A mean free path theory of void diffusion in a porous medium with surface diffusion. II. Numerical evaluation of the effective diffusivity for arbitrary Knudsen number

1982 ◽  
Vol 76 (1) ◽  
pp. 673-677 ◽  
Author(s):  
Fun Gau Ho ◽  
William Strieder
Keyword(s):  
Porous Medium ◽  
Free Path ◽  
Surface Diffusion ◽  
Knudsen Number ◽  
Numerical Evaluation ◽  
Effective Diffusivity ◽  
Mean Free Path

A Note on Modeling of Nano-Scale Thermal Flow via the Lattice Boltzmann Method

2012 ◽  
Author(s):  
P. Lopez ◽  
Y. Bayazitoglu
Keyword(s):  
Distribution Function ◽  
Free Path ◽  
Knudsen Number ◽  
Reference Data ◽  
Mean Free Path ◽  
Knudsen Layer ◽  
High Order ◽  
Distance Functions ◽  
Transition Flow ◽  
Wall Distance

Lattice Boltzmann (LB) method models have been demonstrated to provide an accurate representation of the flow characteristics in rarefied flows. Conditions in such flows are characterized by the Knudsen number (Kn), defined as the ratio between the gas molecular Mean Free Path ( MFP, λ) and the device characteristic length (L). As the Knudsen number increases, the behavior of the flow near the walls is increasingly dominated by interactions between the gas molecules and the solid surface. Due to this, linear constitutive relations for shear stress and heat flux, which are assumed in the Navier-Stokes-Fourier (NSF) system of equations, are not valid within the Knudsen Layer (KL). Fig. 1 illustrates the characteristics of the velocity field within the Knudsen layer in a shear-driven flow. It is easily observed that although the NSF equations with slip flow boundary conditions (represented by dashed line) can predict the velocity profile in the bulk flow region, they fail to capture the flow characteristics inside the Knudsen layer. Slip flow boundary conditions have also been derived using the integral transform technique [1]. Various methods have been explored to extend the applicability of LB models to higher Knudsen number flows, including using higher order velocity sets, and using wall-distance functions to capture the effect of the walls on the mean free path by incorporating such functions on the determination of the local relaxation parameters. In this study, a high order velocity model which contains a two-dimensional, thirteen velocity direction set (e.g., D2Q13), as shown in Fig. 2, is used as the basis of the current LB model. The LB model consists of two independent distribution functions to simulate the density and temperature fields, while the Diffuse Scattering Boundary Condition (DSBC) method is used to simulate the fluid interaction with the walls. To further improve the characterization of transition flow conditions expected in nano-scale heat transfer, we explored the implementation of two wall-distance functions, derived recently based on an integrated form of a probability distribution function, to the high-order LB model. These functions are used to determine the effective mean free path values throughout the height of the micro/nano-channel, and the resulting effect is first normalized and then used to determine local relaxation times for both momentum and energy using a relationship based on the local Knudsen number. The two wall-distance functions are based on integral forms of 1) the classical probability distribution function, ψ(r) = λ0−1e−r/λ0, derived by Arlemark et al [2], in which λ0represents the reference gas mean free path, and 2) a Power-Law probability distribution function, derived by Dongari et al [3]. Thus, the probability that a molecule travels a distance between r and r+dr between two successive collisions is equal to ψ(r)dr. The general form of the integral of the two functions used can be described by ψ(r) = C − f(r), where f(r) represents the base function (exponential or Power Law), and C is set to 1 so that the probability that a molecule will travel a distance r+dr without a collision ranges from zero to 1. The performance of the present LB model coupled with the implementation of the two wall-distance functions is tested using two classical flow cases. The first case considered is that of isothermal, shear-driven Couette flow between two parallel, horizontal plates separated by a distance H, moving in opposite directions at a speed of U0. Fig. 3 shows the normalized velocity profiles across the micro-channel height for various Knudsen numbers in the transition flow regime based on our LB models as compared to data based on the Linearized Boltzmann equation [4]. The results show that our two LB models provide results that are in excellent agreement with the reference data up to the high end of the transition flow regime, with Knudsen numbers greater than 1. The second case is rarefied Fourier flow within horizontal, parallel plates, with the plates being stationary and set to a constant temperature (TTop > TBottom), and the Prandtl number is set to 0.67 to match the reference data based on the Direct Simulation Monte Carlo (DSMC) method [5]. Fig. 4 shows the normalized temperature profiles across the microchannel height for various Knudsen numbers in the slip/transition How regime. For the entire Knudsen number range studied, our two LB models provide temperature profiles that are in excellent agreement with the non-linear profile seen in the reference data. The results obtained show that the effective MFP relationship based on the exponential function improves the results obtained with the high order LB model for both shear-driven and Fourier flows up to Kn∼1. The results also show that the effective MFP relationship based on the Power Law distribution function greatly enhances the results obtained with the high order LB model for the two cases addressed, up to Kn∼3. In conclusion, the resulting LB models represent an effective tool in modeling non-equilibrium gas flows expected within micro/nano-scale devices.

scholarly journals Unified gas-kinetic wave-particle methods IV: multi-species gas mixture and plasma transport

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Chang Liu ◽  
Kun Xu
Keyword(s):  
Free Path ◽  
Cell Size ◽  
Knudsen Number ◽  
Mean Free Path ◽  
Plasma Transport ◽  
Gas Mixture ◽  
Time Step ◽  
Flow Physics ◽  
Asymptotic Complexity ◽  
The Continuum

AbstractIn this paper, we extend the unified gas-kinetic wave-particle (UGKWP) methods to the multi-species gas mixture and multiscale plasma transport. The construction of the scheme is based on the direct modeling on the mesh size and time step scales, and the local cell’s Knudsen number determines the flow physics. The proposed scheme has the multiscale and asymptotic complexity diminishing properties. The multiscale property means that according to the cell’s Knudsen number the scheme can capture the non-equilibrium flow physics when the cell size is on the kinetic mean free path scale, and preserve the asymptotic Euler, Navier-Stokes, and magnetohydrodynamics (MHD) when the cell size is on the hydrodynamic scale and is much larger than the particle mean free path. The asymptotic complexity diminishing property means that the total degrees of freedom of the scheme reduce automatically with the decreasing of the cell’s Knudsen number. In the continuum regime, the scheme automatically degenerates from a kinetic solver to a hydrodynamic solver. In the UGKWP, the evolution of microscopic velocity distribution is coupled with the evolution of macroscopic variables, and the particle evolution as well as the macroscopic fluxes is modeled from a time accumulating solution of kinetic scale particle transport and collision up to a time step scale. For plasma transport, the current scheme provides a smooth transition from particle-in-cell (PIC) method in the rarefied regime to the magnetohydrodynamic solver in the continuum regime. In the continuum limit, the cell size and time step of the UGKWP method are not restricted by the particle mean free path and mean collision time. In the highly magnetized regime, the cell size and time step are not restricted by the Debye length and plasma cyclotron period. The multiscale and asymptotic complexity diminishing properties of the scheme are verified by numerical tests in multiple flow regimes.

Inelastic mean free path and phase-shift determinations in NiO, using EXELFS

1993 ◽  
Vol 3 (7) ◽  
pp. 1649-1659
Author(s):  
Mohammad A. Tafreshi ◽  
Stefan Csillag ◽  
Zou Wei Yuan ◽  
Christian Bohm ◽  
Elisabeth Lefèvre ◽  
...  
Keyword(s):  
Phase Shift ◽  
Free Path ◽  
Mean Free Path ◽  
Inelastic Mean Free Path

The measurement of mean free path λs using a slowing down time lead spectrometer in the 1–10 eV energy region

1968 ◽  
Vol 22 (4) ◽  
pp. 261-262
Author(s):  
M.P. Navalkar ◽  
K. Chandramoleshwar ◽  
D.V.S. Ramkrishna
Keyword(s):  
Free Path ◽  
Energy Region ◽  
Mean Free Path ◽  
Slowing Down ◽  
Time Lead
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