Skip to main content
SpringerLink
Log in

Boundary Layer Flow and Thermal Analysis of a Cu-Nanoliquid Past a Stretching Cylinder

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

We study the steady, two-dimensional boundary layer flow and heat transport caused by a stretching cylinder immersed in an incompressible, viscous nanoliquid. The thermal analysis has been done for prescribed surface temperature as well as for prescribed heat flux. The governing partial differential equations in cylindrical form for momentum and heat transfer are reduced to non-linear ordinary differential equations by using similarity transformation. The differential transform method is used to solve these non-linear ordinary differential equations, under appropriate boundary conditions, in the form of power series. The inverse Domb–Sykes plots between number of terms against ratio of consecutive coefficients of the power series are sketched to determine the minimum number of terms required in the series to ensure convergence. A rational approximation in the form of Pad\(\acute{e}\) approximant is then applied to increase the convergence of the power series. Results for skin friction coefficient and surface heat transfer have been presented as a property of nanoparticle volume fraction, curvature parameter and Prandtl number. Dilute concentration \((0<\phi \le 0.2)\) of copper-nanoparticle is considered. Comparison of numerical results are made with previously published works in some limiting cases. The problem is shown to reduce to the stretching sheet case in the absence of transverse curvature.

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
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

Abbreviations

\(C_p\) :

Specific heat at constant pressure

\(C_f\) :

Skin friction coefficient

\(Nu_x\) :

Local Nusselt number

Pr :

Prandtl number

R :

Radius of the cylinder

\(Re_x\) :

Local Reynolds number

t :

Dimensional local temperature of the nanoliquid

T :

Dimensionless local temperature of the nanoliquid

u, v :

Axial and radial velocity components

x, r :

Axial and radial coordinates

\(\alpha _{nl}\) :

Thermal diffusivity of nanoliquid

\(\eta \) :

Similarity variable

\(\gamma \) :

Transverse curvature parameter

\(\kappa \) :

Temperature-dependent-thermal conductivity

\(\phi \) :

Volume fraction of solid nanoparticle

\(\psi \) :

Stream function

\(\rho \) :

Density

\(\tau _w\) :

Surface shear stress

\(\theta \) :

Dimensionless temperature

\(\vartheta _{nl}\) :

Kinematic viscosity of nanoliquid

\(\textit{nl}\) :

Nanoliquid

\(\textit{bl}\) :

Baseliquid

\(\textit{p}\) :

Solid nanoparticle

\(*\) :

Dimensional quantities

References

  1. Ashorynejad, H.R., Sheikholeslami, M., Pop, I., Ganji, D.D.: Nanofluid flow and heat transfer due to a stretching cylinder in the presence of magnetic field. Heat Mass Transf. 49, 427–436 (2013). doi:10.1007/s00231-012-1087-6

    Article  Google Scholar 

  2. Bachok, N., Ishak, A.: Mixed convection boundary layer flow over a permeable vertical cylinder with prescribed surface heat flux. Eur. J. Sci. Res. 34(1), 46–54 (2009)

    Google Scholar 

  3. Bachok, N., Ishak, A.: Flow and heat transfer over a stretching cylinder with prescribed surface heat flux. Malays. J. Math. Sci. 4(2), 159–169 (2010)

    MATH  Google Scholar 

  4. Baker, G.A.: Essentials of Padé approximants. Academic Press, London (1975)

    MATH  Google Scholar 

  5. Buongiorno, J.: Convective transport in nanofluids. J. Heat Transf. 128, 240–250 (2006)

    Article  Google Scholar 

  6. Buongiorno, J., Hu, W.: Nanofluid coolants for advanced nuclear power plants. In: Proceedings of ICAPP’05, Seoul (2005)

  7. Choi, S.U.S., Eastman J.: Enhancing thermal conductivity of fluids with nanoparticles. In: Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, vol. FED 231/MD 66 (ASME, San Francisco, USA, 1995), pp. 99–105

  8. Crane, L.J.: Boundary layer flow due to a stretching cylinder. J. Appl. Math. Phys. (ZAMP) 26, 619–622 (1975)

    Article  MATH  Google Scholar 

  9. Daungthongsuk, W., Wongwises, S.: A critical review of convective heat transfer nanofluids. Renew. Sustain. Energy Rev. 11, 797–817 (2007)

    Article  Google Scholar 

  10. Domb, C., Sykes, M.F.: On the susceptibility of a ferromagnetic above the curie point. In: Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 240 (The Royal Society, London, 1957), pp. 214–228

  11. Elbashbeshy, E.M.A.: Heat transfer over an exponentially stretching continuous surface with suction. Arch. Mech. 53(6), 643–651 (2001)

    MATH  Google Scholar 

  12. Gorla, R.S.R., Chamkha, A., Al-Meshaiei, E.: Melting heat transfer in a nanofluid boundary layer on a stretching circular cylinder. J. Naval Archit. Mar. Eng. 9, 1–10 (2012). doi:10.3329/jname.v9i1.7416. http://www.banglajol.info

  13. Gorla, R.S.R., El-Kabeir, S.M.M., Rashad, A.M.: Boundary-layer heat transfer from a stretching circular cylinder in a nanofluid. J. Thermophys. Heat Transf. 25(1), 183–186 (2011). doi:10.2514/1.51615

    Article  Google Scholar 

  14. Hamad, M.A.A.: Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field. Int. Comm. Heat. Mass. Transf. 38(4), 487–492 (2011)

    Article  Google Scholar 

  15. Ishak, A., Nazar, R.: Laminar boundary layer flow along a stretching cylinder. Eur. J. Sci. Res. 36, 22–29 (2009)

    Google Scholar 

  16. Ishak, A., Nazar, R., Pop, I.: Uniform suction/blowing effect on flow and heat transfer due to a stretching cylinder. Appl. Math. Model. 2, 2059–2060 (2008)

    Article  MATH  Google Scholar 

  17. Khanafer, K., Vafai, K., Lightstone, M.: Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int. J. Heat Mass Transf. 46, 3639–3653 (2003)

    Article  MATH  Google Scholar 

  18. Kuznetsov, A.V., Nield, D.A.: Natural convective boundary-layer flow of a nanofluid past a vertical plate. Int. Thermal Sci. 49, 243–247 (2010)

    Article  Google Scholar 

  19. Lin, H.T., Shih, Y.P.: Laminar boundary layer heat transfer along static and moving cylinders. J. Chin. Inst. Eng. 3, 73–79 (1980)

    Article  Google Scholar 

  20. Malik, M.Y., Naseer, M., Nadeem, S., Rehman, A.: The boundary layer flow of casson nanofluid over a vertical exponentially stretching cylinder. Appl. Nanosci. 4, 869–873 (2014). doi:10.1007/s13204-013-0267-0

    Article  Google Scholar 

  21. Masuda, H., Ebata, A., Teramae, K., Hishinuma, N.: Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei 7(4), 227–233 (1993)

    Article  Google Scholar 

  22. Maxwell, J.C.: Treatise on Electricity and Magnetism, vol. 1, 2nd edn. Oxford University Press, London (1904)

    MATH  Google Scholar 

  23. Oztop, H.F., Abu-Nada, E.: Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int. J. Heat Fluid Flow 29, 1326–1336 (2008)

    Article  Google Scholar 

  24. Rangi, R.R., Ahmed, N.: Boundary layer flow past a stretching cylinder and heat transfer with variable thermal conductivity. Appl. Math. 3, 205–209 (2012)

    Article  MathSciNet  Google Scholar 

  25. Rasekh, A., Ganji, D.D., Tavakoli, S.: Numerical solutions for a nanofluid past over a stretching circular cylinder with non-uniform heat source. Front. Heat Mass Transf. (FHMT) 3(043003), 1–6 (2012). doi:10.5098/hmt.v3.4.3003

    Google Scholar 

  26. Rashad, A.M., Chamkha, A.J., Modather, M.: Mixed convection boundary-layer flow past a horizontal circular cylinder embedded in a porous medium filled with a nanofluid under convective boundary condition. Comput. Fluids 86, 380–388 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sakiadis, B.C.: Boundary-layer behavior on continuous solid surfaces iii: the boundary layer on a continuous cylindrical surface. AIChE J. 7, 467–471 (1961)

    Article  Google Scholar 

  28. Siddheshwar, P.G., Meenakshi, N.: Amplitude equation and heat transport for Rayleigh–Bénard convection in Newtonian liquids with nanoparticles. Int. J. Appl. Comp. Math. (2015). doi:10.1007/s40819-015-0106-y

  29. Sinha, D., Jain, P., Tomer, N.S.: Computer-assisted power series solution for MHD boundary layer flow of a weakly electrically conducting nanoliquid past a stretching sheet. Open J. Heat Mass Momentum Transf. 2(2), 47–57 (2014)

    Article  Google Scholar 

  30. Sinha, D., Jain, P., Siddheshwar, P.G., Tomer, N.S.: Forced convective flow of a nanoliquid due to a stretching cylinder with free stream. J. Appl Fluid Mech. (JAFM) 9(1), 463–474 (2016)

    Article  Google Scholar 

  31. Tiwari, R.K., Das, M.K.: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 50, 2002–2018 (2007)

    Article  MATH  Google Scholar 

  32. Wang, C.Y.: Fluid flow due to a stretching cylinder. Phys. Fluids 31, 466–468 (1988)

    Article  Google Scholar 

  33. Wang, C.Y.: Stagnation flow on a cylinder with partial slip—an exact solution of the Navier–Stokes equations. IMA J. Appl. Math. 72, 271–277 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, C.Y.: Natural convection on a vertical stretching cylinder. Commun. Nonlinear Sci. Numer. Simulat. 17, 1098–1103 (2012)

    Article  MathSciNet  Google Scholar 

  35. Wang, C.Y., Ng, C.-O.: Slip flow due to a stretching cylinder. Int. J. Non-linear Mech. 46(9), 1191–1194 (2011)

    Article  Google Scholar 

  36. Wang, L., Wei, X.: Heat conduction in nanofluids. Chaos Solitons Fractals 39, 2211–2215 (2009)

    Article  Google Scholar 

  37. Zhou, J.K.: Differential transformation and its applications for electrical circuits. Huazhong University Press, Wuhan (1986). (in Chinese)

    Google Scholar 

Download references

Acknowledgments

Research is supported by the Department of Science and Technology (Government of India), New Delhi, India under the project SR/S4/MS: 409 / 06. The project is being implemented at the Centre for Mathematical Sciences, Banasthali University, Rajasthan, India. The work was initiated during the visit of one of the authors (PGS) to Banasthali University.

Author information

Authors and Affiliations

  1. Department of Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, India

    Deepa Sinha

  2. Department of Mathematics and Statistics, Banasthali University, Banasthali, Rajasthan, India

    Preeti Jain

  3. Department of Mathematics, Bangalore University, Jnana Bharathi Campus, Bangalore, India

    P. G. Siddheshwar

  4. Department of Mathematics, F.G.M. Govt. College, Adampur, Hisar, Haryana, India

    N. S. Tomer

Authors
  1. Deepa Sinha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Preeti Jain
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. G. Siddheshwar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. N. S. Tomer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Deepa Sinha.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sinha, D., Jain, P., Siddheshwar, P.G. et al. Boundary Layer Flow and Thermal Analysis of a Cu-Nanoliquid Past a Stretching Cylinder. Int. J. Appl. Comput. Math 3, 2559–2572 (2017). https://doi.org/10.1007/s40819-016-0255-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40819-016-0255-7

Keywords

Search

Navigation