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Generalized mathematical novel model of thermoelastic diffusion with four phase lags and higher-order time derivative

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Abstract

In this work, taking into account the influences of thermal and diffusion, a new four-phase-lag model comprising the macroscopic and microscopic is constructed. The introduced model is an extension of the papers of Nowacki (Bull Acad Pol Sci Ser Sci Tech 22:55–64, 1974; Bull Acad Pol Sci Ser Sci Tech 22:129–135, 1974; Bull Acad Pol Sci Ser Sci Tech 22:257–266, 1974; Proc Vib Prob 15:105–128, 1974), Sherief et al. (Int J Eng Sci 42:591–608, 2004) and Aouadi (J Therm Stress 30:665–678, 2007). In this model, the Fourier’s and the Fick’s laws have been modified to include higher-order time derivatives of heat flow vector, the gradient of temperature, diffusing mass flux and gradient of chemical potential. Many models in the thermoelastic-diffusion field have been deduced as special cases from the current investigation. Using the resulting formulation, we have studied a thermoelastic-diffusion interaction in a half-space exposed to thermal and chemical shock. Also, the sensitivity of the studied field variables to the variation of the parameters of higher-order time derivative has been investigated.

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Abbreviations

\(\lambda , \mu \) :

Lamé’s constants

\(\alpha _{t}\) :

Thermal expansion coefficient

\(\alpha _{c}\) :

Coefficient of linear diffusion

\(\beta _{1}=(3\lambda +2\mu )\alpha _{t}\) :

Thermal coupling parameter

\(T_{0}\) :

Environmental temperature

\(\theta =T-T_{0}\) :

Temperature increment

T :

Absolute temperature

\(C_{e}\) :

Specific heat

\(e=\mathrm{div}\,{{\varvec{u}}}\) :

Cubical dilatation

\(\sigma _{ij}\) :

Stress tensor

\(e_{ij}\) :

Strain tensor

\({{\varvec{u}}}\) :

Displacement vector

\({{\varvec{q}}}\) :

Heat flux vector

\({\varvec{\eta }}\) :

Flow of diffusing mass vector

l, m, n, h :

Higher-order time derivatives

K :

Thermal conductivity

\(\rho \) :

Material density

Q :

Heat source

\(\beta _{2}=(3\lambda +2\mu )\alpha _{c}\) :

Diffusion coupling parameter

\(\delta _{ij}\) :

Kronecker’s delta function

\(\nabla ^{2}\) :

Laplacian operator

\(\tau _{q}\) :

Phase lag of heat flux

\(\tau _{\theta }\) :

Phase lag of temperature gradient

\(\tau _{\eta }\) :

Phase lag of diffusing mass

\(\tau _{p}\) :

Phase lag of chemical potential gradient

\(\alpha \) :

Fractional order

D :

Diffusion coefficient

P :

Chemical potential

C :

Concentration of diffusion material

a :

Thermoelastic diffusion effect

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

    Ahmed E. Abouelregal

  2. Department of Mathematics, College of Science and Arts, Jouf University, Al-Qurayyat, Saudi Arabia

    Ahmed E. Abouelregal

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Correspondence to Ahmed E. Abouelregal.

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Abouelregal, A.E. Generalized mathematical novel model of thermoelastic diffusion with four phase lags and higher-order time derivative. Eur. Phys. J. Plus 135, 263 (2020). https://doi.org/10.1140/epjp/s13360-020-00282-2

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