SpringerLink
Log in

Thin double-layer film development over a flat stretching sheet

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

Development of thin double-layer film over a stretching sheet is studied. Similarity variables have been applied to transform the guiding Navier–Stokes equations into a set of coupled unsteady nonlinear partial differential equations. These equations along with moving boundary conditions are solved using the asymptotic method. We have presented analytically obtained long time solution for smaller values of Reynolds number. For small and moderate values of Reynolds number, guiding equations were solved using the finite difference method. Effects of density ratio, viscosity ratio, thickness ratio and Reynolds number are analyzed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Crane, L.J.: Flow past a stretching plate. Z. Angew. Math. Phys. (ZAMP) 21, 645–647 (1970)

    Article  Google Scholar 

  2. Pavlov, K.B.: Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a plane surface. Magn. Gidrodin. 4, 146–147 (1974)

    Google Scholar 

  3. Gupta, P.S., Gupta, A.S.: Heat and mass transfer on a stretching sheet with suction or blowing. Can. J. Chem. Eng. 55, 744–746 (1977)

    Article  Google Scholar 

  4. Chappidi, P.R., Gunnerson, F.S.: Analysis of heat and momentum transport along a moving surface. Int. J. Heat Mass Transf. 32, 1383–1386 (1989)

    Article  Google Scholar 

  5. Cortell, R.: Flow and heat transfer in a moving fluid over a moving flat surface. Theor. Comput. Fluid Dyn. 21, 435–446 (2007)

    Article  MATH  Google Scholar 

  6. Andersson, H.I., Hansen, O.R., Holmedal, B.: Diffusion of a chemically reactive species from a stretching sheet. Int. J. Heat Mass Transf. 37, 659–664 (1994)

    Article  MATH  Google Scholar 

  7. Hayat, T., Javed, T., Sajid, M.: Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface. Phys. Lett. A 372, 3264–3273 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Uddin, M.J., Anwar Beg, O., Amin, N.: Hydromagnetic transport phenomena from a stretching or shrinking nonlinear nanomaterial sheet with Navier slip and convective heating: a model for bio-nano-materials processing. J. Magn. Magn. Mater. 368, 252–261 (2014)

    Article  Google Scholar 

  9. Lin, Wu: Mass transfer induced slip effect on viscous gas flows above a shrinking/stretching sheet. Int. J. Heat Mass Transf. 93, 17–22 (2016)

    Article  Google Scholar 

  10. Wang, C.Y.: Liquid film on an unsteady stretching surface. Q. Appl. Math. 48, 601–610 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Andersson, H.I., Aarseth, J.B., Braud, N., Dandapat, B.S.: Flow of a power-law fuid film on an unsteady stretching surface. J. Non-Newton. Fluid Mech. 62, 1–8 (1996)

    Article  Google Scholar 

  12. Andersson, H.I., Aarseth, J.B., Dandapat, B.S.: Heat transfer in a liquid film on an unsteady stretching surface. Int. J. Heat Mass Transf. 43, 69–74 (2000)

    Article  MATH  Google Scholar 

  13. Hayat, T., Saif, S., Abbas, Z.: The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet. Phys. Lett. A 372, 5037–5045 (2008)

    Article  MATH  Google Scholar 

  14. Dandapat, B.S., Santra, B., Andersson, H.I.: Thermocapillarity in a liquid film on an unsteady stretching surface. Int. J. Heat Mass Transf. 46, 3009–3015 (2003)

    Article  MATH  Google Scholar 

  15. Wang, C.: Analytical solutions for a liquid film on an unsteady stretching surface. Heat Mass Transf. 42, 759–766 (2006)

    Article  Google Scholar 

  16. Abel, M.S., Tawadea, J., Nandeppanavarb, M.M.: Effect of non-uniform heat source on MHD heat transfer in a liquid film over an unsteady stretching sheet. Int. J. Non-Linear Mech. 44, 990–998 (2009)

    Article  Google Scholar 

  17. Noor, N.F.M., Hashim, I.: Thermocapillarity and magnetic field effects in a thin liquid film on an unsteady stretching surface. Int. J. Heat Mass Transf. 53, 2044–2051 (2010)

    Article  MATH  Google Scholar 

  18. Lin, Y., Zheng, L., Chen, G.: Unsteady flow and heat transfer of pseudo-plastic nanoliquid in a finite thin film on a stretching surface with variable thermal conductivity and viscous dissipation. Powder Technol. 274, 324–332 (2015)

    Article  Google Scholar 

  19. Pal, D., Saha, P.: Influence of nonlinear thermal radiation and variable viscosity on hydromagnetic heat and mass transfer in a thin liquid film over an unsteady stretching surface. Int. J. Mech. Sci. 119, 208–216 (2016)

    Article  Google Scholar 

  20. Zhang, Y., Zhang, M., Bai, Yu.: Unsteady flow and heat transfer of power-law nanofluid thin film over a stretching sheet with variable magnetic field and power-law velocity slip effect. J. Taiwan Inst. Chem. Eng. 70, 104–110 (2017)

    Article  Google Scholar 

  21. Dandapat, B.S., Kitamura, A., Santra, B.: Transient film profile of thin liquid film flow on a stretching surface. ZAMP 57, 623–635 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Dandapat, B.S., Maity, S.: Flow of a thin liquid film on an unsteady stretching sheet. Phys. Fluids 18, 102101 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Santra, B., Dandapat, B.S.: Unsteady thin-film flow over a heated stretching sheet. Int. J. Heat Mass Transf. 52, 1965–1970 (2009)

    Article  MATH  Google Scholar 

  24. Dandapat, B.S., Singh, S.K.: Thin film flow over a heated nonlinear stretching sheet in presence of uniform transverse magnetic field. Int. Commun. Heat Mass Transf. 38, 324–328 (2011)

    Article  Google Scholar 

  25. Dandapat, B.S., Maity, S., Kitamura, A.: Liquid film flow due to an unsteady stretching sheet. Int. J. Non-Linear Mech. 43, 880–886 (2008)

    Article  MATH  Google Scholar 

  26. Maity, S.: Thermocapillary flow of thin liquid film over a porous stretching sheet in presence of suction/injection. Int. J. Heat Mass Transf. 70, 819–826 (2014)

    Article  Google Scholar 

  27. Maity, S., Ghatani, Y., Dandapat, B.S.: Thermocapillary Flow of a thin nanoliquid film over an unsteady stretching sheet. ASME J. Heat Transf. 138, 041501 (2016)

    Google Scholar 

  28. Sikder, U., Asif Zaman, M.: Optimization of multilayer antireflection coating for photovoltaic applications. Opt. Laser Technol. 79, 88–94 (2016)

    Article  Google Scholar 

  29. Kruk, T., Szczepanowicz, K., Kregiel, D., Szyk-Warszynska, L., Warszynski, P.: Nanostructured multilayer polyelectrolyte films with silver nanoparticles as antibacterial coatings. Colloids Surf. B Biointerfaces 137, 158–166 (2016)

    Article  Google Scholar 

  30. Mishra, S.K., Kumar, V., Tiwari, S.K., Mishra, T., Angula, G., Adhikari, S.: Development and degradation behavior of protective multilayer coatings for aluminum reflectors for solar thermal applications. Thin Solid Films 619, 202–207 (2016)

    Article  Google Scholar 

  31. Kitamura, A.: Application of thin film in potentially hazardous environment. In: Dandapat, B.S., Mazumder, B.S. (eds.) ICA Fluid Mechanics in Industry and Environment, p. 50. Research Publishing, Chennai (2006)

    Google Scholar 

  32. Fletcher, C.A.J.: Computational Techniques for Fluid Dynamics, vol. II. Springer, Berlin (1988)

    Book  MATH  Google Scholar 

  33. Rehg, T.J., Higgins, B.G.: The effects of inertia and interfacial shear on film flow on a rotating disk. Phys. Fluids 31, 1360–1371 (1998)

    Article  Google Scholar 

  34. Ferziger, J.H., Peric̀, M.: Computational Methods for Fluid Dynamics. Springer, Berlin (2002)

    Book  Google Scholar 

Download references

Acknowledgements

We express our sincere thanks to the referee for his valuable comments and suggestions to improve the contents of this article. S. K. Singh expresses his sincere thanks to the Science and Engineering Research Board, India for financial support under Young Scientist Scheme (S. No. YSS/2015/002085).

Author information

Authors and Affiliations

  1. Air Control Engineering Pvt. Ltd., 21 Hemanta Bose Sarani, Kolkata, 700001, India

    B. S. Dandapat

  2. Department of Basic and Applied Science, National Institute of Technology, Yupia, Papumpare, Arunachal Pradesh, 791112, India

    S. Maity

  3. Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore, 560064, India

    S. K. Singh

Authors
  1. B. S. Dandapat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Maity
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. K. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. K. Singh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dandapat, B.S., Maity, S. & Singh, S.K. Thin double-layer film development over a flat stretching sheet. Z. Angew. Math. Phys. 69, 109 (2018). https://doi.org/10.1007/s00033-018-1003-0

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-018-1003-0

Keywords

Mathematics Subject Classification

Search

Navigation