Sinc-collocation method for solving sodium alginate (SA) non-Newtonian nanofluid flow between two vertical flat plates

  • Abbas SaadatmandiEmail author
  • Saeid Shateri
Technical Paper


This paper seeks to develop an efficient method for solving natural convection of a non-Newtonian nanofluid flow between two vertical flat plates. Sodium alginate is considered as the non-Newtonian fluid, and then two distinct types of nanoparticles, namely silver and copper, are added to it. To do so, the governing boundary layer and temperature equations are reduced to a set of ordinary differential equations. This approach is based on a global collocation method using Sinc basis functions, and the resulting set of ordinary differential equations are replaced by a system of algebraic equations. It is well known that the Sinc procedure converges to the solution at an exponential rate. Numerical results are included to demonstrate the validity and applicability of the method, and a comparison is made with the existing results. Also, the effect of various parameters such as Prandtl number (Pr), dimensionless non-Newtonian viscosity number \((\delta )\) and nanoparticle volume fraction (\(\phi \)) on non-dimensional velocity and temperature profiles are discussed. It was concluded from this study that velocity and temperature increased with increasing Pr. Moreover, our results indicate that both the velocity and temperature decrease as \(\delta \) increases. Finally, the results demonstrated that, when \(\phi \) increases, the velocity increases but the temperature values decrease.


Natural convection Sinc-collocation Non-Newtonian fluid Nanoparticles 

List of symbols


Distance between the plates (m)


Specific heat at constant pressure (J kg K\(^{-1}\))


Eckert number


Thermal conductivity (W m\(^{-1}\) K\(^{-1}\))


Prandtl number


Temperature (K)

Greek symbols

\(\mu \)

Dynamic viscosity    (N s m\(^{-2}\))

\(\rho \)

Density (kg m\(^{-3}\))

\(\theta \)

Dimensionless temperature

\(\phi \)

Solid volume fraction

\(\delta \)

Dimensionless non-Newtonian viscosity



Pure fluid







Authors would like to thank Prof. Ahmad Tavasoli (School of Chemistry, College of Science, University of Tehran, Tehran, Iran) for his useful comments. Also, the authors would like to thank the anonymous reviewers for their careful reading of the manuscript and their comments which substantially improved the quality of the paper.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesUniversity of KashanKashanIran
  2. 2.School of Chemistry, College of ScienceUniversity of TehranTehranIran

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