Homogenous–heterogenous reactions in Carreau fluid flow with heat generation/absorption: multiple solution

  • Masood Khan
  • Sana Ejaz
  • Humara SardarEmail author
Technical Paper


This research article investigates the two-dimensional stagnation boundary layer flow over a permeable stretching sheet with variable thickness and heat generation/absorption. The heat generation/absorption is taking place due to a homogenous/heterogenous reactions. Viscous dissipation and radiation effects are neglected here. We have utilized the appropriate dimensionless transformations to alter the basic conservation equations into a set of partially coupled ODEs. The related system of reduced ODEs together with physical boundary restrictions is numerically integrated via versatile and extensively validated, MATLAB bvp4c package with Labatto III A scheme. To obtain the numerical solution, problem is governed by active physical parameters, such as viscosity ratio parameter (\(\beta ^{{*}}\)), local Weissenberg number (We), heat generation/absorption parameter (\({\gamma }^{*})\), homogeneous reaction parameter ( K), Schmidt number (Sc), heterogeneous reaction parameter (Ks), ratio of mass diffusion coefficient (\({\delta }\)), mass transfer parameter with suction (\({s>0}\)) and injection (\({s<0}\)). We exhibit and explain the impacts of these active parameters on dimensionless fluid velocity, fluid temperature, fluid concentration, skin friction, Nusselt number and Sherwood number by means of tables and graphs. From this study, it is observed that fluid concentration is depressed by higher heterogeneous reaction parameter (Ks). However, Prandtl number (Pr) and heat absorption/generation parameter \({(\gamma }^{*})\) increase the fluid temperature.


Homogenous–heterogenous reaction Heat generation/absorption Stagnation point flow Carreau fluid Dual solutions Labatto III A scheme 



  1. 1.
    Carreau PJ (1972) Rheological equations from molecular network theories. Trans Soc Rheol 116:99–127CrossRefGoogle Scholar
  2. 2.
    Terrones G, Smith PA, Armstrong TR, Soltesz TJ (1997) Application of the Carreau model to tape-casting fluid mechanics. J Am Ceram Soc 80:3151–3156CrossRefGoogle Scholar
  3. 3.
    Chhabra RP, Uhlherr PHT (1980) Creeping motion of spheres through shear thinning elastic fluids described by the Carreau viscosity equation. Rheol Acta 19:187–95CrossRefGoogle Scholar
  4. 4.
    Bush MB, Phan-Thein N (1984) Drag force on a sphere in creeping motion through a Carreau model fluid. J Non Newton Fluid Mech 16:303–313CrossRefGoogle Scholar
  5. 5.
    Mas R, Magnin A (1993) Rheology of colloidal suspensions: case of lubricating greases. Soc Rheol 38:889–907CrossRefGoogle Scholar
  6. 6.
    Hyun YH, Lim ST, Choi HJ, John MS (2001) Rheology of poly(ethylene oxide)/organoclay nanocomposites. Macromolecules 34:8084–8093CrossRefGoogle Scholar
  7. 7.
    Hiemenz K (1972) Die Grenzschicht an einem in den gleichfo “rmigen Flu” essigkeitsstrom eingetauchten geraden Kreiszylinder. Ding Polym J 326:321–324Google Scholar
  8. 8.
    Akbar NS, Nadeem S, Haq RUI, Ye S (2014) MHD stagnation point flow of Carreau fluid toward a permeable shrinking sheet: dual solutions. Ain Shams Eng J 5:1233–1239CrossRefGoogle Scholar
  9. 9.
    Sulochana C, Ashwinkumar GP, Sandeep N (2016) Transpiration effect on stagnation-point flow of a Carreau nanofluid in the presence of thermophoresis and Brownian motion. Alex Eng J 55(2):1151–1157CrossRefGoogle Scholar
  10. 10.
    Sulochana C, Ashwinkumar GP, Sandeep N (2017) Joule heating effect on a continuously moving thin needle in MHD Sakiadis flow with thermophoresis and Brownian moment. Eur Phys J Plus 132(9):387CrossRefGoogle Scholar
  11. 11.
    Khan M, Sardar H, Gulzar MM (2018) On radiative heat transfer in stagnation point flow of MHD Carreau fluid over a stretched surface. Results Phys 8:524–531CrossRefGoogle Scholar
  12. 12.
    Erickson LE, Fan LT, Fox VG (1966) Heat and mass transfer on moving continuous flat plate with suction or injection. Ind Eng Chem Fundam 5(1):19–25CrossRefGoogle Scholar
  13. 13.
    Merkin JH (1996) A model for isothermal homogeneous–heterogeneous reactions in boundary-layer flow. Math Comput Model 24:125–136MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chaudhary MA, Merkin JH (1995) A simple isothermal model for homogeneous–heterogeneous reactions in boundary layer flow I. Equal diffusivities. Fluid Dyn Res 16:311–333MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sulochana C, Ashwinkumar GP (2017) Carreau model for liquid thin film flow of dissipative magnetic-nanofluids over a stretching sheet. Int J Hybrid Inf Technol 10(1):239–254CrossRefGoogle Scholar
  16. 16.
    Khan WA, Pop I (2012) Effects of homogeneous–heterogeneous reactions on the viscoelastic fluid towards a stretching sheet. ASME J Heat Transf 134:1–5Google Scholar
  17. 17.
    Kameswaran K, Shaw S, Sibanda P, Murthy PVSN (2013) Homogeneous–heterogeneous reactions in a nanofluid flow due to a porous stretching sheet. Int J Heat Mass Transf 57:465–472CrossRefGoogle Scholar
  18. 18.
    Shaw S, Kameswaran PK, Sibanda P (2013) Homogeneous–heterogeneous reactions in micropolar fluid flow from a permeable stretching or shrinking sheet in a porous medium. Bound Value Probl 2013:1–10MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sardar H, Khan M, Ahmad L (2018) Local non-similar solutions of Carreau fluid flow with radiative heat transfer in the presence of MHD mixed convection flow and stagnation point. J Braz Soc Mech Eng. CrossRefGoogle Scholar
  20. 20.
    Samrat SP, Sulochana C, Ashwinkumar GP (2019) Impact of thermal radiation on an unsteady Casson nanofluid flow over a stretching surface. Int J Appl Comput Math 5(2):31MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sulochana C, Ashwinkumar GP (2018) Impact of Brownian moment and thermophoresis on magnetohydrodynamic flow of magnetic nanofluid past an elongated sheet in the presence of thermal diffusion. Multidiscip Model Mater Struct 14(4):744–755CrossRefGoogle Scholar
  22. 22.
    Sulochana C, Ashwinkumar GP, Sandeep N (2017) Effect of thermophoresis and Brownian moment on 2D MHD nanofluid flow over an elongated sheet. Defect Diffus Forum 377:111–126CrossRefGoogle Scholar
  23. 23.
    Khan M, Sardar H, Hashim (2018) Heat generation/absorption and thermal radiation impacts on three-dimensional flow of Carreau fluid with convective heat transfer. J Mol Liq 272:474–480CrossRefGoogle Scholar
  24. 24.
    Vajravelu K, Hadjinicolaou A (1997) Convective heat transfer in an electrically conducting fluid at a stretching surface with uniform free stream. Int J Eng Sci 35(12–13):1237–1244CrossRefGoogle Scholar
  25. 25.
    Chamkha AJ, Issa C (2000) Effects of heat generation/absorption and thermophoresis on hydromagnetic flow with heat and mass transfer over a flat surface. Int J Numer Methods Heat Fluid Flow 10(4):432–449CrossRefGoogle Scholar
  26. 26.
    Khan M, Sardar H (2018) On steady two-dimensional Carreau fluid flow over a wedge in the presence of infinite shear rate viscosity. Results Phys 8:516–523CrossRefGoogle Scholar
  27. 27.
    Khan M, Sardar H (2019) On steady two-dimensional Carreau nanofluid flow in the presence of infinite shear rate viscosity. Can J Phys 97(4):400-407. CrossRefGoogle Scholar
  28. 28.
    Soomro FA, Usman M, Haq RUl, Wang W (2018) Melting heat transfer analysis of Sisko fluid over a moving surface with nonlinear thermal radiation via collocation method. Int J Heat Mass Transf 126(A):1034–1042CrossRefGoogle Scholar
  29. 29.
    Haq RUl, Noor NFM, Khan ZH (2016) Numerical simulation of water based magnetite nanoparticles between two parallel disks. Adv Powder Technol 27(4):1568–1575CrossRefGoogle Scholar
  30. 30.
    Soomro FA, Haq R Ul, Al-Mdallal QM, Zhang Q (2018) Heat generation/absorption and nonlinear radiation effects on stagnation point flow of nanofluid along a moving surface. Results Phys 8:404–414CrossRefGoogle Scholar
  31. 31.
    Usman M, Hamid M, Zubair T, Haq RUl, Wang W (2018) \(\text{Cu}{-}\text{Al}_2\text{O}_3\)/water hybrid nanofluid through a permeable surface in the presence of nonlinear radiation and variable thermal conductivity via LSM. Int J Heat Mass Transf 126(A):1347–1356CrossRefGoogle Scholar
  32. 32.
    Usman M, Soomro FA, Haq R Ul, Wang W, Defterli O (2018) Thermal and velocity slip effects on Casson nanofluid flow over an inclined permeable stretching cylinder via collocation method. Int J Heat Mass Transf 122:1255–1263CrossRefGoogle Scholar
  33. 33.
    Sheikholeslami M, Haq R Ul, Shafee A, Li Z (2019) Heat transfer behavior of nanoparticle enhanced PCM solidification through an enclosure with V shaped fins. Int J Heat Mass Transf 130:1322–1342CrossRefGoogle Scholar
  34. 34.
    Ur Rehman F, Nadeem S, Ur Rehman H, Haq R Ul (2018) Thermo physical analysis for three-dimensional MHD stagnation-point flow of nano-material influenced by an exponential stretching surface. Results Phys 8:316–323CrossRefGoogle Scholar
  35. 35.
    Akbar NS, Nadeem S, Haq R Ul, Khan ZH (2013) Radiation effects on MHD stagnation point flow of nano fluid towards a stretching surface with convective boundary condition. Chin J Aerona 26(6):1389–1397CrossRefGoogle Scholar
  36. 36.
    Haq R Ul, Khan ZH, Khan WA (2014) Thermo physical effects of carbon nanotubes on MHD flow over a stretching surface. Physica E 63:215–222CrossRefGoogle Scholar
  37. 37.
    Shampine LF, Kierzenka J, Reichelt MW (2000) Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c. Tutor Notes 2000:1–27Google Scholar
  38. 38.
    Khan M, Sardar H, Gulzar MM, Alshomrani AS (2018) On multiple solutions of non-Newtonian Carreau fluid flow over an inclined shrinking sheet. Results Phys 8:926–932CrossRefGoogle Scholar
  39. 39.
    Ul Haq R, Nadeem S, Khan ZH, Akbar NS (2015) Thermal radiation and slip effects on MHD stagnation point flow of nanofluid over a stretching sheet. Physica E 65:17–23CrossRefGoogle Scholar
  40. 40.
    Khan M, Sardar H (2018) Secrutinization of 2D and mixed convection flow of generalized Newtonian fluid with nanoparticles and magnetic field. Can J Phys. CrossRefGoogle Scholar
  41. 41.
    Sardar H, Khan M, Ahmad L, Alshomrani AS (2019) Investigation of the mixed convection flow of Carreau nano fluid over a wedge in the presence of Soret and Dufor effects. Int J Heat Mass Transf 137:809–822CrossRefGoogle Scholar
  42. 42.
    Bachok A, Pop I, Ishak A (2010) Melting heat transfer in boundary layer stagnation point flow towards a stretching/shrinking sheet. Phys Lett A 374:4075–4079CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

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

Personalised recommendations