Supplementary balance laws for Cattaneo heat propagation

Serge Preston

Communications in Mathematics (2013)

  • Volume: 21, Issue: 2, page 161-171
  • ISSN: 1804-1388

Abstract

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In this work we determine for the Cattaneo heat propagation system all the supplementary balance laws (shortly SBL) of the same order (zero) as the system itself and extract the constitutive relations (expression for the internal energy) dictated by the Entropy Principle. The space of all supplementary balance laws (having the functional dimension 8) contains four original balance laws and their deformations depending on 4 functions of temperature ( λ 0 ( ϑ ) , K A ( ϑ ) , A = 1 , 2 , 3 ). The requirements of the II law of thermodynamics leads to the exclusion of three functional degrees ( K A = 0 , A = 1 , 2 , 3 ) and to further restriction to the form of internal energy. In its final formulation, entropy balance represents the deformation of the energy balance law by the functional parameter λ 0 ( ϑ ) .

How to cite

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Preston, Serge. "Supplementary balance laws for Cattaneo heat propagation." Communications in Mathematics 21.2 (2013): 161-171. <http://eudml.org/doc/260808>.

@article{Preston2013,
abstract = {In this work we determine for the Cattaneo heat propagation system all the supplementary balance laws (shortly SBL) of the same order (zero) as the system itself and extract the constitutive relations (expression for the internal energy) dictated by the Entropy Principle. The space of all supplementary balance laws (having the functional dimension 8) contains four original balance laws and their deformations depending on 4 functions of temperature ($\lambda ^0(\vartheta ), K^A(\vartheta ), A=1,2,3 $). The requirements of the II law of thermodynamics leads to the exclusion of three functional degrees ($K^A=0, A=1,2,3$) and to further restriction to the form of internal energy. In its final formulation, entropy balance represents the deformation of the energy balance law by the functional parameter $\lambda ^\{0\}(\vartheta )$.},
author = {Preston, Serge},
journal = {Communications in Mathematics},
keywords = {Cattaneo balance equations; conservation laws; entropy; Cattaneo balance equations; conservation laws; entropy},
language = {eng},
number = {2},
pages = {161-171},
publisher = {University of Ostrava},
title = {Supplementary balance laws for Cattaneo heat propagation},
url = {http://eudml.org/doc/260808},
volume = {21},
year = {2013},
}

TY - JOUR
AU - Preston, Serge
TI - Supplementary balance laws for Cattaneo heat propagation
JO - Communications in Mathematics
PY - 2013
PB - University of Ostrava
VL - 21
IS - 2
SP - 161
EP - 171
AB - In this work we determine for the Cattaneo heat propagation system all the supplementary balance laws (shortly SBL) of the same order (zero) as the system itself and extract the constitutive relations (expression for the internal energy) dictated by the Entropy Principle. The space of all supplementary balance laws (having the functional dimension 8) contains four original balance laws and their deformations depending on 4 functions of temperature ($\lambda ^0(\vartheta ), K^A(\vartheta ), A=1,2,3 $). The requirements of the II law of thermodynamics leads to the exclusion of three functional degrees ($K^A=0, A=1,2,3$) and to further restriction to the form of internal energy. In its final formulation, entropy balance represents the deformation of the energy balance law by the functional parameter $\lambda ^{0}(\vartheta )$.
LA - eng
KW - Cattaneo balance equations; conservation laws; entropy; Cattaneo balance equations; conservation laws; entropy
UR - http://eudml.org/doc/260808
ER -

References

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  1. Glensdorf, P., Prigogine, I., Thermodynamical Theory of Structure, Stability and fluctuations, 1971, Wiley, Brussels. (1971) 
  2. Jou, D., Casas-Vasquez, J., Lebon, G., Extended Irreversible Thermodynamics, 3rd ed, 2001, Springer. (2001) 
  3. Liu, I-Shish, Method of Lagrange multipliers for exploatation of the entropy principle, Arch. Rational Mech. Anal., 46, 1972, 131-148. (1972) MR0337164
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  5. Muller, I., Ruggeri, T., Rational Extended Thermodynamics, 2nd ed, 1998, Springer. (1998) Zbl0895.00005MR1632151
  6. Olver, P., Applications of Lie Groups to Differential Equations, 2nd ed, 2000, Springer-Verlag New York. (2000) MR1240056
  7. Preston, S., Geometrical Theory of Balance Systems and the Entropy Principle, Proceedings of GCM7, Lancaster, UK, J. Phys.: Conf. Ser., 62, 2007, 102-154, DOI: 10.1088/1742-6596/62/1/007. (2007) 
  8. Preston, S., 10.1142/S0219887810004543, Int. J. Geom. Methods of Mod. Phys., 7, 2010, DOI: 10.1142/S0219887810004543. (2010) Zbl1205.80041MR2720544DOI10.1142/S0219887810004543
  9. Preston, S., Supplementary balance laws for the Navier-Stokes-Fourier Fluid, Manuscript, unpublished. 
  10. Ruggeri, T., 10.1007/BF01125883, Cont. Mech. Thermodyn., 1, 1, 1989, 3-20. (1989) Zbl0759.35039MR1001434DOI10.1007/BF01125883
  11. Ruggeri, T., 10.3390/e10030319, Entropy, 10, 2008, 319-333. (2008) Zbl1179.82004DOI10.3390/e10030319

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