Wavelet analysis of stagnation point flow of non-Newtonian nanofluid

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


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 



Buoyancy effect due to the temperature difference


Buoyancy effect due to the concentration difference


coefficient of the mean absorption


coefficient of the concentration expansion


density of the fluid


electrical conductivity


Hartmann number


magnetic field strength


mean fluid temperature


ratio of the thermal diffusion


Reynolds number


stagnation point parameter


Soret number


thermal conductivity


unsteadiness parameter


Boltzmann constant


concentration susceptibility


coefficient of the Williamson fluid


coefficient of thermal expansion


Dufour number


gravitational acceleration


kinematic viscosity


mass diffusivity coefficient


Prandtl number


ratio of λ* to λ


specific heat


suction parameter


Schmidt number


thermal radiation


Williamson fluid parameter

Chinese Library Classification


2010 Mathematics Subject Classification

42C40 76A05 76W05 


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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.


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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

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