Skip to main content

Advertisement

SpringerLink
Log in

Carreau fluid flow due to nonlinearly stretching sheet with thermal radiation, heat flux, and variable conductivity

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

A sophisticated theoretical and mathematical model is proposed. It is verified that this model can estimate and monitor the detailed behavior for the steady Carreau fluid flow past a nonlinear stretching surface and the predicted phenomena due to the presence of heat flux, thermal radiation, and viscous dissipation. Despite the fact that some properties of the fluid do not depend on the temperature, the fluid thermal conductivity is assumed to depend on the temperature. Based on accelerating the fluid elements, some of the kinetic energy for the fluid can be turned to the internal heating energy in the form of viscous dissipation phenomena. The contribution in this study is that a similar solution is obtained, in spite of the high nonlinearity of the Carreau model, especially, with the heat flux, variable conductivity, and viscous dissipation phenomena. Some of the major significant findings of this study can be observed from the reduction in the fluid velocity with enhancing the Weissenberg number. Likewise, the increase in the sheet temperature is noted with increasing the Eckert number while the reverse behavior is observed for increasing both the radiation parameter and the conductivity parameter. Finally, the accuracy and trust in the proposed numerical method are validated after benchmarking for our data onto the earlier results.

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

Similar content being viewed by others

Abbreviations

b :

positive constant

m :

stretching constant

C f x :

local skin-friction coefficient

n :

power-law index

c p :

specific heat at constant pressure (m2·s−2·K−1)

Nu x :

Nusselt number

Ec :

Eckert number

Pr :

Prandtl number

f :

dimensionless stream function

q w :

heat flux (kg·s−3)

q r :

radiation heat flux (kg·m−2)

η :

dimensionless variable

R :

radiation parameter

θ :

dimensionless temperature

Re x :

local Reynolds number

κ :

fluid conductivity (kg·m·s−3·K−1)

T :

fluid temperature (K)

κ ∞ :

ambient conductivity (kg·m·s−3·K−1)

T ∞ :

ambient temperature (K)

μ :

fluid viscosity (kg·m−1·s−1)

u :

velocity component in the x-axis (m·s−1)

ρ :

fluid density (kg·m−3)

U w :

velocity of the sheet (m·s−1)

ε :

thermal conductivity parameter

v :

velocity components in the y-axis (m·s−1)

σ*:

Stefan-Boltzmann constant (W·m−2·K−4)

We :

Weissenberg number

ψ :

stream function (m2·s−1)

References

  1. CRANE, L. J. Flow past a stretching plate. Zeitschrift für Angewandte Mathematik und Physik, 21, 645–647 (1970)

    Article  Google Scholar 

  2. GRUBKA, L. G. and BOBBA, K. M. Heat transfer characteristics of a continuous stretching surface with variable temperature. Journal of Heat Transfer, 107, 248–250 (1985)

    Article  Google Scholar 

  3. CHEN, C. K. and CHAR, M. Heat transfer on a continuous stretching surface with suction or blowing. Journal of Mathematical Analysis and Applications, 35, 568–580 (1988)

    Article  MathSciNet  Google Scholar 

  4. CHEN, C. H. Laminar mixed convection adjacent to vertical continuously stretching sheets. Heat and Mass Transfer, 33, 471–476 (1998)

    Article  Google Scholar 

  5. MAGYARI, E. and KELLER, B. Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. Journal of Physics D: Applied Physics, 32, 577–585 (1999)

    Article  Google Scholar 

  6. LIU, I. C. and ANDERSON, H. I. Heat transfer over a bidirectional stretching sheet with variable thermal conditions. International Journal of Heat and Mass Transfer, 51, 4018–4024 (2008)

    Article  Google Scholar 

  7. GNANESWARA REDDY, M. Influence of thermal radiation, viscous dissipation and Hall current on MHD convection flow over a stretched vertical flat plate. Ain Shams Engineering Journal, 5, 169–175 (2014)

    Article  Google Scholar 

  8. POP, I. and NA, T. Unsteady flow past a stretching sheet. Mechanics Research Communications, 23, 413–422 (1996)

    Article  Google Scholar 

  9. TSAI, R., HUANG, K. H., and HUANG, J. S. Flow and heat transfer over an unsteady stretching surface with a non-uniform heat source. International Communications in Heat and Mass Transfer, 35, 1340–1343 (2008)

    Article  Google Scholar 

  10. PAVITHRA, G. M. and GIREESHA, B. J. Unsteady flow and heat transfer of a fluid-particle suspension over an exponentially stretching sheet. Ain Shams Engineering Journal, 5, 613–624 (2014)

    Article  Google Scholar 

  11. MEGAHED, A. M. Variable heat flux effect on MHD flow and heat transfer over an unsteady stretching sheet in the presence of thermal radiation. Canadian Journal of Physics, 92, 86–91 (2014)

    Article  Google Scholar 

  12. HAYAT, T., SHAFIQ, A., ALSAEDI, A., and SHAHZAD, S. A. Unsteady MHD flow over exponentially stretching sheet with slip conditions. Applied Mathematics and Mechanics (English Edition), 37(2), 193–208 (2016) https://doi.org/10.1007/s10483-016-2024-8

    Article  MathSciNet  Google Scholar 

  13. GNANESWARA REDDY, M. Chemically reactive species and radiation effects on MHD convective flow past a moving vertical cylinder. Ain Shams Engineering Journal, 4, 879–888 (2013)

    Article  Google Scholar 

  14. MEGAHED, A. M. Numerical solution for variable viscosity and internal heat generation effects on boundary layer flow over an exponentially stretching porous sheet with constant heat flux and thermal radiation. Journal of Mechanics, 30, 395–402 (2014)

    Article  Google Scholar 

  15. GNANESWARA REDDY, M. Thermal radiation and chemical reaction effects on MHD mixed convective boundary layer slip flow in a porous medium with heat source and Ohmic heating. The European Physical Journal Plus, 129, 41 (2014)

    Article  Google Scholar 

  16. GNANESWARA REDDY, M. Cattaneo-Christov heat flux effect on hydromagnetic radiative Oldroyd-B liquid flow across a cone/wedge in the presence of cross-diffusion. The European Physical Journal Plus, 133, 24 (2018)

    Article  Google Scholar 

  17. CARREAU, P. J. Rheological equations from molecular network theories. Transactions of the Society of Rheology, 16, 99–127 (1972)

    Article  Google Scholar 

  18. ANRADE, L. C. F., PETRONILIO, J. A., and MANESCHY, C. E. A. The Carreau-Yasuda fluids: a skin friction equation for turbulent flow in pipes and kolmogorov dissipative scales. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 29, 162–167 (2007)

    Google Scholar 

  19. HAYAT, T., ASAD, S., MUSTAFA, M., and ALSAEDI, A. Boundary layer flow of Carreau fluid over a convectively heated stretching sheet. Applied Mathematics and Computation, 246, 12–22 (2014)

    Article  MathSciNet  Google Scholar 

  20. KHAN, M. and AZAM, M. Unsteady boundary layer flow of Carreau fluid over a permeable stretching surface. Results in Physics, 6, 1168–1174 (2016)

    Article  Google Scholar 

  21. WAQAS, M., KHAN, M. I., HAYAT, T., and ALSAEDI, A. Numerical simulation for magneto Carreau nanofluid model with thermal radiation: a revised model. Computer Methods in Applied Mechanics and Engineering, 324, 640–653 (2017)

    Article  MathSciNet  Google Scholar 

  22. ALI, U., UR REHMANA, K., MALIKA, M. Y., and ZEHRA, I. Thermal aspects of Carreau fluid around a wedge. Case Studies in Thermal Engineering, 12, 462–469 (2018)

    Article  Google Scholar 

  23. HASHIM, A. and MASOOD, K. Numerical simulation for flow and heat transfer to Carreau fluid with magnetic field effect: dual nature study. Journal of Magnetism and Magnetic Materials, 443, 13–21 (2017)

    Article  Google Scholar 

  24. HASHIM, KHAN, M., ALSHOMRANI, A. S., and UL HAQ, R. Investigation of dual solutions in flow of a non-Newtonian fluid with homogeneousheterogeneous reactions: critical points. European Journal of Mechanics-B/Fluids, 68, 30–38 (2018)

    Article  MathSciNet  Google Scholar 

  25. MASOOD, K., HUMARA, S., and HASHIM. Heat generation/absorption and thermal radiation impacts on three-dimensional flow of Carreau fluid with convective heat transfer. Journal of Molecular Liquids, 272, 474–480 (2018)

    Article  Google Scholar 

  26. MACHIREDDY, G. R. and NARAMGARI, S. Heat and mass transfer in radiative MHD Carreau fluid with cross diffusion. Ain Shams Engineering Journal, 9, 1189–1204 (2018)

    Article  Google Scholar 

  27. HASHIM, KHAN, M., UL HUDA, N., and HAMID, A. Non-linear radiative heat transfer analysis during the flow of Carreau nanofluid due to wedge-geometry: a revised model. International Journal of Heat and Mass Transfer, 131, 1022–1031 (2019)

    Article  Google Scholar 

  28. KHAN, M. and HASHIM. Boundary layer flow and heat transfer to Carreau fluid over a nonlinear stretching sheet. AIP Advances, 5, 107203 (2015)

    Article  Google Scholar 

  29. RAPTIS, A. Flow of a micropolar fluid past a continuously moving plate by the presence of radiation. International Journal of Heat and Mass Transfer, 41, 2865–2866 (1998)

    Article  Google Scholar 

  30. RAPTIS, A. Radiation and viscoelastic flow. International Communications in Heat and Mass Transfer, 26, 889–895 (1999)

    Article  Google Scholar 

  31. MAHMOUD, M. A. A. Variable fluid properties effects on hydromagnetic fluid flow over an exponentially stretching sheet. Open Science Journal of Mathematics and Application, 3, 26–33 (2015)

    Google Scholar 

  32. MAHMOUD, M. A. A. and MEGAHED, A. M. MHD flow and heat transfer in a non-Newtonian liquid film over an unsteady stretching sheet with variable fluid properties. Canadian Journal of Physics, 87, 1065–1071 (2009)

    Article  Google Scholar 

Download references

Acknowledgements

The suggestions and constructive comments of the honorable editors and anonymous reviewers are appreciated from the author.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt

    A. M. Megahed

Authors
  1. A. M. Megahed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. M. Megahed.

Additional information

Citation: MEGAHED, A. M. Carreau fluid flow due to nonlinearly stretching sheet with thermal radiation, heat flux, and variable conductivity. Applied Mathematics and Mechanics (English Edition), 40(11), 1615–1624 (2019) https://doi.org/10.1007/s10483-019-2534-6

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Megahed, A.M. Carreau fluid flow due to nonlinearly stretching sheet with thermal radiation, heat flux, and variable conductivity. Appl. Math. Mech.-Engl. Ed. 40, 1615–1624 (2019). https://doi.org/10.1007/s10483-019-2534-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-019-2534-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation