Two Temperature Extension of Phonon Hydrodynamics

  • 1 Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of 4 Engineering Mechanics, Tsinghua University, Beijing, China
  • 2 École Polytechnique de Montréal, C.P.6079 suc. Centre-ville, Montréal, Canada
Bing-Yang Cao
  • Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of 4 Engineering Mechanics, 12442Tsinghua University, Beijing, 100084, China
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, Miroslav Grmela
  • Corresponding author
  • École Polytechnique de Montréal, C.P.6079 suc. Centre-ville, Montréal, H3C 3A7, Québec, Canada
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, Zeng-Yuan Guo
  • Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of 4 Engineering Mechanics, 12442Tsinghua University, Beijing, 100084, China
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, Yu-Chao Hua
  • Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of 4 Engineering Mechanics, 12442Tsinghua University, Beijing, 100084, China
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and Ben-Dian Nie
  • Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of 4 Engineering Mechanics, 12442Tsinghua University, Beijing, 100084, China
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Abstract

Phonon hydrodynamics uses the fields of the total energy and the heat flux as state variables. We extend it by promoting the microscopic internal energy field into the status of an extra independent state variable. The governing equations of both the phonon and the extended (two temperature) phonon hydrodynamics are formulated as particular realizations of the abstract GENERIC equation. Such unified formulation makes both theories manifestly compatible with mechanics and thermodynamics. Also differences and similarities (in the physical content, in the mathematical structure, and in qualitative properties of solutions) between the two heat transfer theories, as well as their mutual compatibility, become manifestly displayed.

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