Simultaneous impact of nonlinear radiative heat flux and Arrhenius activation energy in flow of chemically reacting Carreau nanofluid

  • M. IrfanEmail author
  • M. Khan
Original Article


This notion reports the properties of nanofluid on Carreau fluid for dual nature study. Nanofluids are stable colloidal suspensions of metals or non-metals element in a disreputable liquid, which intensify the heat transfer of the solution and exaggerate the storage aptitude. The forthright intention of this communication is to scrutinize the impact of binary chemical reaction and activation energy on dual nature study for unsteady flow of Carreau magnetite nanofluid owing to porous shrinking/stretching sheet. Additionally, the aspects of convective heat transport with nonlinear thermal radiation and heat sink/source have been established for heat transport phenomenon. With the assistance of compatible conversions the partial differential equations (PDEs) are condensed into ordinary differential equations (ODEs) which are then elucidated numerically via bvp4c scheme and the impacts of intricate parameters are exposed explicitly. It is noted that the intensifying values of activation energy parameter and fitted rate constant causes to an enhancement in the nanofluid concentration.


Carreau nanofluid Binary chemical reaction Arrhenius activation energy Nonlinear thermal radiation Heat sink/source Convective condition 

List of symbols


Cauchy stress tensor




Identity tensor

\(\dot {\gamma }\)

Shear rate


Zero shear rate viscosity


Infinity shear rate viscosity

A 1

First Riviln–Erickson tensor

u, v

Velocity components (m/s)

x, y

Space coordinates (m/s)


Nanofluid temperature (K)


Nanoparticles concentration (K)


Kinematic viscosity (K)


Material rate constant (m2/s)


Power law index


Magnetic parameter (A/m)


Electrical conductivity (S/m)


Fluid density (kg/m3)


Heat capacity of fluid (J/K/m3)


Thermal diffusivity (m/s)


Effective heat capacity ratio


Specific heat of fluid (J/kg K)


Radiative heat flux (W/m2)


Mean absorption coefficient (m−1)


Stefan–Boltzmann constant (W/m2 K4)


Heat generation/absorption coefficient


Nanofluid ambient temperature (K)


Nanofluid ambient concentration (K)


Brownian diffusion coefficient (m2 s)


Thermophoresis diffusion coefficient (m2 s)


Reaction rate


Fitted rate constant


Activation energy


Boltzmann constant

a, b

Positive constants

Uw(x, t)

Stretching velocity (m/s)

Ue(x, t)

Potential flow velocity (m/s)

Vw(x, t)

Mass fluid velocity (m/s)

− V

Mass transfer coefficient


Heat transfer coefficient

(Tf, Cw)

Temperature of fluid and concentration of wall 


Dimensional unsteadiness parameter


Time (s)


Dimensionless variable


Local Wessinberg number


Unsteadiness parameter


Magnetic parameter


Radiation parameter


Temperature ratio parameter


Brownian motion parameter


Thermophoresis parameter


Prandtl number

δ > 0

Heat source parameter

δ < 0

Heat sink parameter


Lewis number


Reaction rate parameter


Temperature difference parameter


Activation energy parameter


Thermal Biot number


Mass transfer parameter


Shrinking/stretching parameter


Skin friction coefficients


Local Nusselt number


Local Sherwood number


Local Reynolds number


Dimensionless velocities


Dimensionless temperature


Dimensionless concentration



The authors wish to convey their true thanks to the reviewers for their substantial suggestions and comments to progress the superiority of this manuscript. This work has the financial support from Higher Education Commission (HEC) of Pakistan under the project number 6210.


  1. Ahmed J, Khan M, Ahmad L (2019) Transient thin-film spin-coating flow of chemically reactive and radiative Maxwell nanofluid over a rotating disk. Appl Phys A. Google Scholar
  2. Alshomrani AS, Irfan M, Saleem A, Khan M (2018) Chemically reactive flow and heat transfer of magnetite Oldroyd-B nanofluid subject to stratifications. Appl Nanosci 8:1743–1754CrossRefGoogle Scholar
  3. Anwar MS, Rasheed A (2017) Simulations of a fractional rate type nanofluid flow with non-integer Caputo time derivatives. Comput Math Appl 74:2485–2502CrossRefGoogle Scholar
  4. Arrhenius S (1889) Über die Dissociationswärme und den Einfluss der Temperatur auf den Dissociationsgrad der Elektrolyte. Z Phys Chem 4:96–116Google Scholar
  5. Astanina MS, Sheremet MA, Oztop HF, Abu-Hamdeh N (2018) MHD natural convection and entropy generation of ferrofluid in an open trapezoidal cavity partially filled with a porous medium. Int J Mech Sci 136:493–502CrossRefGoogle Scholar
  6. Bestman AR (1990) Natural convection boundary layer with suction and mass transfer in a porous medium. Int J Energy Res 14:389–396CrossRefGoogle Scholar
  7. Bhattacharyya K (2011) Dual solutions in unsteady stagnation-point flow over a shrinking sheet. Chin Phys Lett. Google Scholar
  8. Carreau PJ (1972) Rheological equations from molecular network theories. Trans Soc Rheol 16:99–127CrossRefGoogle Scholar
  9. Choi SUS (1995) Enhancing thermal conductivity of fluids with nanoparticles. ASME Int Mech Eng 66:99–105Google Scholar
  10. Delgado MV, Pérez ASO, Fuentes WF, Zanoguera MEB, Ramírez AA, Díaz JDO, Balbuena DH, López MR, Sergiyenko O (2018) Theoretical and experimental study of low conducting fluid MHD flow in an open annular channel. Int J Heat Mass Transf 127(C):322–331CrossRefGoogle Scholar
  11. Devendiran DK, Amirtham VA (2016) A review on preparation, characterization, properties and applications of nanofluids. Renew Sustain Energy Rev 60:21–40CrossRefGoogle Scholar
  12. Hamid A, Hashim, Khan M (2018) Impacts of binary chemical reaction with activation energy on unsteady flow of magneto-Williamson nanofluid. J Mol Liq 262:435–442CrossRefGoogle Scholar
  13. Haq RU, Soomro FA, Mekkaoui T, Mdallal QMA (2018) MHD natural convection flow enclosure in a corrugated cavity filled with a porous medium. Int J Heat Mass Transf 121:1168–1178CrossRefGoogle Scholar
  14. Hayat T, Rashid M, Alsaedi A (2017) MHD convective flow of magnetite-Fe3O4 nanoparticles by curved stretching sheet. Results Phys 7:3107–3115CrossRefGoogle Scholar
  15. Hayat T, Rashid M, Alsaedi A (2018) Three dimensional radiative flow of magnetite-nanofluid with homogeneous-heterogeneous reactions. Results Phys 8:268–275CrossRefGoogle Scholar
  16. Irfan M, Khan M, Khan WA (2017) Numerical analysis of unsteady 3D flow of Carreau nanofluid with variable thermal conductivity and heat source/sink. Results Phys 7:3315–3324CrossRefGoogle Scholar
  17. Irfan M, Khan M, Khan WA, Ayaz M (2018) Modern development on the features of magnetic field and heat sink/source in Maxwell nanofluid subject to convective heat transport. Phys Lett A 382:1992–2002CrossRefGoogle Scholar
  18. Irfan M, Khan M, Khan WA, Ahmad L (2019a) Influence of binary chemical reaction with Arrhenius activation energy in MHD nonlinear radiative flow of unsteady Carreau nanofluid: dual solutions. Appl Phys A. Google Scholar
  19. Irfan M, Khan WA, Khan M, Gulzar MM (2019b) Influence of Arrhenius activation energy in chemically reactive radiative flow of 3D Carreau nanofluid with nonlinear mixed convection. J Phys Chem Solids 125:141–152CrossRefGoogle Scholar
  20. Khan M, Irfan M, Khan WA (2017) Impact of forced convective radiative heat and mass transfer mechanisms on 3D Carreau nanofluid: a numerical study. Eur Phys J Plus. Google Scholar
  21. Khan M, Irfan M, Ahmad L, Khan WA (2018a) Simultaneous investigation of MHD and convective phenomena on time-dependent flow of Carreau nanofluid with variable properties: dual solutions. Phys Lett A 382:2334–2342CrossRefGoogle Scholar
  22. Khan MI, Qayyum S, Hayat T, Waqas M, Khan MI, Alsaedi A (2018b) Entropy generation minimization and binary chemical reaction with Arrhenius activation energy in MHD radiative flow of nanomaterial. J Mol Liq 259:274–283CrossRefGoogle Scholar
  23. Lok YY, Ishak A, Pop I (2011) MHD stagnation-point flow towards a shrinking sheet. Int J Numer Methods Heat Fluid Flow 21:61–72CrossRefGoogle Scholar
  24. Mahanthesh B, Gireesha BJ (2018) Dual solutions for unsteady stagnation-point flow of Prandtl nanofluid past a stretching/shrinking plate. Defect Diffus Forum 388:124–134CrossRefGoogle Scholar
  25. Mahanthesh B, Gireesha BJ, Gorla RSR, Abbasi FM, Shehzad SA (2016) Numerical solutions for magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary. J Mag Mag Mater 417:189–196CrossRefGoogle Scholar
  26. Mahanthesh B, Gireesha BJ, Prasannakumara BC, Kumar PBS (2017) Magneto-thermo-marangoni convective flow of Cu–H2O nanoliquid past an infinite disk with particle shape and exponential space based heat source effects. Results Phys 7:2990–2996CrossRefGoogle Scholar
  27. Mahanthesh B, Gireesha BJ, Gorla RSR, Makinde OD (2018a) Magnetohydrodynamic three-dimensional flow of nanofluids with slip and thermal radiation over a nonlinear stretching sheet: a numerical study. Neural Comput Appl 30:1557–1567CrossRefGoogle Scholar
  28. Mahanthesh B, Gireesha BJ, Shehzad SA, Rauf A, Kumar PBS (2018b) Nonlinear radiated MHD flow of nanoliquids due to a rotating disk with irregular heat source and heat flux condition. Physica B 537:98–104CrossRefGoogle Scholar
  29. Rashid M, Hayat T, Alsaedi A (2019) Entropy generation in Darcy–Forchheimer flow of nanofluid with five nanoarticles due to stretching cylinder. Appl Nanosci. Google Scholar
  30. Sajid T, Sagheer M, Hussain S, Bilal M (2018) Darcy–Forchheimer flow of Maxwell nanofluid flow with nonlinear thermal radiation and activation energy. AIP Adv. Google Scholar
  31. Shafique Z, Mustafa M, Mushtaq A (2016) Boundary layer flow of Maxwell fluid in rotating frame with binary chemical reaction and activation energy. Results Phys 6:627–633CrossRefGoogle Scholar
  32. Tian XY, Li BW, Hu ZM (2018) Convective stagnation point flow of a MHD non-Newtonian nanofluid towards a stretching plate. Int J Heat Mass Transf 127(A):768–780CrossRefGoogle Scholar
  33. Wang CY (2008) Stagnation flow towards a shrinking sheet. Int J Nonlinear Mech 43:377–382CrossRefGoogle Scholar
  34. Waqas M, Khan MI, Hayat T, Alsaedi A (2017) Numerical simulation for magneto Carreau nanofluid model with thermal radiation: a revised model. Comput Methods Appl Mech Eng 324:640–653CrossRefGoogle Scholar
  35. Zeeshan A, Shehzad N, Ellahi R (2018) Analysis of activation energy in Couette-Poiseuille flow of nanofluid in the presence of chemical reaction and convective boundary conditions. Results Phys 8:502–512CrossRefGoogle Scholar

Copyright information

© King Abdulaziz City for Science and Technology 2019

Authors and Affiliations

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

Personalised recommendations