scholarly journals Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows

2014 ◽  
Vol 746 ◽  
pp. 53-84 ◽  
Author(s):  
Lei Wu ◽  
Jason M. Reese ◽  
Yonghao Zhang
Keyword(s):  
Boltzmann Equation ◽  
Aspect Ratio ◽  
Flow Pattern ◽  
Knudsen Number ◽  
Rectangular Channel ◽  
Velocity Profiles ◽  
Thermal Creep ◽  
Flow Rates ◽  
Linearized Boltzmann Equation ◽  
The Boltzmann Equation

AbstractBased on the fast spectral approximation to the Boltzmann collision operator, we present an accurate and efficient deterministic numerical method for solving the Boltzmann equation. First, the linearized Boltzmann equation is solved for Poiseuille and thermal creep flows, where the influence of different molecular models on the mass and heat flow rates is assessed, and the Onsager–Casimir relation at the microscopic level for large Knudsen numbers is demonstrated. Recent experimental measurements of mass flow rates along a rectangular tube with large aspect ratio are compared with numerical results for the linearized Boltzmann equation. Then, a number of two-dimensional microflows in the transition and free-molecular flow regimes are simulated using the nonlinear Boltzmann equation. The influence of the molecular model is discussed, as well as the applicability of the linearized Boltzmann equation. For thermally driven flows in the free-molecular regime, it is found that the magnitudes of the flow velocity are inversely proportional to the Knudsen number. The streamline patterns of thermal creep flow inside a closed rectangular channel are analysed in detail: when the Knudsen number is smaller than a critical value, the flow pattern can be predicted based on a linear superposition of the velocity profiles of linearized Poiseuille and thermal creep flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearized Poiseuille and thermal creep velocity profiles at the critical Knudsen number. The critical Knudsen number is found to be related to the aspect ratio of the rectangular channel.

A Database for Interpolation of Poiseuille Flow Rates for High Knudsen Number Lubrication Problems

1990 ◽  
Vol 112 (1) ◽  
pp. 78-83 ◽  
Author(s):  
S. Fukui ◽  
R. Kaneko
Keyword(s):  
Boltzmann Equation ◽  
Knudsen Number ◽  
Flow Rate ◽  
Poiseuille Flow ◽  
Interpolation Method ◽  
Numerical Calculations ◽  
Flow Rates ◽  
Lubrication Equation ◽  
Linearized Boltzmann Equation ◽  
Rapid Calculation

This paper proposes the use of a Poiseuille flow rate database for rapid calculation of a generalized lubrication equation for high Knudsen number gas films. The database is created by numerical calculations based on the linearized Boltzmann equation. The proposed interpolation method is verified to reduce calculation time to several tenths of that required to perform rigorous calculations with the same accuracy.

scholarly journals Non-equilibrium dynamics of dense gas under tight confinement

2016 ◽  
Vol 794 ◽  
pp. 252-266 ◽  
Author(s):  
Lei Wu ◽  
Haihu Liu ◽  
Jason M. Reese ◽  
Yonghao Zhang
Keyword(s):  
Boltzmann Equation ◽  
Flow Regime ◽  
Mass Flow Rate ◽  
Knudsen Number ◽  
Mass Flow ◽  
Flow Rate ◽  
Transitional Flow ◽  
Molecular Flow ◽  
Parallel Plates ◽  
The Boltzmann Equation

The force-driven Poiseuille flow of dense gases between two parallel plates is investigated through the numerical solution of the generalized Enskog equation for two-dimensional hard discs. We focus on the competing effects of the mean free path ${\it\lambda}$, the channel width $L$ and the disc diameter ${\it\sigma}$. For elastic collisions between hard discs, the normalized mass flow rate in the hydrodynamic limit increases with $L/{\it\sigma}$ for a fixed Knudsen number (defined as $Kn={\it\lambda}/L$), but is always smaller than that predicted by the Boltzmann equation. Also, for a fixed $L/{\it\sigma}$, the mass flow rate in the hydrodynamic flow regime is not a monotonically decreasing function of $Kn$ but has a maximum when the solid fraction is approximately 0.3. Under ultra-tight confinement, the famous Knudsen minimum disappears, and the mass flow rate increases with $Kn$, and is larger than that predicted by the Boltzmann equation in the free-molecular flow regime; for a fixed $Kn$, the smaller $L/{\it\sigma}$ is, the larger the mass flow rate. In the transitional flow regime, however, the variation of the mass flow rate with $L/{\it\sigma}$ is not monotonic for a fixed $Kn$: the minimum mass flow rate occurs at $L/{\it\sigma}\approx 2{-}3$. For inelastic collisions, the energy dissipation between the hard discs always enhances the mass flow rate. Anomalous slip velocity is also found, which decreases with increasing Knudsen number. The mechanism for these exotic behaviours is analysed.

The Effects of Geometry and Knudsen Numbers on Micro- and Nanochannel Flows

2011 ◽  
Author(s):  
J. H. Kim ◽  
A. J. H. Frijns ◽  
S. V. Nedea ◽  
A. A. van Steenhoven
Keyword(s):  
Flow Velocity ◽  
Knudsen Number ◽  
Gas Flow ◽  
Rectangular Channel ◽  
Maximum Velocity ◽  
Velocity Profiles ◽  
Dynamics Simulation ◽  
Circular Channel ◽  
Knudsen Numbers ◽  
Slit Geometry

In this work we use a three dimensional Molecular Dynamics simulation method to study the effect of different geometries and Knudsen number regimes on the gas flow in micro-nanochannels. Argon molecules have been used for the simulations. Thermal wall and diffusive-specular wall types were used for the boundaries of the channels. The velocity profiles in the channel were obtained and analyzed with three different channel geometries that are commonly used in the industry: circular, rectangular (square), and slit channel. We found that when using the same driving force, the maximum velocity of the flow increases when the geometry changes in the order from circular geometry to rectangular geometry to slit geometry, where the latter becomes 1.2∼1.5 times as large compared with either the rectangular or circular channel. While the absolute values of the velocity profiles show a distinct difference according to the different geometries, geometry effect on the shape of the velocity profile also shows interesting features. Rectangular tube shows much flatter profile compared with the other two channels. Also the effect of the size of the channels and different Knudsen numbers on the velocity profiles is investigated. Two different sizes were used here: 100nm and 10nm corresponding to typical sizes of a nano channel and carbon nanotubes. We found that the Knudsen number has an effect on the slip and maximum flow velocity for the slit geometry even for higher Knudsen number. For the Kn higher than approximately 3, it was found that the Knudsen number has a small influence on the slip flow velocity for the circular channel and rectangular channel than for lower Knudsen number.

Mechanism for Formation of Highly Monodisperse Droplet in a Microfluidic T-Junction Device

2007 ◽  
Author(s):  
Zhipeng Liu ◽  
Jinliang Xu
Keyword(s):  
Aspect Ratio ◽  
High Aspect Ratio ◽  
Rectangular Channel ◽  
Dispersed Phase ◽  
Breakup Process ◽  
Flow Rates ◽  
Junction Device ◽  
Micron Size ◽  
Unbounded Fluid ◽  
Monodisperse Droplet

Droplet formation in a microfluidic T-junction device which is high aspect ratio rectangular channel connected by a perpendicular channel were investigated experimentally. This geometry is quite similar to the classic T-junction device, however the perpendicular channel is a slightly narrower with respect to the dispersed phase inlet, leading to remarkably different result. The perfectly controllable droplets were found to be monodispersed with a less than 2% variation in micron size. Experimental results, including the relation between diameter and flow rates, the change of the velocity and pressure at drop break-up process, had been analyzed in detail. The single breakup process and the quasistatic character were described by evolution of the width and length of the liquid-liquid interface. Finally, in contrast to the capillary instability in an unbounded fluid, the breakup process was explained in term of absolute instabilities.

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