Skip to main content
SpringerLink
Log in

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

  • Technical Paper
  • Published:
Journal of the Brazilian Society of Mechanical Sciences and Engineering Aims and scope Submit manuscript

Abstract

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.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Abbreviations

2b :

Distance between the plates (m)

\(C_{\mathrm{p}}\) :

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

Ec :

Eckert number

k :

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

Pr :

Prandtl number

T :

Temperature (K)

\(\mu \) :

Dynamic viscosity    (N s m\(^{-2}\))

\(\rho \) :

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

\(\theta \) :

Dimensionless temperature

\(\phi \) :

Solid volume fraction

\(\delta \) :

Dimensionless non-Newtonian viscosity

f:

Pure fluid

s:

Nanoparticle

nf:

Nanofluid

References

  1. Sheikholeslami M (2018) Influence of magnetic field on Al2O3–H2O nanofluid forced convection heat transfer in a porous lid driven cavity with hot sphere obstacle by means of LBM. J Mol Liq 263:472–488

    Article  Google Scholar 

  2. Sheikholeslami M, Jafaryar M, Li Z (2018) Second law analysis for nanofluid turbulent flow inside a circular duct in presence of twisted tape turbulators. J Mol Liq 263:489–500

    Article  Google Scholar 

  3. Sheikholeslami M, Shehzad SA, Li Z, Shafee A (2018) Numerical modeling for alumina nanofluid magnetohydrodynamic convective heat transfer in a permeable medium using Darcy law. Int J Heat Mass Transf 127:614–622

    Article  Google Scholar 

  4. Sheikholeslami M, Li Z, Shafee A (2018) Lorentz forces effect on NEPCM heat transfer during solidification in a porous energy storage system. Int J Heat Mass Transf 127:665–674

    Article  Google Scholar 

  5. Sheikholeslami M (2018) Solidification of NEPCM under the effect of magnetic field in a porous thermal energy storage enclosure using CuO nanoparticles. J Mol Liq 263:303–315

    Article  Google Scholar 

  6. Sheikholeslami M, Rokni HB (2017) Influence of melting surface on MHD nanofluid flow by means of two phase model. Chin J Phys 55:1352–1360

    Article  Google Scholar 

  7. Sheikholeslami M, Rokni HB (2017) Numerical modeling of nanofluid natural convection in a semi annulus in existence of Lorentz force. Comput Methods Appl Mech Eng 317:419–430

    Article  MathSciNet  Google Scholar 

  8. Sheikholeslami M, Rokni HB (2017) Nanofluid two phase model analysis in existence of induced magnetic field. Int J Heat Mass Transf 107:288–299

    Article  Google Scholar 

  9. Sheikholeslami M, Rokni HB (2017) Simulation of nanofluid heat transfer in presence of magnetic field: a review. Int J Heat Mass Transf 115:1203–1233

    Article  Google Scholar 

  10. Shahmohamadi H, Rashidi MM (2016) VIM solution of squeezing MHD nanofluid flow in a rotating channel with lower stretching porous surface. Adv Powder Technol 27:171–178

    Article  Google Scholar 

  11. Ma Y, Mohebbi R, Rashidi MM, Yang Z (2018) Study of nanofluid forced convection heat transfer in a bent channel by means of lattice Boltzmann method. Phys Fluids 30:032001

    Article  Google Scholar 

  12. Abbas MA, Bai YQ, Rashidi MM, Bhatti MM (2015) Application of drug delivery in magnetohydrodynamics peristaltic blood flow of nanofluid in a non-uniform channel. J Mech Med Biol 16:1650052

    Article  Google Scholar 

  13. Makulati N, Kasaeipoor A, Rashidi MM (2016) Numerical study of natural convection of a water-alumina nanofluid in inclined C-shaped enclosures under the effect of magnetic field. Adv Powder Technol 27:661–672

    Article  Google Scholar 

  14. Hatami M, Ganji DD (2014) Natural convection of sodium alginate (SA) non-Newtonian nanofluid flow between two vertical flat plates by analytical and numerical methods. Case Stud Therm Eng 2:14–22

    Article  Google Scholar 

  15. Etbaeitabari A, Barakat M, Imani AA, Domairryd G, Jalili P (2013) An analytical heat transfer assessment and modeling in a natural convection between two infinite vertical parallel flat plates. J Mol Liq 188:252–257

    Article  Google Scholar 

  16. Seyedi SH, Saray BN, Nobari MRH (2015) Using interpolation scaling functions based on Galerkin method for solving non-Newtonian fluid flow between two vertical flat plates. Appl Math Comput 269:488–496

    MathSciNet  Google Scholar 

  17. Sahebi SAR, Pourziaei H, Feizi AR, Taheri MH, Rostamiyan Y, Ganji DD (2015) Numerical analysis of natural convection for non-Newtonian fluid conveying nanoparticles between two vertical parallel plates. Eur Phys J Plus 130:238–250

    Article  Google Scholar 

  18. Bruce RW, Na TY (1967) Natural convection flow of Powell–Eyring fluids between two vertical flat plates. ASME, New York

    Google Scholar 

  19. Rajagopal KR, Na TY (1985) Natural convection flow of a non-Newtonian fluid between two vertical plates. Acta Mech 54:239–246

    Article  Google Scholar 

  20. Minkowycz WJ, Sparrow EM, Abraham JP (2013) Nanoparticle heat transfer and fluid flow. CRC Press, Boca Raton

    Google Scholar 

  21. Ziabakhsh Z, Domairry G (2009) Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method. Commun Nonlinear Sci Numer Simul 14:1868–1880

    Article  Google Scholar 

  22. Sheikholeslami M, Ganji DD, Rashidi MM (2016) Magnetic field effect on unsteady nanofluid flow and heat transfer using Buongiorno model. J Magn Magn Mater 416:164–173

    Article  Google Scholar 

  23. Garoosi F, Jahanshaloo L, Rashidi MM, Badakhsh A, Ali ME (2015) Numerical simulation of natural convection of the nanofluid in heat exchangers using a Buongiorno model. Appl Math Comput 254:183–203

    MathSciNet  MATH  Google Scholar 

  24. Stenger F (1993) Numerical methods based on Sinc and analytic functions. Springer, New York

    Book  Google Scholar 

  25. Stenger F (2011) Handbook of Sinc numerical methods. CRC Press, Boca Raton

    MATH  Google Scholar 

  26. Lund J, Bowers K (1992) Sinc methods for quadrature and differential equations. Siam, Philadelphia

    Book  Google Scholar 

  27. Babolian E, Eftekhari A, Saadatmandi A (2015) A Sinc-Galerkin technique for the numerical solution of a class of singular boundary value problems. Comput Appl Math 34:45–63

    Article  MathSciNet  Google Scholar 

  28. Saadatmandi A, Razzaghi M, Dehghan M (2005) Sinc-Galerkin solution for nonlinear two-point boundary value problems with application to chemical reactor theory. Math Comput Model 42:1237–1244

    Article  MathSciNet  Google Scholar 

  29. Saadatmandi A, Asadi A, Eftekhari A (2016) Collocation method using quintic B-spline and sinc functions for solving a model of squeezing flow between two infinite plates. Int J Comput Math 93:1921–1936

    Article  MathSciNet  Google Scholar 

  30. Saadatmandi A, Razzaghi M, Dehghan M (2005) Sinc-collocation methods for the solution of Hallen’s integral equation. J Electromagn Waves Appl 19:245–256

    Article  MathSciNet  Google Scholar 

  31. Rashidinia J, Zarebnia M (2007) The numerical solution of integro-differential equation by means of the Sinc method. Appl Math Comput 188:1124–1130

    MathSciNet  MATH  Google Scholar 

  32. Winter DF, Bowers K, Lund J (2000) Wind-driven currents in a sea with a variable Eddy viscosity calculated via a Sinc-Galerkin technique. Int J Numer Methods Fluids 33:1041–1073

    Article  MathSciNet  Google Scholar 

  33. Parand K, Dehghan M, Pirkhedri A (2012) The use of Sinc-collocation method for solving Falkner–Skan boundary-layer equation. Int J Numer Methods Fluids 68:36–47

    Article  MathSciNet  Google Scholar 

  34. Parand K, Dehghan M, Pirkhedri A (2013) The Sinc-collocation method for solving the Thomas–Fermi equation. J Comput Appl Math 237:244–252

    Article  MathSciNet  Google Scholar 

  35. Parand K, Pirkhedri A (2010) Sinc-collocation method for solving astrophysics equations. New Astron 15:533–537

    Article  Google Scholar 

  36. Xuan Y, Roetzel W (2000) Conceptions for heat transfer correlations of nanofluids. Int J Heat Mass Transf 43:3701–3707

    Article  Google Scholar 

  37. Pawar SS, Sunnapwar VK (2013) Experimental studies on heat transfer to Newtonian and non-Newtonian fluids in helical coils with laminar and turbulent flow. Exp Therm Fluid Sci 44:792–804

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, 87317-53153, Iran

    Abbas Saadatmandi

  2. School of Chemistry, College of Science, University of Tehran, Tehran, Iran

    Saeid Shateri

Authors
  1. Abbas Saadatmandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saeid Shateri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abbas Saadatmandi.

Additional information

Technical Editor: Cezar Negrao, PhD.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saadatmandi, A., Shateri, S. Sinc-collocation method for solving sodium alginate (SA) non-Newtonian nanofluid flow between two vertical flat plates. J Braz. Soc. Mech. Sci. Eng. 41, 158 (2019). https://doi.org/10.1007/s40430-019-1665-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40430-019-1665-3

Keywords

Search

Navigation