Skip to main content
Springer Nature Link
Log in

Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face

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

Abstract

Recently, in Tarzia (Thermal Sci 21A:1–11, 2017) for the classical two-phase Lamé–Clapeyron–Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction was obtained. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in Zhou et al. (J Eng Math 2017. https://doi.org/10.1007/s10665-017-9921-y).

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
$34.99 /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

Explore related subjects

Discover the latest articles and news from researchers in related subjects, suggested using machine learning.

Abbreviations

\(d_\mathrm{l}\), \(d_\mathrm{s}\) :

Diffusivity coefficient (m\(^2\)/s)

\(h_0\) :

Coefficient that characterizes the heat transfer in condition (1.6) [kg/(\(^{\circ }\)Cs\(^{5/2}\))]

\(k_\mathrm{l}\),\(k_\mathrm{s}\) :

Thermal conductivity [W/(m \(^{\circ }\)C)]

s :

Position of the free front (m)

t :

Time (s)

\(T_{\infty }\) :

Coefficient that characterizes the bulk temperature in condition (1.6) [\(^{\circ }\)C/s\(^{\alpha /2}\)]

\(T_i\) :

Coefficient that characterizes the initial temperature of the material in condition (1.7), [\(^{\circ }\)C/m\(^{\alpha }\)]

x :

Spatial coordinate (m)

\(\alpha \) :

Power of the position that characterizes the latent heat per unit volume (dimensionless)

\(\gamma \) :

Coefficient that characterizes the latent heat per unit volume [kg/(s\(^2\)m\(^{\alpha +1}\))]

\(\nu \) :

Coefficient that characterizes the free interface (dimensionless)

\(\eta \) :

Similarity variable in expression (2.1) (dimensionless)

\(\Psi _\mathrm{l},\Psi _\mathrm{s}\) :

Temperature (\(^{\circ }\)C).

l :

Liquid phase

s :

Solid phase

References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards, Washington (1964)

    MATH  Google Scholar 

  2. Alexiades, V., Solomon, A.D.: Mathematical Modelling of Melting and Freezing Processes. Hemisphere-Taylor Francis, Washington (1993)

    Google Scholar 

  3. Bollati, J., Tarzia, D.A.: Explicit solution for Stefan problem with latent heat depending on the position and a convective boundary condition at the fixed face using Kummer functions. Comm. Appl. Anal. (2017). https://arxiv.org/pdf/1610.09338v1.pdf

  4. Briozzo, A.C., Natale, M.F.: Nonlinear Stefan problem with convective boundary condition in Storm’s materials. Z. Angrew. Math. Phys. 67(19), 1–11 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Cannon, J.R.: The One-Dimensional Heat Equation. Addison-Wesley, Menlo Park (1984)

    Book  MATH  Google Scholar 

  6. Carslaw, H.S., Jaeger, C.J.: Conduction of Heat in Solids. Clarendon Press, Oxford (1959)

    MATH  Google Scholar 

  7. Crank, J.: Free and Moving Boundary Problem. Clarendon Press, Oxford (1984)

    MATH  Google Scholar 

  8. Gupta, S.C.: The Classical Stefan Problem. Basic Concepts, Modelling and Analysis. Elsevier, Amsterdam (2003)

    MATH  Google Scholar 

  9. Lorenzo-Trueba, J., Voller, V.R.: Analytical and numerical solution of a generalized Stefan Problem exhibiting two moving boundaries with application to ocean delta deformation. J. Math. Anal. Appl. 366, 538–549 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lunardini, V.J.: Heat Transfer With Freezing and Thawing. Elsevier, London (1991)

    Google Scholar 

  11. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010)

    MATH  Google Scholar 

  12. Perchuk, L.L.: Progress in Metamorphic and Magmatic Petrology. Cambridge University Press, Wallingford, UK (2003)

    Google Scholar 

  13. Primicerio, M.: Stefan-like problems with space-dependent latent heat. Meccanica 5, 187–190 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rogers, C.: Application of a reciprocal transformation to a two-phase Stefan problem. J. Phys. A Math. Gen. 18, L105–L109 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rubinstein, L.I.: The Stefan Problem. American Mathematical Society, Providence (1971)

    Google Scholar 

  16. Salva, N.N., Tarzia, D.A.: Explicit solution for a Stefan problem with variable latent heat and constant heat flux boundary conditions. J. Math. Anal. Appl. 379, 240–244 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tao, L.N.: The exact solutions of some Stefan problems with prescribed heat flux. J. Appl. Mech. 48, 732–736 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tarzia, D.A.: Relationship between Neumann solutions for two phase Lamé-Clapeyron-Stefan problems with convective and temperature boundary conditions. Thermal Sci. 21, 1–11 (2017)

    Article  Google Scholar 

  19. Tarzia D.A.: Explicit and approximated solutions for heat and mass transfer problems with a moving interface. Chapter 20, in Advanced Topics in Mass Transfer, M. El-Amin (Ed.), InTech Open Access Publisher, Rijeka, pp 439–484 (2011)

  20. Tarzia, D.A.: A bibliography on moving-free boundary problems for the heat-diffusion equation. The Stefan and related problems. MAT-Serie A 2, 1–297 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Tarzia, D.A.: An inequality for the coefficient \(\sigma \) of the free boundary \(s(t)=2\sigma \sqrt{t}\) of the Neumann solution for the two-phase Stefan problem. Quart. Appl. Math. 39, 491–497 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. Voller, V.R., Swenson, J.B., Paola, C.: An analytical solution for a Stefan problem with variable latent heat. Int. J. Heat Mass Transf. 47, 5387–5390 (2004)

    Article  MATH  Google Scholar 

  23. Zhou, Y., Shi, X., Zhou, G.: Exact solution for a two-phase problem with power-type latent heat. J. Eng. Math. (2017). https://doi.org/10.1007/s10665-017-9921-y

  24. Zhou, Y., Wang, Y.J., Bu, W.K.: Exact solution for a Stefan problem with latent heat a power function of position. Int. J. Heat Mass Transf. 69, 451–454 (2014)

    Article  Google Scholar 

  25. Zhou, Y., Xia, L.J.: Exact solution for Stefan problem with general power-type latent heat using Kummer function. Int. J. Heat Mass Transf. 84, 114–118 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

The present work has been partially sponsored by the Projects PIP No. 0534 from CONICET-UA and ANPCyT PICTO Austral 2016 No. 0090, Rosario, Argentina.

Author information

Authors and Affiliations

  1. Depto. Matemática- CONICET, FCE, Univ. Austral, Paraguay 1950, S2000FZF, Rosario, Argentina

    Julieta Bollati & Domingo A. Tarzia

Authors
  1. Julieta Bollati
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Domingo A. Tarzia
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Julieta Bollati.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bollati, J., Tarzia, D.A. Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face. Z. Angew. Math. Phys. 69, 38 (2018). https://doi.org/10.1007/s00033-018-0923-z

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-018-0923-z

Mathematics Subject Classification

Keywords