On the representation of solutions for the theory of generalized thermoelasticity with three phase-lags

Mukhopadhyay, Santwana; Kothari, Shweta; Kumar, Roushan

doi:10.1007/s00707-010-0291-z

Cited by 32 publications

(18 citation statements)

References 24 publications

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“…The effect of three-phase-lags on thermoelastic interactions due to the step input in temperature on the stress-free boundary of a cylindrical cavity in an unbounded medium is implemented [273]. The effect of phase-lags on wave propagation in a two-dimensional thick plate due to an axisymmetric temperature distribution is analyzed by employing the DPL and the TPL theories of thermoelasticity [274].…”

Section: Three-phase-lag Modelmentioning

confidence: 99%

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

2015

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The classical model of the Fourier's law is known as the most common constitutive relation for thermal transport in various engineering materials. Although the Fourier's law has been widely used in a variety of engineering application areas, there are many exceptional applications in which the Fourier's law is questionable. This paper gathers together such applications. Accordingly, the paper is divided into two parts. The first part reviews the papers pertaining to the fundamental theory of the phase-lagging models and the analytical and numerical solution approaches. The second part wrap ups the various applications of the phase-lagging models including the biological materials, ultra-high-speed laser heating, the problems involving moving media, micro/nanoscale heat transfer, multi-layered materials, the theory of thermoelasticity, heat transfer in the material defects, the diffusion problems we call as the non-Fick models, and some other applications. It is predicted that the interest in the field of phase-lagging heat transport has grown incredibly in recent years because they show good agreement with the experiments across a wide range of length and time scales.

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Section: Three-phase-lag Modelmentioning

confidence: 99%

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

2015

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“…In this section, we investigate the spatial behavior of the solutions of the problem determined by the systems (10) and (11) with the initial conditions (7) and (12) and the boundary conditions (8), (9), (13), and (14).…”

Section: A Thermoelastic Systemmentioning

confidence: 99%

“…For this reason a big interest has been developed to understand different formal Taylor approximations to these equations. [12][13][14][15][16][17][18][19] In fact, the literature concerning these topics is increasing quickly because many researchers are interested in these proposals. These new theories allow to obtain stability of solutions and the well posedness of the problems, provided that certain conditions on the parameters hold.…”

Section: Introductionmentioning

confidence: 99%

Spatial behavior in high‐order partial differential equations

Leseduarte

Quintanilla

2018

Math Methods in App Sciences

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In this paper, we study the spatial behavior of solutions to the equations obtained by taking formal Taylor approximations to the heat conduction dual‐phase‐lag and 3‐phase‐lag theories, reflecting Saint‐Venant's principle. In a recent paper, 2 families of cases for high‐order partial differential equations were studied. Here, we investigate a third family of cases, which corresponds to the fact that a certain condition on the time derivative must be satisfied. We also study the spatial behavior of a thermoelastic problem. We obtain a Phragmén‐Lindelöf alternative for the solutions in both cases. The main tool to handle these problems is the use of an exponentially weighted Poincaré inequality.

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“…Of course, this is not suitable for a heat conduction theory and disagrees with what one would expect a priori. For this reason a big interest has been developed to study different Taylor approximations to these equations [1,8,11,12,13,14,15,16,17,18]. These alternative theories allow to obtain the stability of solutions and the well-posedness of the problems.…”

Section: (12)mentioning

confidence: 99%

Phase-lag heat conduction: decay rates for limit problems and well-posedness

Borgmeyer

Quintanilla²,

Racke

2014

J. Evol. Equ.

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In two recent papers the authors have studied conditions on the relaxation parameters in order to guarantee the stability or instability of solutions for the Taylor approximations to dual-phase-lag and three-phase-lag heat conduction equations. However, for several limit cases relating to the parameters the kind of stability was unclear. Here we analyze these limit cases and clarify whether we can expect exponential or slow decay for the solutions. Moreover, rather general well-posedness results for three-phase-lag models are presented. Finally, the exponential stability expected by spectral analysis is rigorously proved exemplarily.

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On the representation of solutions for the theory of generalized thermoelasticity with three phase-lags

Cited by 32 publications

References 24 publications

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

Macro- to Nanoscale Heat and Mass Transfer: The Lagging Behavior

Spatial behavior in high‐order partial differential equations

Phase-lag heat conduction: decay rates for limit problems and well-posedness

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