Heat transfer at nanoscale and boundary conditions

Abstract

A model of nonlocal heat transfer at nanoscale in rigid bodies is considered. Depending on the relevance of the particular interaction’s mechanism between the heat carriers and the lateral walls, three different strategies for the setting-up of the boundary conditions are analyzed, and the consequent forms of the basic fields have been obtained, as well. From the physical point of view, the possible influence of those interactions on the unknown fields is pointed out. From the mathematical point of view, instead, the well-posedness of the problem is shown.

Phonon-boundary scatterings and second-order boundary conditions: Application to the heat transfer in thin layers

“…The reason why the case ω = 0 has to be discarded is straightforward, as it has been also pointed out in Ref. [17] wherein the compatibility of a similar SL problem is analyzed (see therein Appendix, Theorem 3), i.e., if α = 0 in Eq. ( 16).…”

“…By correctly adding the BCs to Eqs. (1), instead, one can also emphasize the role played by the boundary effects [17,28,29].…”

“…Ref. [17] the authors set θ 0 (x) = 0, ∀x ∈ Ω. Here, instead, we let the initial condition on the basic field θ to be different from zero in order to obtain more general results 1 ; from the mathematical point of view ID indeed do not properly yield any particular complication in the analysis.…”

“…Keeping in mind that the thermal analysis is not properly an easy task from the theoretical point of view, on a macroscopic level it is well-known that the transport of phonons is usually diffusive: it is therefore describable by the classical Fourier's law of heat conduction. In 2D materials, instead, ballistic and/or hydrodynamic phonon transport may also occur [14][15][16][17]: in such cases, as well as in other situations at nano-scale, the classical Fourier's law breaks down [18][19][20][21], that is, more refined models beyond that law have to be taken into account for a more realistic description of the heat-transport phenomenon [14,[22][23][24][25].…”

“…In non-Fourier heat-transport regimes the macroscopic method based on the phonon hydrodynamics, whose characteristic equations naturally arise from the Boltzmann equation for phonons [21,26,27], may not only depict in a clear and intuitive way the heat-transport phenomenon, but also allows an elegant mathematical analysis, especially when one is dealing with the problem of right modeling of the boundary conditions (BCs) [17]. In particular, in the phonon-hydrodynamics approach the heat-flux propagation may be in principle meant as the consequence of moving heat carriers (i.e., mainly the phonons) which behave as the parti-cles of a rarefied gas in a box [28,29].…”