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

Zhukovsky, K. V.; Oskolkov, D.; Gubina, Nadezhda

doi:10.3390/axioms7030048

Cited by 13 publications

(10 citation statements)

References 71 publications

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“…It is relevant when studying elements moving with the fluid flow, as illustrated in Figure 2.1. The substantial derivative applies to any flow-field variable, and can, for example, be defined as [3,86]:…”

Section: Governing Equationsmentioning

confidence: 99%

“…where ∂T ∂t is the local derivative, time rate of change at a fixed point, and ∇v is the convective derivative, time rate of change due to the movement of the fluid from one location to another in the flow field where the flow properties are spatially different [3,86]. The continuity equation can be derived in several ways, obtaining a different form of the equation directly.…”

Section: Governing Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Eulerian CFD model of direct absorption solar collector with nanofluid

Bårdsgård

Kuzmenkov

Kosinski

et al. 2020

Journal of Renewable and Sustainable Energy

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Section: Governing Equationsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

Eulerian CFD model of direct absorption solar collector with nanofluid

Bårdsgård

Kuzmenkov

Kosinski

et al. 2020

Journal of Renewable and Sustainable Energy

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“…The thermal Klein–Gordon equation in the Zhukovsky model [ 8 , 9 ] explicitly includes the heat and thermal wave equations. If Equation (1) is solved according to [ 9 ], the following solution (see Appendix A ) can be obtained:

where:

…”

Section: Zhukovsky Mathematical Modelmentioning

confidence: 99%

“…Here,

is the laser beam intensity and

is the absorption coefficient. In this case, the electron temperature variation of a nano-/micro-particle under ultrashort laser pulse irradiation is [ 8 , 9 ]

…”

Section: The Generalized Lambert–beer Lawmentioning

confidence: 99%

Thermal Nonlinear Klein–Gordon Equation for Nano-/Micro-Sized Metallic Particle–Attosecond Laser Pulse Interaction

et al. 2021

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In this study, a rigorous analytical solution to the thermal nonlinear Klein–Gordon equation in the Kozłowski version is provided. The Klein–Gordon heat equation is solved via the Zhukovsky “state-of-the-art” mathematical techniques. Our study can be regarded as an initial approximation of attosecond laser–particle interaction when the prevalent phenomenon is photon–electron interaction. The electrons interact with the laser beam, which means that the nucleus does not play a significant role in temperature distribution. The particle is supposed to be homogenous with respect to thermophysical properties. This theoretical approach could prove useful for the study of metallic nano-/micro-particles interacting with attosecond laser pulses. Specific applications for Au “nano” particles with a 50 nm radius and “micro” particles with 110, 130, 150, and 1000 nm radii under 100 attosecond laser pulse irradiation are considered. First, the cross-section is supposed to be proportional to the area of the particle, which is assumed to be a perfect sphere of radius R or a rotation ellipsoid. Second, the absorption coefficient is calculated using a semiclassical approach, taking into account the number of atoms per unit volume, the classical electron radius, the laser wavelength, and the atomic scattering factor (10 in case of Au), which cover all the basic aspects for the interaction between the attosecond laser and a nanoparticle. The model is applicable within the 100–2000 nm range. The main conclusion of the model is that for a range inferior to 1000 nm, a competition between ballistic and thermal phenomena occurs. For values in excess of 1000 nm, our study suggests that the thermal phenomena are dominant. Contrastingly, during the irradiation with fs pulses, this value is of the order of 100 nm. This theoretical model’s predictions could be soon confirmed with the new EU-ELI facilities in progress, which will generate pulses of 100 as at a 30 nm wavelength.

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“…Website: http://www.math.uvic.ca/faculty/harimsri/ This volume contains the invited, accepted and published submissions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]) to a Special Issue of the MDPI journal Axioms on the subject-area of "Mathematical Analysis and Applications".…”

mentioning

confidence: 99%