American Institute of Mathematical Sciences

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August  2012, 5(4): 797-808. doi: 10.3934/dcdss.2012.5.797

Stable and unstable initial configuration in the theory wave fronts

Ruediger Landes 1,

1. 

University of Oklahoma, Noman, OK 73019, United States

Received  March 2011 Revised  May 2011 Published  November 2011

In this paper we study the wavefront like phase transition of solutions of a parabolic nonlinear boundary value problem used to model phase transitions in the theory of boiling liquids. Using weak supersolutions we provide bounds for the propagation speed of such a phase transition. Also we construct stable supersolutions to initial configurations which have locally supercritical values.
Keywords: Wavefronts, Parabolic equations, Nonlinear Boundary Conditions, Phastransitions, Supersolutions..
Mathematics Subject Classification: Primary: 35K99, Secondary: 35B35, 35B40, 35D3.
Citation: Ruediger Landes. Stable and unstable initial configuration in the theory wave fronts. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 797-808. doi: 10.3934/dcdss.2012.5.797
References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[2]

H. Auracher, W. Marquardt, M. Buchholz, R. Hohl, T. Lüttich and J. Blum, New experimental results on steady-state and transient pool boiling heat transfer, Therm. Sci. Engng, 9 (2001), 29-39. Google Scholar

[3]

J. Blum, T. Lüttich and W. Marquardt, Temperature Wave Propagation as a Route from Nucleate to Film Boiling?, In "2nd International Symposium on Two-Phase Flow Modelling and Experimentation" (eds. G. P. Celata, P. DiMarco and R. K. Shah), 1, Rome, Edizioni ETS, Pisa, (1999), 137-144. Google Scholar

[4]

V. K. Dhir, Boiling heat transfer, Annu. Rev. Fluid Mech., 30 (1998), 365-401. doi: 10.1146/annurev.fluid.30.1.365.  Google Scholar

[5]

P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[6]

R. Landes, Wavefront solution in the theory of boiling liquids, Analysis (Munich), 29 (2009), 283-298.  Google Scholar

[7]

T. Lüttich, W. Marquardt, M. Buchholz and H. Auracher, "Towards a Unifying Heat Transfer Correlation for the Entire Boiling Curve," 5th International Conference on Boiling Heat Transfer, Montego Bay, Jamaica, May 2003. Google Scholar

[8]

M. Speetjens, A. Reusken and W. Marquardt, Steady-state solutions in a nonlinear pool boiling model, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1475-1494. doi: 10.1016/j.cnsns.2006.11.001.  Google Scholar

[9]

M. Speetjens, A. Reusken and W. Marquardt, Steady-state solutions in a three-dimensional nonlinear pool-boiling heat-transfer model, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1518-1537. doi: 10.1016/j.cnsns.2006.11.002.  Google Scholar

[10]

J. R. Thome, Boiling, in "Handbook of Heat Transfer" (eds. A. Bejan and A. D. Krause), Wiley & Sons, New York, (2003), 635-717. Google Scholar

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar

[2]

H. Auracher, W. Marquardt, M. Buchholz, R. Hohl, T. Lüttich and J. Blum, New experimental results on steady-state and transient pool boiling heat transfer, Therm. Sci. Engng, 9 (2001), 29-39. Google Scholar

[3]

J. Blum, T. Lüttich and W. Marquardt, Temperature Wave Propagation as a Route from Nucleate to Film Boiling?, In "2nd International Symposium on Two-Phase Flow Modelling and Experimentation" (eds. G. P. Celata, P. DiMarco and R. K. Shah), 1, Rome, Edizioni ETS, Pisa, (1999), 137-144. Google Scholar

[4]

V. K. Dhir, Boiling heat transfer, Annu. Rev. Fluid Mech., 30 (1998), 365-401. doi: 10.1146/annurev.fluid.30.1.365.  Google Scholar

[5]

P. Fife, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, 28, Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[6]

R. Landes, Wavefront solution in the theory of boiling liquids, Analysis (Munich), 29 (2009), 283-298.  Google Scholar

[7]

T. Lüttich, W. Marquardt, M. Buchholz and H. Auracher, "Towards a Unifying Heat Transfer Correlation for the Entire Boiling Curve," 5th International Conference on Boiling Heat Transfer, Montego Bay, Jamaica, May 2003. Google Scholar

[8]

M. Speetjens, A. Reusken and W. Marquardt, Steady-state solutions in a nonlinear pool boiling model, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1475-1494. doi: 10.1016/j.cnsns.2006.11.001.  Google Scholar

[9]

M. Speetjens, A. Reusken and W. Marquardt, Steady-state solutions in a three-dimensional nonlinear pool-boiling heat-transfer model, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 1518-1537. doi: 10.1016/j.cnsns.2006.11.002.  Google Scholar

[10]

J. R. Thome, Boiling, in "Handbook of Heat Transfer" (eds. A. Bejan and A. D. Krause), Wiley & Sons, New York, (2003), 635-717. Google Scholar

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