Abstract
Spatially convoluting formulations have been used to describe nonlocal thermal transport, yet there is no related investigation at the microscopic level such as the Boltzmann transport theory. The spatial fractional-order Boltzmann transport equations (BTEs) are first applied to the description of nonlocal phonon heat transport. Constitutive and continuity equations are derived, and two anomalous behaviors are thereafter observed in one-dimensional steady-state heat conduction: one is the power-law length-dependence of the effective thermal conductivity, κeff∝Lβ with L as the system length, and the other is the nonlinear temperature profile, Tx−Tx=0∼x1+η. A connection between the length-dependence and nonlinearity exponents is established, namely, β=−η. Furthermore, we show that the order of these BTEs should be restricted by the ballistic limit. In minimizing problems, the nonlocal models in this work give rise to different results from the case of Fourier heat conduction, namely that the optimized temperature gradient is not uniform.
Highlights
In the past decades, temporal fractional-order derivatives have been applied to various physical processes and systems, i.e., anomalous diffusion1–3 and non-Fourier heat conduction.4–7 In recent years, temporal fractional-order modeling of heat conduction attracts increasing interest.8,9 Generally speaking, heat conduction in three-dimensional bulk materials obeys Fourier’s law, q = −κ∇T. (1)In Eq (1), q = q(x, t) denotes the heat flux, T = T(x, t) is the local temperature, and κ is the thermal conductivity
The corresponding energy continuity equations can be obtained upon multiplying the above Boltzmann transport equations (BTEs) by hω and integrating them over the wave vector space
The spatial fractional-order derivatives are first introduced into the phonon BTEs, which reflects nonlocal effects in heat transport
Summary
We mention that the nonlocality from the integer-order Boltzmann transport theory has been studied, and the nonlocal effects are commonly induced by the second-order derivatives of the heat flux such as (∇2q) and (∇∇ ⋅ q) This class is usually termed phonon hydrodynamics and has been employed to explain nonFourier behaviors observed in experiments.. The two non-Fourier behaviors cannot be expected by previous nonlocal theories such as phonon hydrodynamics, which will generally reproduce Fourier heat conduction in the large size limit It indicates that the fractionalorder nonlocal theory can extend the existing framework of phonon thermal transport. We propose the spatial fractional-order BTEs for the nonlocal effects, which predict non-Fourier anomalies different from the temporal fractional-order situations, i.e., the nonlinear temperature profiles Another difference from our previous works is.
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