Exact harmonic solutions to Guyer–Krumhansl-type equation and application to heat transport in thin films

Zhukovsky, K. V.; Oskolkov, D.

doi:10.1007/s00161-018-0648-4

Cited by 4 publications

(1 citation statement)

References 54 publications

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“…In the distinct ballistic case, Kn = 1, all heat transport terms in GK-type equation contribute more or less equally, in the weak ballistic case, Kn = 0.1, the Cattaneo wave-term prevails. The inhomogeneous system of PDE (5) and (6) for ballistic heat transport thin film was studied numerically in [30]; based on the periodic analytical solutions to GK-type equation [72] some solutions to (5) and (6) were obtained in [73,74].…”

Section: Ballistic Heat Transport Equations With Substantial Derivativementioning

confidence: 99%

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Zhukovsky

Oskolkov

Gubina

2018

Axioms

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Abstract:One-dimensional equations of telegrapher's-type (TE) and Guyer-Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated-fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed.

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Section: Ballistic Heat Transport Equations With Substantial Derivativementioning

confidence: 99%

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Zhukovsky

Oskolkov

Gubina

2018

Axioms

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Two-field mixed hp-finite elements for time-dependent problems in the refined theories of thermodynamics

Tóth,

Molnár,

Kovács

2024

Continuum Mech. Thermodyn.

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Modern manufacturing technologies allow heterogeneous materials with complex inner structures (e.g., foams) to be easily produced. However, their utilization is not straightforward, as the classical constitutive laws are not necessarily valid. According to various experimental observations, the Guyer–Krumhansl equation is a promising candidate for modeling such complex structures. However, practical applications need a reliable and efficient algorithm capable of handling both complex geometries and advanced heat equations. In the present paper, we derive new two-field variational formulations which treat the temperature and the heat flux as independent field variables, and we develop new, advanced hp-type mixed finite element methods, which can be reliably applied. We investigate their convergence properties for various situations, challenging in relation to stability and the treatment of fast propagation speeds. That algorithm is also proved to be outstandingly efficient, providing solutions four magnitudes faster than commercial algorithms.

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Modeling of heat transport and exact analytical solutions in thin films with account for constant non-relativistic motion

Zhukovsky

Oskolkov

2020

International Journal of Heat and Mass Transfer

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Exact harmonic solutions to Guyer–Krumhansl-type equation and application to heat transport in thin films

Cited by 4 publications

References 54 publications

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Two-field mixed hp-finite elements for time-dependent problems in the refined theories of thermodynamics

Modeling of heat transport and exact analytical solutions in thin films with account for constant non-relativistic motion

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