Hydrodynamic approximation to the phonon Boltzmann equation

1974 ◽  
Vol 10 (8) ◽  
pp. 3546-3551 ◽  
Author(s):  
Robert J. Hardy ◽  
Dennis L. Albers
Keyword(s):  
Boltzmann Equation ◽  
Hydrodynamic Approximation ◽  
Phonon Boltzmann Equation

scholarly journals Thermal transport at the nanoscale: A Fourier's law vs. phonon Boltzmann equation study

2017 ◽  
Vol 121 (4) ◽  
pp. 044302 ◽  
Author(s):  
J. Kaiser ◽  
T. Feng ◽  
J. Maassen ◽  
X. Wang ◽  
X. Ruan ◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Thermal Transport ◽  
Fourier’S Law ◽  
Phonon Boltzmann Equation ◽  
Fourier's Law

scholarly journals Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation

2017 ◽  
Vol 27 (12) ◽  
pp. 2261-2296 ◽  
Author(s):  
Yan Guo ◽  
Shuangqian Liu
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Kinetic Equations ◽  
Navier Stokes ◽  
Viscous Heating ◽  
Text Setting ◽  
Hydrodynamic Approximation ◽  
Energy Conversation ◽  
The Boltzmann Equation

The incompressible Navier–Stokes–Fourier (INSF) system with viscous heating was first derived from the Boltzmann equation in the form of the diffusive scaling by Bardos–Levermore–Ukai–Yang [Kinetic equations: Fluid dynamical limits and viscous heating, Bull. Inst. Math. Acad. Sin.[Formula: see text] 3 (2008) 1–49]. The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in [Formula: see text] setting in a periodic box. Based on an odd–even expansion of the solution with respect to the microscopic velocity, the diffusive coefficients are determined by the INSF system with viscous heating and the super-Burnett functions. More importantly, the remainder of the expansion is proven to decay exponentially in time via an [Formula: see text] approach on the condition that the initial data satisfies the mass, momentum and energy conversation laws.

Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation

2005 ◽  
Vol 127 (3) ◽  
pp. 298-306 ◽  
Author(s):  
Ronggui Yang ◽  
Gang Chen ◽  
Marine Laroche ◽  
Yuan Taur
Keyword(s):  
Heat Conduction ◽  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Heat Generation ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Computational Time ◽  
Discrete Ordinates ◽  
Fourier Law ◽  
Phonon Boltzmann Equation

Heat conduction in micro- and nanoscale and in ultrafast processes may deviate from the predictions of the Fourier law, due to boundary and interface scattering, the ballistic nature of the transport, and the finite relaxation time of heat carriers. The transient ballistic-diffusive heat conduction equations (BDE) were developed as an approximation to the phonon Boltzmann equation (BTE) for nanoscale heat conduction problems. In this paper, we further develop BDE for multidimensional heat conduction, including nanoscale heat source term and different boundary conditions, and compare the simulation results with those obtained from the phonon BTE and the Fourier law. The numerical solution strategies for multidimensional nanoscale heat conduction using BDE are presented. Several two-dimensional cases are simulated and compared to the results of the transient phonon BTE and the Fourier heat conduction theory. The transient BTE is solved using the discrete ordinates method with a two Gauss-Legendre quadratures. Special attention has been paid to the boundary conditions. Compared to the cases without internal heat generation, the difference between the BTE and BDE is larger for the case studied with internal heat generation due to the nature of the ballistic-diffusive approximation, but the results from BDE are still significantly better than those from the Fourier law. Thus we conclude that BDE captures the characteristics of the phonon BTE with much shorter computational time.

Solution of the Linearized Phonon Boltzmann Equation

1966 ◽  
Vol 148 (2) ◽  
pp. 766-778 ◽  
Author(s):  
R. A. Guyer ◽  
J. A. Krumhansl
Keyword(s):  
Boltzmann Equation ◽  
Phonon Boltzmann Equation
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