Second-Order Approximation to Unsteady One-Dimensional Heterogeneous Heat Conduction

2021 ◽  
pp. 1-5
Author(s):  
Madiha Bouafia ◽  
Hamou Sadat
Keyword(s):  
Heat Conduction ◽  
Order Approximation ◽  
Second Order ◽  
One Dimensional ◽  
Second Order Approximation

scholarly journals Second-order approximation for heat conduction: dissipation principle and free energies

2016 ◽  
Vol 472 (2186) ◽  
pp. 20150707 ◽  
Author(s):  
Giovambattista Amendola ◽  
Mauro Fabrizio ◽  
Murrough Golden ◽  
Barbara Lazzari
Keyword(s):  
Free Energy ◽  
Heat Conduction ◽  
Order Approximation ◽  
Minimum Free Energy ◽  
Second Order ◽  
Fading Memory ◽  
Free Energies ◽  
Second Order Approximation ◽  
Energy Functionals ◽  
With Memory

In the context of new models of heat conduction, the second-order approximation of Tzou's theory, derived by Quintanilla and Racke, has been studied recently by two of the present authors, where it was proved equivalent to a fading memory material. The importance of determining free energy functionals for such materials, and indeed for any material with memory, is emphasized. Because the kernel does not satisfy certain convexity restrictions that allow us to obtain various traditional free energies for materials with fading memory, it is necessary to restrict the study to the minimum and related free energies, which do not require these restrictions. Thus, the major part of this work is devoted to deriving an explicit expression for the minimum free energy. Simple modifications of this expression also give an intermediate free energy and the maximum free energy for the material. These derivations differ in certain important respects from earlier work on such free energies.

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