Skip to main content
Log in

Integral formulation for a Stefan problem with spherical symmetry

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

Abstract

A one-dimensional Stefan problem with spherical symmetry corresponding to the evaporation process of a droplet is considered. An equivalent integral formulation is obtained, and through a fixed point theorem, the existence and uniqueness of the solution are proved.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Briozzo, A.C., Tarzia, D.A.: A one-phase Stefan problem for a non-classical heat equation with a heat flux condition on the fixed face. Appl. Math. Comput. 182, 809–819 (2006)

    MathSciNet  MATH  Google Scholar 

  2. Briozzo, A.C., Tarzia, D.A.: Existence and uniqueness for one-phase Stefan problems of non-classical heat equation with temperature boundary condition at a fixed face. Electron. J. Differ. Equ. 2006, 1–16 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Briozzo, A.C., Tarzia, D.A.: A Stefan problem for a non-classical heat equation with a convective condition. Appl. Math. Comput. 217, 4051–4060 (2010)

    MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  5. Case, E., Tausch, J.: An integral equation method for spherical Stefan problems. Appl. Math. Comput. 218, 11451–11460 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Ehrich, O., Chuang, Y.K., Schwerdtfeger, K.: The melting of metal spheres involving the initially frozen shells with different material properties. Int. J. Heat Mass Transf. 24, 341–349 (1978)

    Article  Google Scholar 

  7. Friedman, A.: Free boundary problems for parabolic equations I. Melting of solids. J. Math. Mech. 8, 499–517 (1959)

    MathSciNet  MATH  Google Scholar 

  8. Herrero, M.A., Velazquez, J.J.L.: On the melting of ice balls. SIAM J. Math. Anal. 28, 1–32 (1997)

    Article  MathSciNet  Google Scholar 

  9. McCue, S.W., Hsieh, M., Moroney, T.J., Nelson, M.I.: Asymptotic and numerical results for a model of solvent-dependent drug diffusion through polymeric spheres. SIAM J. Appl. Math. 71, 2287–2311 (2011)

    Article  MathSciNet  Google Scholar 

  10. Mitchell, S.L., Vynnycky, M., Gusev, I.G., Sazhin, S.S.: An accurate numerical solution for the transient heating of an evaporating droplet. Appl. Math. Comput. 217, 9219–9233 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Pedroso, R.I., Domoto, G.A.: Perturbation solutions for spherical solidification of saturated liquids. J. Heat Transf. 95, 42–46 (1973)

    Article  Google Scholar 

  12. Rubinstein, L.I.: The Stefan Problem. Translations of Mathematical Monographs, vol. 27. American Mathematical Society, Providence (1971)

    Google Scholar 

  13. Sherman, B.: A free boundary problem for the heat equation with prescribed flux at both fixed face and melting interface. Q. Appl. Math. 25, 53–63 (1967)

    Article  MathSciNet  Google Scholar 

  14. Sherman, B.: General one-phase Stefan problems and free boundary problems for the heat equation with Cauchy data prescribed on the free boundary. SIAM J. Appl. Math. 20, 555–63 (1971)

    Article  MathSciNet  Google Scholar 

  15. Soward, A.M.: A unified approach to Stefan’s problem for spheres. Proc. R. Soc. A 373, 131–147 (1980)

    MathSciNet  Google Scholar 

  16. Vynnycky, M., Mitchell, S.L.: On the numerical solution of a Stefan problem with finite extinction time. J. Comput. Appl. Math. 279, 98–109 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The present work has been partially supported by the Project PIP No 0275 from CONICET-UA, Rosario, Argentina, ANPCyT PICTO Austral 2016 No 0090 and the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement 823731 CONMECH

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Adriana C. Briozzo.

Additional information

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

Bollati, J., Briozzo, A.C. & Gutierrez, M.S. Integral formulation for a Stefan problem with spherical symmetry. Z. Angew. Math. Phys. 72, 98 (2021). https://doi.org/10.1007/s00033-021-01527-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01527-5

Keywords

Mathematics Subject Classification

Navigation