Advertisement

Wavelet analysis of stagnation point flow of non-Newtonian nanofluid

  • M. Hamid
  • M. Usman
  • R. U. Haq
  • Z. H. KhanEmail author
  • Wei Wang
Article

Abstract

The wavelet approach is introduced to study the influence of the natural convection stagnation point flow of the Williamson fluid in the presence of thermophysical and Brownian motion effects. The thermal radiation effects are considered along a permeable stretching surface. The nonlinear problem is simulated numerically by using a novel algorithm based upon the Chebyshev wavelets. It is noticed that the velocity of the Williamson fluid increases for assisting flow cases while decreases for opposing flow cases when the unsteadiness and suction parameters increase, and the magnetic effect on the velocity increases for opposing flow cases while decreases for assisting flow cases. When the thermal radiation parameter, the Dufour number, and Williamson’s fluid parameter increase, the temperature increases for both assisting and opposing flow cases. Meanwhile, the temperature decreases when the Prandtl number increases. The concentration decreases when the Soret parameter increases, while increases when the Schmidt number increases. It is perceived that the assisting force decreases more than the opposing force. The findings endorse the credibility of the proposed algorithm, and could be extended to other nonlinear problems with complex nature.

Key words

Williamson nanofluid heat and mass transfer stagnation point flow assisting and opposing flow Chebyshev wavelet method 

Nomenclature

λ

Buoyancy effect due to the temperature difference

λ*

Buoyancy effect due to the concentration difference

k*

coefficient of the mean absorption

βC

coefficient of the concentration expansion

ρ

density of the fluid

σ

electrical conductivity

M

Hartmann number

B

magnetic field strength

Tm

mean fluid temperature

cs

ratio of the thermal diffusion

Re

Reynolds number

r

stagnation point parameter

Sr

Soret number

k

thermal conductivity

A

unsteadiness parameter

σ*

Boltzmann constant

kT

concentration susceptibility

Γ

coefficient of the Williamson fluid

βT

coefficient of thermal expansion

Du

Dufour number

g

gravitational acceleration

ν

kinematic viscosity

DB

mass diffusivity coefficient

Pr

Prandtl number

R

ratio of λ* to λ

cp

specific heat

S

suction parameter

Sc

Schmidt number

Rd

thermal radiation

Λ

Williamson fluid parameter

Chinese Library Classification

O361 

2010 Mathematics Subject Classification

42C40 76A05 76W05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are grateful to the reviewers and editor for suggesting suitable changes in the original manuscript. The first author is also grateful to the China Scholarship Council (CSC) for the financial assistance.

References

  1. [1]
    HIEMENZ, K. Die grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Dingler’s Polytechnic Journal, 326, 321–324 (1911)Google Scholar
  2. [2]
    HOMANN, F. and ANGEW, Z. Der einfluss grosser Zähigkeit bei der Strömung um den zylinder und um die Kugel. Journal of Applied Mathematics and Mechanics, 16, 153–164 (1936)zbMATHGoogle Scholar
  3. [3]
    MAKINDE, O. D., KHAN, W. A., and KHAN, Z. H. Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet. International Journal of Heat and Mass Transfer, 62, 526–533 (2013)Google Scholar
  4. [4]
    HAQ, R. U., NADEEM, S., KHAN, Z. H., and AKBAR, N. S. Thermal radiation and slip effects on MHD stagnation point flow of nanofluid over a stretching sheet. Physica E: Low-Dimensional Systems and Nanostructures, 65, 17–23 (2015)Google Scholar
  5. [5]
    HAYAT, T., KHAN, M. I., WAQAS, M., ALSAEDI, A., and FAROOQ, M. Numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon-water nanofluid. Computer Methods in Applied Mechanics and Engineering, 315, 1011–1024 (2017)MathSciNetGoogle Scholar
  6. [6]
    TARAKARAMU, N. and NARAYANA, P. V. Nonlinear thermal radiation and Joule heating effects on MHD stagnation point flow of a nanofluid over a convectively heated stretching surface. Journal of Nanofluids, 8, 1066–1075 (2019)Google Scholar
  7. [7]
    PAL, D. Heat and mass transfer in stagnation-point flow towards a stretching surface in the presence of buoyancy force and thermal radiation. Meccanica, 44, 145–158 (2009)zbMATHGoogle Scholar
  8. [8]
    USMAN, M., ZUBAIR, T., HAMID, M., HAQ, R. U., and WANG, W. Wavelets solution of MHD 3-D fluid flow in the presence of slip and thermal radiation effects. Physics of Fluids, 30, 023104 (2018)Google Scholar
  9. [9]
    MOHYUD-DIN, S. T., USMAN, M., AFAQ, K., HAMID, M., and WANG, W. Examination of carbon-water nanofluid flow with thermal radiation under the effect of Marangoni convection. Engineering Computations, 34, 2330–2343 (2017)Google Scholar
  10. [10]
    HAYAT, T., QASIM, M., SHEHZAD, S. A., and ALSAEDI, A. Unsteady stagnation point flow of second grade fluid with variable free stream. Alexandria Engineering Journal, 53, 455–461 (2014)Google Scholar
  11. [11]
    BOULAHIA, Z., WAKIF, A., and SEHAQUI, R. Heat transfer and cu-water nanofluid flow in a ventilated cavity having central cooling cylinder and heated from the below considering three different outlet port locations. Frontiers in Heat and Mass Transfer, 11, 11 (2018)Google Scholar
  12. [12]
    SAIDUR, R., LEONG, K. Y., and MOHAMMAD, H. A review on applications and challenges of nanofluids. Renewable and Sustainable Energy Reviews, 15, 1646–1668 (2011)Google Scholar
  13. [13]
    CHOI, S. U. S. Enhancing thermal conductivity of fluids with nanoparticle. Developments and Applications of Non-Newtonian Flows, Springer, New York (1995)Google Scholar
  14. [14]
    BUONGIORNO, J. Convective transport in nanofluids. Journal of Heat Transfer, 128, 240–250 (2006)Google Scholar
  15. [15]
    TIWARI, R. K. and DAS, M. K. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. International Journal of Heat and Mass Transfer, 50, 2002–2018 (2007)zbMATHGoogle Scholar
  16. [16]
    HAYAT, T., ABBASI, F. M., AL-YAMI, M., and MONAQUEL, S. Slip and Joule heating effects in mixed convection peristaltic transport of nanofluid with Soret and Dufour effects. Journal of Molecular Liquids, 194, 93–99 (2014)Google Scholar
  17. [17]
    REDDY, P. S. and CHAMKHA, A. J. Soret and Dufour effects on MHD convective flow of Al2O3-water and TiO2-water nanofluids past a stretching sheet in porous media with heat generation/absorption. Advanced Powder Technology, 27, 1207–1218 (2016)Google Scholar
  18. [18]
    HAYAT, T., FAROOQ, S., ALSAEDI, A., and AHMAD, B. Numerical study for Soret and Dufour effects on mixed convective peristalsis of Oldroyd 8-constants fluid. International Journal of Thermal Sciences, 112, 68–81 (2017)Google Scholar
  19. [19]
    HAMID, M., USMAN, M., ZUBAIR, T., HAQ, R. U., and WANG, W. Shape effects of MoS2 nanoparticles on rotating flow of nanofluid along a stretching surface with variable thermal conductivity: a Galerkin approach. International Journal of Heat and Mass Transfer, 124, 706–714 (2018)Google Scholar
  20. [20]
    USMAN, M., HAMID, M., MOHYUD-DIN, S. T., WAHEED, A., and WANG, W. Exploration of uniform heat flux on the flow and heat transportation of ferrofluids along a smooth plate: comparative investigation. International Journal of Biomathematics, 11, 1850048 (2018)MathSciNetzbMATHGoogle Scholar
  21. [21]
    SHEIKHOLESLAMI, M. and ROKNI, H. B. Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation. International Journal of Heat and Mass Transfer, 118, 823–831 (2018)Google Scholar
  22. [22]
    WILLIAMSON, R. V. The flow of pseudoplastic materials. Industrial and Engineering Chemistry, 21, 1108–1111 (1929)Google Scholar
  23. [23]
    AMANULLA, C. H., NAGENDRA, N., RAO, A. S., BEG, O. A., and KADIR, A. Numerical exploration of thermal radiation and Biot number effects on the flow of a non-Newtonian MHD Williamson fluid over a vertical convective surface. Heat Transfer-Asian Research, 47, 286–304 (2018)Google Scholar
  24. [24]
    JAIN, S. and PARMAR, A. Radiation effect on MHD Williamson fluid flow over stretching cylinder through porous medium with heat source. Applications of Fluid Dynamics, Springer, Singapore (2018)Google Scholar
  25. [25]
    HAMID, M., USMAN, M., KHAN, Z. H., HAQ, R. U., and WANG, W. Numerical study of unsteady MHD flow of Williamson nanofluid in a permeable channel with heat source/sink and thermal radiation. The European Physical Journal Plus, 133, 527 (2018)Google Scholar
  26. [26]
    MUSTAFA, M., KHAN, J. A., HAYAT, T., and ALSAEDI, A. Buoyancy effects on the MHD nanofluid flow past a vertical surface with chemical reaction and activation energy. International Journal of Heat and Mass Transfer, 108, 1340–1346 (2017)Google Scholar
  27. [27]
    NOOR, N. F., HAQ, R. U., NADEEM, S., and HASHIM, I. Mixed convection stagnation flow of a micropolar nanofluid along a vertically stretching surface with slip effects. Meccanica, 50, 2007–2022 (2015)MathSciNetGoogle Scholar
  28. [28]
    HAQ, R. U., NADEEM, S., AKBAR, N. S., and KHAN, Z. H. Buoyancy and radiation effect on stagnation point flow of micropolar nanofluid along a vertically convective stretching surface. IEEE Transactions on Nanotechnology, 14, 42–50 (2015)Google Scholar
  29. [29]
    AKBAR, N. S., TRIPATHI, D., KHAN, Z. H., and BEG, O. A. A numerical study of magnetohydrodynamic transport of nanofluids over a vertical stretching sheet with exponential temperature-dependent viscosity and buoyancy effects. Chemical Physics Letters, 661, 20–30 (2016)Google Scholar
  30. [30]
    DOHA, E. H., ABD-ELHAMEED, W. M., and ALSUYUTI, M. M. On using third and fourth kinds Chebyshev polynomials for solving the integrated forms of high odd-order linear boundary value problems. Journal of the Egyptian Mathematical Society, 23, 397–405 (2015)MathSciNetzbMATHGoogle Scholar
  31. [31]
    ZHOU, F. and XU, X. The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients. Applied Mathematics and Computation, 280, 11–29 (2016)MathSciNetzbMATHGoogle Scholar
  32. [32]
    ABD-ELHAMEED, W. M., DOHA, E. H., and YOUSSRI, Y. H. New wavelets collocation method for solving second-order multipoint boundary value problems using Chebyshev polynomials of third and fourth kinds. Abstract and Applied Analysis, 2013, 542839 (2013)MathSciNetzbMATHGoogle Scholar
  33. [33]
    MUKHOPADHYAY, S. Effect of thermal radiation on unsteady mixed convection flow and heat transfer over a porous stretching surface in a porous medium. International Journal of Heat and Mass Transfer, 52, 3261–3265 (2009)zbMATHGoogle Scholar
  34. [34]
    GRUBKA, L. J. and BOBBA, K. M. Heat transfer characteristics of a continuous stretching surface with variable temperature. Journal of Heat Transfer, 107, 248–250 (1985)Google Scholar
  35. [35]
    CHEN, C. H. Laminar mixed convection adjacent to vertical, continuously stretching sheets. Heat and Mass Transfer, 33, 471–476 (1998)Google Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • M. Hamid
    • 1
  • M. Usman
    • 2
    • 3
  • R. U. Haq
    • 4
  • Z. H. Khan
    • 5
    • 6
    Email author
  • Wei Wang
    • 1
  1. 1.School of Mathematical SciencesPeking UniversityBeijingChina
  2. 2.BIC-ESAT, College of EngineeringPeking UniversityBeijingChina
  3. 3.State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering SciencePeking UniversityBeijingChina
  4. 4.Department of Electrical EngineeringBahria UniversityIslamabadPakistan
  5. 5.State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource & HydropowerSichuan UniversityChengduChina
  6. 6.Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of EducationTsinghua UniversityBeijingChina

Personalised recommendations