Numerical Study of Carreau Nanofluid Flow Under Slips


The numerical exploration of three dimensional Carreau magneto Nanofluid flow through a stretching sheet has been bestowed with considering nonlinear thermal radiation, velocity, thermal and mass slips. The heat and mass transfer attributes have been reported under the existing important variables in this work. The elementary equations which influence flow are remoulded to a system through the similarity transmutations. The remoulded system together with the boundary restrictions are procured deploying Runge–Kutta 4th order process and Shooting approach numerically. The impacts of involving physical variables on heat and mass transfer features are examined. The impacts of the important variables on skin friction factors have been tackled via tables. The Nusselt number’s estimates and Sherwood number’s estimates have been obtained and tackled via graphs and tables. This study predicts that the velocity distribution is suppressed by greater values of Hartmann number. The temperature profile and concentration profile both upgrade with uplifted estimates of thermophoresis parameter, while they decay with augmented values of thermal slip parameter. The skin friction coefficients improve with hiking estimates of permeability parameter, while depreciate with the ratio of infinite shear rate viscosity to the zero shear rate viscosity. Mounting estimates of Lewis number enhance the temperature distribution.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19


\( a \) :

Positive fixed number

\( b \) :

Positive fixed number

\( B_{0} \) :

Constant magnetic field

\( C \) :

Fluid concentration

\( (C_{p} )_{f} \) :

Specific heat at constant pressure

\( Cf_{x} \) :

Skin friction coefficient along x-direction

\( Cf_{y} \) :

Skin friction coefficient along y-direction

\( C_{\infty } \) :

Ambient fluid concentration

\( C_{W} \) :

Wall surface concentration

\( D_{B} \) :

Brownian diffusion coefficient

\( D_{T} \) :

Thermophoretic diffusion coefficient

\( f \) :

Boundary-layer stream function

\( g \) :

Gravitational acceleration

\( k^{*} \) :

Mean absorption coefficient

\( K_{p} \) :

Permeability of the porous medium

\( K_{r} \) :

Chemical reaction factor

\( K_{1} \) :

Permeability parameter

\( Le \) :

Lewis number

\( M \) :

Hartmann number

\( n \) :

Power law index

\( N^{*} \) :

Concentration to thermal buoyancy forces ratio

\( Nb \) :

Brownian motion parameter

\( N_{R} \) :

The nonlinear radiation parameter

\( Nt \) :

Thermophoresis parameter

\( Nu_{{\bar{x}}} \) :

Local Nusselt number

\( N_{1} \) :

Velocity slip factor for the u velocity component

\( N_{2} \) :

Thermal slip factor

\( N_{3} \) :

Mass slip factor

\( \Pr \) :

Prandtl number

\( q_{r} \) :

Radiative heat flux

\( \text{Re}_{x} \) :

Local Reynolds number in x-direction

\( \text{Re}_{y} \) :

Local Reynolds number in y-direction

\( Sh_{{\bar{x}}} \) :

Local Sherwood number

\( T \) :

Fluid temperature

\( T_{W} \) :

Cone surface temperature

\( T_{\infty } \) :

Free stream temperature

\( u \) :

Component of velocity along x direction

\( U_{w} ,\,V_{w} \) :

Stretching velocities of the surface

\( v \) :

Component of velocity along y direction

\( w \) :

Component of velocity along z direction

\( We \) :

The local Weissenberg number

\( x,\,y,\,z \) :

Coordinate axes

\( \alpha \) :

Thermal diffusivity

\( \alpha^{*} \) :

Ratio of stretching sheet

\( \alpha_{1}^{ * } \) :

Mixed convection parameter

\( \beta^{ * } \) :

The ratio of infinite shear rate viscosity to the zero shear rate viscosity

\( \beta_{c} \) :

Nonlinear convection parameter due to concentration

\( \delta_{c} \) :

Mass slip parameter

\( \delta_{t} \) :

Thermal slip parameter

\( \delta_{u} \) :

Tangential slip parameter

\( \eta \) :

Similarity variable

\( \gamma \) :

Local Biot number

\( \mu_{f} \) :

Dynamic viscosity

\( \mu_{0} \) :

Zero shear rate viscosity

\( \mu_{\infty } \) :

Infinite shear rate viscosity

\( \nu_{f} \) :

Kinematic viscosity

\( \lambda_{0} \) :

Ratio of thermal expansion coefficient

\( \lambda_{1} ,\,\lambda_{2} \) :

Ratio of concentration expansion coefficients

\( \phi \) :

Boundary-layer concentration

\( \rho_{f} \) :

Density of the fluid

\( \sigma \) :

Electrical charge density

\( \sigma * \) :

Stefan Boltzmann constant

\( \theta \) :

Boundary-layer temperature

\( \theta_{w} \) :

Temperature ratio parameter


  1. 1.

    Hayat, T., Bashir, G., Waqas, M., Alsadi, A.: MHD flow of Jeffrey liquid due to a non-linear radially stretched sheet in presence of Newtonian heating. Results Phys. 6, 817–823 (2016)

    Google Scholar 

  2. 2.

    Ramana Reddy, G.V., Raja Sekhar, K., Sita Maha Lakshmi, A.: MHD free convection fluid flow past a semi-infinite vertical porous plate with heat absorption and chemical reaction. Int. J. Chem. Sci. 13, 525–540 (2015)

    Google Scholar 

  3. 3.

    Jhansi Rani, K., Ramana Reddy, G.V., Ramana Murthy, Ch.V, Ramana Murthy, M.V.: Heat and mass transfer effects on MHD free convective flow over an inclined plate embedded in a porous Medium. Int. J. Chem. Sci. 13, 1998–2016 (2015)

    Google Scholar 

  4. 4.

    Jabar, K.K.: Joule heating and viscous dissipation effects on MHD flow over a stretching porous sheet subjected to power law heat Flux in presence of Heat source. Open J. Fluid Dyn. 6, 156–165 (2016)

    Google Scholar 

  5. 5.

    Sandeep, N., Sulochana, C., Rushi Kumar, B.: Unsteady MHD radiative flow and heat transfer of a dusty nanofluid over an exponentially stretching surface. Eng. Sci. Technol. Int. J. 19, 227–240 (2016)

    Google Scholar 

  6. 6.

    Gnaneswara Reddy, M., Padma, P., Shankar, B.: Effects of viscous dissipation and heat source on unsteady MHD flow over a stretching sheet. Ain Shams Eng. J. 6, 1195–1201 (2015)

    Google Scholar 

  7. 7.

    Ramana Reddy, G.V., Hari Krishna, Y.: Numerical solutions of unsteady MHD flow heat transfer over a stretching surface with suction or injection. FDMP. 14, 213–222 (2018)

    Google Scholar 

  8. 8.

    Manjula, J., Padma, P., Gnaneswara Reddy, M., Venkateswarlu, M.: Influence of thermal radiation and chemical reaction on MHD flow, Heat mass transfer over a stretching surface. Proc. Eng. 127, 1315–1322 (2015)

    Google Scholar 

  9. 9.

    Sreedevi, G., Prasada Rao, D.R.V., Makinde, O.D., Venkata Ramana Reddy, G.: Soret and Dufour effects on MHD flow with heat and mass transfer past a permeable stretching sheet in presence of thermal radiation. Indian J. Pure Appl. Phys. 55, 551–563 (2017)

    Google Scholar 

  10. 10.

    Gnaneswara Reddy, M., Padma, P., Shankar, B., Gireesha, B.J.: Thermal radiation effects on MHD stagnation point flow of Nanofluid over a stretching sheet in a porous Medium. J. Nanofluids 5, 753–764 (2016)

    Google Scholar 

  11. 11.

    Sreelakshmi, K., Nagendramma, V., Sarojamma, G.: Unsteady boundary layer flow induced by a stretching sheet in a rotating fluid with thermal radiation. Proc. Eng. 127, 678–685 (2015)

    Google Scholar 

  12. 12.

    Choi, S.U.S., Eastman, J.A.: Enhancing thermal conductivity fluids with nanoparticles. J. Am. Soc. Mech. Eng. 231, 99–106 (1995)

    Google Scholar 

  13. 13.

    Rudraswamy, N.G., Gireesha, B.J.: Influence of chemical reaction and thermal radiation on MHD boundary layer flow and heat transfer of a nanofluid over an exponentially stretching sheet. J. Appl. Math. Phys. 2, 24–32 (2014)

    Google Scholar 

  14. 14.

    Reddy, M.G.: Boundary layer flow of a nanofluid past a stretching sheet. J. Sci. Res. 6, 257–272 (2014)

    Google Scholar 

  15. 15.

    El-Hakeem, M.A., Ramzan, M., Chung, J.D.: A numerical study of magnetohydrodynamic stagnation point flow of nanofluid with Newtonian heating. J. Comput. Theor. Nanosci. 13, 8419–8426 (2016)

    Google Scholar 

  16. 16.

    Ellahi, R., Tariq, M.H., Hassan, M.: On boundary layer nano-ferro liquid flow under the influence of low oscillating stretchable rotating disk. J. Mol. Liq. 229, 339–345 (2017)

    Google Scholar 

  17. 17.

    Sayehvand, H.O., Parsa, A.B.: A new numerical method for investigation of thermophoresis and Bronian motion effects MHD nanofluid flow and heat transfer between parallel plates partially filled with porous medium. J. Results Phys. 7, 1595–1607 (2017)

    Google Scholar 

  18. 18.

    Hayat, T., Khan, M.I., Waqas, M., Alsaedi, A., Farooq, M.: Numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon–water nanofluid. J. Comput. Method Appl. Mech. Eng. 315, 1011–1024 (2017)

    MathSciNet  Google Scholar 

  19. 19.

    Ramzan, M., Bilal, M., Chung, J.D.: Radiative flow of Powell-Eyring magneto-nanofluid over a stretching cylinder with chemical reaction and double stratification near a stagnation point. PLoS ONE 12(1), e0170790 (2017).

    Article  Google Scholar 

  20. 20.

    Rashidi, S., Akar, S., Bovand, M.: Volume of fluid model to simulate the nanofluid flow and entropy generation in a single solar still. Renew. Energy 115, 400–410 (2018)

    Google Scholar 

  21. 21.

    Zeeshan, A., Shehzad, N., Ellahi, R.: Analysis of activation energy in Couette–Poiseuille flow of nanofluid in the presence of chemical reaction and convective boundary conditions. J. Results Phys. 8, 502–512 (2018)

    Google Scholar 

  22. 22.

    Mallikarjuna, B., Rasheed, A.M., Hussein, A.K., Hariprasad Raju, S.: Transpiration and thermophoresis effects on non-darcy convective flow past a rotating cone with thermal radiation. Arab. J. Sci. Eng. 41, 1319–8025 (2016)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Narayana, M., Awad, F.G., Sibanda, P.: Free magneto hydrodynamic flow and convection from a vertical spinning cone with cross-diffusion effects. Appl. Math. Modell. 37, 2662–2678 (2013)

    MATH  Google Scholar 

  24. 24.

    Vanita, A.K.: Numerical study of effect of induced magnetic field on transient natural convection over a vertical cone. Alexandria Eng. J. 55, 1211–1223 (2016)

    Google Scholar 

  25. 25.

    Gnaneswara Reddy, M., Manjula, J., Padma, P.: Mass transfer flow of MHD radiative tangent hyperbolic fluid over a cylinder: A numerical study. Int. J. Appl. Comput. Math. 3, 447–472 (2017)

    MathSciNet  Google Scholar 

  26. 26.

    Ramana Reddy, G.V., Jaya Rami Reddy, K., Lakshmi, R.: Radiation and mass transfer effects on nonlinear MHD boundary layer flow of liquid metal over a porous stretching surface embedded in porous medium with heat generation. WSEAS Trans. Fluid Mech. 10, 1–12 (2015)

    Google Scholar 

  27. 27.

    Gnaneswaara Reddy, M., Padma, P., Shankar, B.: Effects of magnetic field and ohmic heating on viscous flow of a nanofluid towards a nonlinear permeable stretching sheet. J. Nano Fluids 5, 459–470 (2016)

    Google Scholar 

  28. 28.

    Sreelakshmi, K., Sarojamma, G., Murthy, J.V.R.: Homotopy analysis of an unsteady flow heat transfer of a Jeffrey nanofluid over a radially stretching convective surface. J. Nanofluids 7, 1–10 (2018)

    Google Scholar 

  29. 29.

    Sreelakshmi, K., Sarojamma, G., Makinde, O.D.: Dual stratification on the Darcy–Forchheimer flow of a maxwell nanofluid over a stretching surface. Defect Diffus. Forum 387, 207–217 (2018)

    Google Scholar 

  30. 30.

    Hayat, T., Waleed Ahmed Khan, M., Khan, M.I., Alsaedi, A.: Non-linear radiative heat flux and heat source/sink on entropy generation minimization rate. Phys. B: Condens. Matter. 538, 95–103 (2018)

    Google Scholar 

  31. 31.

    Suneetha, K., Ibrahim, S.M., Ramana Reddy, G.V.: Radiation and heat source effects on MHD flow over a permeable stretching sheet through porous stratum with chemical reaction. Multidiscip. Model. Mater. Struct. 14, 1101–1114 (2018)

    Google Scholar 

  32. 32.

    Luo, Z., Xu, H.: Numerical simulation of heat and mass transfer through micro porous media with lattice Boltzmann method. Therm. Sci. Eng. Prog. 9, 44–51 (2019)

    Google Scholar 

  33. 33.

    Ramana Reddy, G.V., Bhaskar Reddy, N., Gorla, R.S.R.: Radiation and chemical reaction effects on MHD flow along a moving vertical porous plate. Int. J. Appl. Mech. Eng. 21, 157–168 (2016)

    Google Scholar 

  34. 34.

    Khan, M., Hasim, A.S., Hussain, M., Azam, M.: Magnetohydrodynamic flow of Carreau fluid over a convectively heated surface in the presence of non-linear radiation. J Magn. Magn. Mater. 412, 63–68 (2016)

    Google Scholar 

  35. 35.

    Khan, M., Azam, M.: Unsteady heat and mass transfer mechanism in MHD Carreaunanofluid flow. J. Mol. Liq. 225, 554–562 (2017)

    Google Scholar 

  36. 36.

    Lu, D., Ramzan, M., Huda, N.U.: Nonlinear radiation effect on MHD Carreau nanofluid flow over a radially stretching surface with zero mass flux at the surface. J. Sci. 8, 3709 (2018)

    Google Scholar 

  37. 37.

    Khan, M., Sardar, H., Hasim.: Heat generation/absorption and thermal radiation impacts on three dimensional flow of Carreau fluid with convective heat transfer. J. Mol. Liq. 272, 474–480 (2018)

    Google Scholar 

  38. 38.

    Hakeem, A.K., Ganesh, N.V., Ganga, B.: Magnetic field effects on second order slip flow of nanofluid over a stretching/shrinking sheet with thermal radiation effect. Magn. Magn. Mater. 381, 243–257 (2015)

    Google Scholar 

  39. 39.

    Gnaneswara Reddy, M., Manjula, J., Padma, P.: Influence of second-order velocity slip and double stratification on MHD 3D Casson nanofluid flow over a stretching sheet. J. Nanofluids 6, 436–446 (2017)

    Google Scholar 

  40. 40.

    Das, K., Jana, S., Kundu, P.K.: Thermophoretic MHD slip flow over a permeable surface with variable fluid properties. J. Alex. Eng. J. 54, 35–44 (2015)

    Google Scholar 

  41. 41.

    Ali, R., Shahzad, A., Khan, M., Ayub, M.: Analytic and numerical solutions for axisymmetric flow with partial slip. Eng. Comput. 32, 149–154 (2016)

    Google Scholar 

  42. 42.

    Gnaneswara Reddy, M., Sudha Rani, M.V.V.N.L., Makinde, O.D.: Effects of nonlinear radiation and thermo-diffusion on MHD Carreau fluid past a stretching surface with slip. Diffus. Found. 11, 57–71 (2018)

    Google Scholar 

  43. 43.

    Samad Khan, A., Nie, Y., Nie, Z., Shah, Z., Dawar, A., Khan, W., Islam, S.: Three-dimensional nanofluid flow with heat and mass transfer analysis over a linear stretching surface with convective boundary conditions. Appl. Sci. 8, 2244 (2018)

    Google Scholar 

  44. 44.

    Ariel, P.D.: Three-Dimensional flow past a stretching sheet and the homotopy perturbation method. Int. J. Comput. Math. Appl. 54, 920–925 (2007)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Malik, M.Y., Bilal, S., Salahuddin, T., Rahman, K.U.: Three dimensional williamson fluid flow over a linear stretching surface. Math. Sci. 6(1), 53–61 (2017)

    Google Scholar 

  46. 46.

    Raju, C.S.K., Sandeep, N.: Unsteady three-dimensional flow of Casson–Carreau fluids past a stretching surface. Collection of Alex. Eng. J. 1–25 (2017)

Download references

Author information



Corresponding author

Correspondence to Penumudi Naga Santoshi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Naga Santoshi, P., Venkata Ramana Reddy, G. & Padma, P. Numerical Study of Carreau Nanofluid Flow Under Slips. Int. J. Appl. Comput. Math 5, 122 (2019).

Download citation


  • Radiation
  • MHD
  • Carreau nanofluid
  • Stretching sheet
  • Convection condition