.
    Anwar MS, Rasheed A (2017) Simulations of a fractional rate type nanofluid flow with non-integer Caputo time derivatives. Comput Math Appl.
  8. 8.
    Carreau PJ (1972) Rheological equations from molecular network theories. Trans Soc Rheol 116:99–127CrossRefGoogle Scholar
  9. 9.
    Hayat T, Waqas M, Shehzad SA, Alsaedi A (2016) Stretched flow of Carreau nanofluid with convective boundary condition. Pramana J Phys 86:3–17CrossRefGoogle Scholar
  10. 10.
    Hayat T, Khan MI, Waqas M, Alsaedi A (2017) Mathematical modeling of non-Newtonian fluid with chemical aspects: a new formulation and results by numerical technique. Colloids Surf A Physicochem Eng Asp 518:263–272CrossRefGoogle Scholar
  11. 11.
    Raju CSK, Sandeep N, Saleem S (2016) Effects of induced magnetic field and homogeneous-heterogeneous reactions on stagnation flow of a Casson fluid. Eng Sci Tech Int J 19:875–887CrossRefGoogle Scholar
  12. 12.
    Khan MI, Waqas M, Hayat T, Alsaedi A (2017) A comparative study of Casson fluid with homogeneous–heterogeneous reactions. J Colloid Interface Sci 498:85–90CrossRefGoogle Scholar
  13. 13.
    Khan M, Ahmad L, Khan WA, Alshomrani AS, Alzahrani AK, Alghamdi MS (2017) A 3D Sisko fluid flow with Cattaneo-Christov heat flux model and heterogeneous–homogeneous reactions: a numerical study. J Mol Liq 238:19–26CrossRefGoogle Scholar
  14. 14.
    Hayat T, Rashid M, Imtiaz M, Alsaedi A (2017) Nanofluid flow due to rotating disk with variable thickness and homogeneous–heterogeneous reactions. Int J Heat Mass Transf 113:96–105CrossRefGoogle Scholar
  15. 15.
    Khan MI, Waqas M, Hayat T, Khan MI, Alsaedi A (2017) Numerical simulation of nonlinear thermal radiation and homogeneous–heterogeneous reactions in convective flow by a variable thicked surface. J Mol Liq 246:259–267CrossRefGoogle Scholar
  16. 16.
    Khan WA, Irfan M, Khan M, Alshomrani AS, Alzahrani AK, Alghamdi MS (2017) Impact of chemical processes on magneto nanoparticle for the generalized Burgers fluid. J Mol Liq 234:201–208CrossRefGoogle Scholar
  17. 17.
    Hang Xu (2017) A homogeneous–heterogeneous reaction model for heat fluid flow in the stagnation region of a plane surfac. Int Commun Heat Mass Transf 87:112–117CrossRefGoogle Scholar
  18. 18.
    Qayyum S, Khan MI, Hayat T, Alsaedi A (2017) A framework for nonlinear thermal radiation and homogeneous–heterogeneous reactions flow based on silver-water and copper-water nanoparticles: a numerical model for probable error. Res Phys 7:1907–1914Google Scholar
  19. 19.
    Hayat T, Sajjad R, Ellahi R, Alsaedi A, Muhammad T (2017) Homogeneous–heterogeneous reactions in MHD flow of micropolar fluid by a curved stretching surface. J Mol Liq 240:209–220CrossRefGoogle Scholar
  20. 20.
    Merkin JH (1996) A model for isothermal homogeneous–heterogeneous reactions in boundarylayer flow. Math Comput Model 24:125–136MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Khan M, Irfan M, Khan WA, Alshomrani AS (2017) A new modeling for 3D Carreau fluid flow considering nonlinear thermal radiation. Res Phys 7:2692–2704Google Scholar
  22. 22.
    Sohail A, Khan WA, Khan M, Shah SIA (2017) Consequences of non-Fourier’s heat conduction relation and chemical processes for viscoelastic liquid. Res Phys 7:3281–3286Google Scholar
  23. 23.
    Hayat T, Rashid M, Alsaedi A (2017) MHD convective flow of magnetite-Fe3O4 nanoparticles by curved stretching sheet. Res Phys 7:3107–3115Google Scholar
  24. 24.
    Khan M, Irfan M, Khan WA (2017) Impact of nonlinear thermal radiation and gyrotactic microorganisms on the Magneto-Burgers nanofluid. Int J Mech Sci 130:375–382CrossRefGoogle Scholar
  25. 25.
    Khan WA, Irfan M, Khan M (2017) An improved heat conduction and mass diffusion models for rotating flow of an Oldroyd-B fluid. Res Phys.
  26. 26.
    Waqas M, Hayat T, Shehzad SA, Alsaedi A (2017) Transport of magnetohydrodynamic nanomaterial in a stratified medium considering gyrotactic microorganisms. Phys B Cond Matter.
  27. 27.
    Khan M, Irfan M, Khan WA, Ahmad L (2017) Modeling and simulation for 3D magneto Eyring–Powell nanomaterial subject to nonlinear thermal radiation and convective heating. Res Phys 7:1899–1906Google Scholar
  28. 28.
    Khan M, Irfan M, Khan WA (2017) Numerical assessment of solar energy aspects on 3D magneto-Carreau nanofluid: a revised proposed relation. Int J Hydro Energy 42:22054–22065CrossRefGoogle Scholar
  29. 29.
    Sharidan S, Mahmood T, Pop I (2006) Similarity solutions for the unsteady boundary layer flow and heat transfer due to a stretching sheet. Int J Appl Mech Eng 11:647–654MATHGoogle Scholar
  30. 30.
    Chamkha AJ, Aly AM, Mansour MA (2010) Similarity solution for unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and chemical reaction effects. Chem Eng Commun 197:846–858CrossRefGoogle Scholar
  31. 31.
    Irfan M, Khan M, Khan WA (2017) Numerical analysis of unsteady 3D flow of Carreau nanofluid with variable thermal conductivity and heat source/sink. Res Phys 7:3315–3324Google Scholar
  32. 32.
    Liu IC, Anderson HI (2008) Heat transfer over a bidirectional stretching sheet with variable thermal conditions. Int J Heat Mass Transf 51:4018–4024CrossRefMATHGoogle Scholar
  33. 33.
    Munir A, Shahzad A, Khan M (2015) Convective flow of Sisko fluid over a bidirectional stretching surface. PLOS One 10:e0130342CrossRefGoogle Scholar
  34. 34.
    Khan WA, Pop I (2010) Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transf 53:2477–2483CrossRefMATHGoogle Scholar
  35. 35.
    Wang CY (1989) Free convection on a vertical stretching surface. J Appl Math Mech (ZAMM) 69:418–420CrossRefMATHGoogle Scholar
  36. 36.
    Gorla RSR, Sidawi I (1994) Free convection on a vertical stretching surface with suction and blowing. Appl Sci Res 52:247–257CrossRefMATHGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan

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