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<p>In a previous study, the authors proposed a new nonlinear system of governing equations of a three-phase-lag model for a rigid thermal conductor, where a thermodynamically consistent free energy function and dissipation function were formulated. To illustrate the applicability of this model, a simple numerical one-dimensional example is presently solved using Mathematica, and the results are plotted and discussed. The thermal conductivity is taken to vary linearly with temperature. For the sake of conciseness, contributions of higher order than quadratic are disregarded. Comparisons are carried out graphically, to put in evidence the differences between the present model and the classical heat equation for temperature in three-phase-lag model, as well as the effect of nonlinearity. This allows to validate the obtained results. The novelty of this work lies in extending the TPL framework to nonlinear regimes coupling temperature, heat flux, and thermal displacement, while preserving thermodynamic consistency, thus providing new insights into advanced heat conduction modeling.</p>

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Nonlinear equations for the determination of temperature and heat flux in the three-phase-lag theory with application

  • 期刊论文

  • 发表日期：

    2026年6月28日

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  • Asmaa Fawzy,
  • E. K. Rawy,
  • W. Mahmoud,
  • A. F. Ghaleb

摘要

In a previous study, the authors proposed a new nonlinear system of governing equations of a three-phase-lag model for a rigid thermal conductor, where a thermodynamically consistent free energy function and dissipation function were formulated. To illustrate the applicability of this model, a simple numerical one-dimensional example is presently solved using Mathematica, and the results are plotted and discussed. The thermal conductivity is taken to vary linearly with temperature. For the sake of conciseness, contributions of higher order than quadratic are disregarded. Comparisons are carried out graphically, to put in evidence the differences between the present model and the classical heat equation for temperature in three-phase-lag model, as well as the effect of nonlinearity. This allows to validate the obtained results. The novelty of this work lies in extending the TPL framework to nonlinear regimes coupling temperature, heat flux, and thermal displacement, while preserving thermodynamic consistency, thus providing new insights into advanced heat conduction modeling.