The linearized Boltzmann equation with Cercignani–Lampis boundary conditions: Heat transfer in a gas confined by two plane-parallel surfaces

2015 ◽  
Vol 86 ◽  
pp. 45-54 ◽  
Author(s):  
R.D.M. Garcia ◽  
C.E. Siewert
Keyword(s):  
Heat Transfer ◽  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Plane Parallel ◽  
Linearized Boltzmann Equation ◽  
Parallel Surfaces

Inlet and outlet boundary conditions for the discrete velocity direction model

2018 ◽  
Vol 32 (04) ◽  
pp. 1850048 ◽  
Author(s):  
Zhenyu Zhang ◽  
Wei Zhao ◽  
Qingjun Zhao ◽  
Guojing Lu ◽  
Jianzhong Xu
Keyword(s):  
Heat Transfer ◽  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Kinetic Method ◽  
Distribution Functions ◽  
Molecular Flow ◽  
Discrete Velocity ◽  
Flow And Heat Transfer ◽  
Linearized Boltzmann Equation ◽  
Direction Model

The discrete velocity direction model is an approximate method to the Boltzmann equation, which is an optional kinetic method to microgas flow and heat transfer. In this paper, the treatment of the inlet and outlet boundary conditions for the model is proposed. In the computation strategy, the microscopic molecular speed distribution functions at inlet and outlet are indirectly determined by the macroscopic gas pressure, mass flux and temperature, which are all measurable parameters in microgas flow and heat transfer. The discrete velocity direction model with the pressure correction boundary conditions was applied into the plane Poiseuille flow in microscales and the calculations cover all flow regimes. The numerical results agree well with the data of the NS equation near the continuum regime and the date of linearized Boltzmann equation and the DSMC method in the transition regime and free molecular flow. The Knudsen paradox and the nonlinear pressure distributions have been accurately captured by the discrete velocity direction model with the present boundary conditions.

Theory of Boundary Conditions for the Boltzmann Equation

1977 ◽  
Vol 32 (6) ◽  
pp. 521-531 ◽  
Author(s):  
L. Waldmann
Keyword(s):  
Boltzmann Equation ◽  
Boundary Conditions ◽  
Temperature Jump ◽  
Functional Relation ◽  
Linearized Boltzmann Equation ◽  
Kinetic Boundary ◽  
The Difference ◽  
Molecular Distribution Function ◽  
Transport Relaxation ◽  
Slip Coefficients

Abstract In preceding papers, Refs. 1,2, boundary conditions were developed for transport-relaxation equations by aid of a general reciprocity postulate for the interface. The same method is now used for the linearized Boltzmann equation. A new scheme emerges: the kinetic boundary conditions consist in a linear functional relation between interfacial "forces and fluxes" - in the sense of non-equilibrium thermodynamics - which are, broadly speaking, given by the sum and the difference of the molecular distribution function and its time-reversed, at the wall. The general properties of the kernels occurring in this atomistic boundary law are studied. The phenomenological surface coefficients of (generalized) linear thermo-hydrodynamics, as e. g. temperature jump, slip coefficients etc., can in a simple way be expressed by the kernel of the atomistic boundary law. This kernel is explicitly worked out for completely thermalizing wall collisions.

Giant Enhancement in Radiative Heat Transfer in Sub-30 nm Gaps of Plane Parallel Surfaces

Nano Letters ◽  
2018 ◽  
Vol 18 (6) ◽  
pp. 3711-3715 ◽  
Author(s):  
Anthony Fiorino ◽  
Dakotah Thompson ◽  
Linxiao Zhu ◽  
Bai Song ◽  
Pramod Reddy ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Radiative Heat Transfer ◽  
Radiative Heat ◽  
Plane Parallel ◽  
Parallel Surfaces

scholarly journals Differential Equations and Boundary Conditions for Higher Gas Kinetic Moments. Heat Transfer Between Parallel Plates

1977 ◽  
Vol 32 (7) ◽  
pp. 667-677 ◽  
Author(s):  
H. Vestner ◽  
L. Waldmann
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Parallel Plates ◽  
Formula One ◽  
Linearized Boltzmann Equation ◽  
Transport Relaxation ◽  
Monatomic Gas

Abstract Transport-relaxation equations for seventeen moments are derived from the linearized Boltzmann equation for a monatomic gas. Besides the well-known thirteen moments (i. e. density n, tempera­ture T, velocity ν, heat flux q, friction pressure tensor ), one additional scalar A and one ad­ditional vector A are taken into account. In steady state, differential equations for T, v. A and constitutive laws for q, , A follow from the transport-relaxation equations. Boundary conditions for T, v, A are obtained by the thermodynamical method from the interfacial entropy production. The role of the higher moments A and A for heat transfer in a gas between parallel plates is dis­cussed. The heat flux has the correct low pressure limit. Due to the presence of A and A, expo­nential terms occur in the temperature profile near the boundary.

scholarly journals On the Equations and Boundary Conditions Governing Phonon-Mediated Heat Transfer in the Small Mean Free Path Limit: An Asymptotic Solution of the Boltzmann Equation

2014 ◽  
Author(s):  
Jean-Philippe M. Péraud ◽  
Nicolas G. Hadjiconstantinou
Keyword(s):  
Heat Transfer ◽  
Relaxation Time ◽  
Boltzmann Equation ◽  
Asymptotic Solution ◽  
Boundary Conditions ◽  
Interaction Model ◽  
Material Model ◽  
Fourier Law ◽  
Boundary Interaction ◽  
The Boltzmann Equation

Using an asymptotic solution procedure, we construct solutions of the Boltzmann transport equation in the relaxation-time approximation in the limit of small Knudsen number, Kn << 1, to obtain continuum equations and boundary conditions governing phonon-mediated heat transfer in this limit. Our results show that, in the bulk, heat transfer is governed by the Fourier law of heat conduction, as expected. However, this description does not hold within distances on the order of a few mean free paths from the boundary; fortunately, this deviation from Fourier behavior can be captured by a universal boundary-layer solution of the Boltzmann equation that depends only on the material model and the phonon-boundary interaction model (Boltzmann boundary condition). Boundary conditions for the Fourier description follow from matching this inner solution to the outer (Fourier) solution. This procedure shows that the traditional no-jump boundary conditions are appropriate only to zeroth order in Kn. Solution to first order in Kn shows that the Fourier law needs to be complemented by jump boundary conditions with jump coefficients that depend on the material model and the phonon-boundary interaction model. In this work, we calculate these coefficients and the form of the jump conditions for an adiabatic-diffuse and a prescribed-temperature boundary in contact with a constant-relaxation-time material. Extension of this work to variable relaxation-time models is straightforward and will be discussed elsewhere. Our results are validated via comparisons with low-variance deviational Monte Carlo simulations.

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