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.

Plane Thermal Transpiration of a Rarefied Gas Between Two Walls of Maxwell-Type Boundaries With Different Accommodation Coefficients

2014 ◽  
Vol 136 (8) ◽  
Author(s):  
Toshiyuki Doi
Keyword(s):  
Boltzmann Equation ◽  
Mass Flow Rate ◽  
Knudsen Number ◽  
Mass Flow ◽  
Flow Rate ◽  
Poiseuille Flow ◽  
Rarefied Gas ◽  
Thermal Transpiration ◽  
Wide Range ◽  
Accommodation Coefficients

Plane thermal transpiration of a rarefied gas between two walls of Maxwell-type boundaries with different accommodation coefficients is studied based on the linearized Boltzmann equation for a hard-sphere molecular gas. The Boltzmann equation is solved numerically using a finite difference method, in which the collision integral is evaluated by the numerical kernel method. The detailed numerical data, including the mass and heat flow rates of the gas, are provided over a wide range of the Knudsen number and the entire range of the accommodation coefficients. Unlike in the plane Poiseuille flow, the dependence of the mass flow rate on the accommodation coefficients shows different characteristics depending on the Knudsen number. When the Knudsen number is relatively large, the mass flow rate of the gas increases monotonically with the decrease in either of the accommodation coefficients like in Poiseuille flow. When the Knudsen number is small, in contrast, the mass flow rate does not vary monotonically but exhibits a minimum with the decrease in either of the accommodation coefficients. The mechanism of this phenomenon is discussed based on the flow field of the gas.

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.

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.

Experimental Investigation of Externally Pressurized Bearings Under High Knudsen Number Conditions

1988 ◽  
Vol 110 (1) ◽  
pp. 144-147 ◽  
Author(s):  
S. Fukui ◽  
R. Kaneko
Keyword(s):  
Experimental Investigation ◽  
Boltzmann Equation ◽  
Knudsen Number ◽  
Numerical Results ◽  
Experimental Results ◽  
Molecular Flow ◽  
Free Molecular Flow ◽  
Transition Flow ◽  
Lubrication Equation ◽  
Medium Vacuum

The characteristics of the externally pressurized bearings under high Knudsen number conditions were investigated experimentally by the use of surface restriction bearings in a medium vacuum on the order of 0.1 kPa (10−3 atm.). The experimental results agreed well with the numerical results calculated from a generalized lubrication equation based on the Boltzmann equation. Therefore, it would appear that this generalized lubrication equation is valid even when flows are categorized into transition flow or free molecular flow.

A Gas Lubrication Equation for High Knudsen Number Flows and Striated Rough Surfaces

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
James White
Keyword(s):  
Surface Roughness ◽  
Knudsen Number ◽  
Poiseuille Flow ◽  
Rough Surfaces ◽  
Multiple Scale ◽  
Scale Analysis ◽  
Recording Head ◽  
Lubrication Equation ◽  
Highly Nonlinear ◽  
Multiple Scale Analysis

This paper describes the derivation and numerical solution of a lubrication equation appropriate for high Knudsen number flows and certain types of striated rough surfaces. The derivation begins with the compressible form of the lubrication equation together with the nonlinear series form of the Poiseuille flow reported by Fukui and Kaneko (1990, “A Database for Interpolation of Poiseuille Flow Rates for High Knudsen Number Lubrication Problems,” ASME J. Tribol., 112, pp. 78–83.). A multiple-scale analysis is performed on the lubrication equation for a finite-width time-dependent bearing and is limited to either stationary-transverse or longitudinal striated surface roughness of very short length scale. The rough surface averaging that takes place within the multiple-scale analysis includes a fully coupled treatment of the Poiseuille flow. What results is an especially nonlinear lubrication equation with averaged surface roughness effects that is appropriate for high Knudsen number analysis. A rotational transformation is also introduced to provide the roughness averaged lubrication equation in a form that allows analysis of the skewed orientation of a recording head slider with roughness defined relative to the direction of disk motion but with the lubrication equation conveniently expressed in the coordinate system of the slider. A factored-implicit numerical algorithm is described that provides the solution of the roughness averaged lubrication equation. Even though the lubrication equation is highly nonlinear, the numerical scheme is crafted to be fully second-order, time-accurate, and noniterative for tracking the solution in time either to an asymptotic steady-state or in response to a dynamic event. Numerical solutions of several simple geometry bearings are presented that utilize parameters that are typical of the slider-disk interface of current hard disk drives. It is anticipated that the primary benefit of this work may be the ability to accurately and efficiently include the influence of discrete disk data tracks in the air bearing design of very low clearance recording head sliders.

Surface Roughness Effects on Air Bearing Performance Over a Wide Range of Knudsen and Wave Numbers

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
James White
Keyword(s):  
Surface Roughness ◽  
Data Storage ◽  
Knudsen Number ◽  
Poiseuille Flow ◽  
Air Bearing ◽  
Roughness Effects ◽  
Averaging Methods ◽  
Lubrication Equation ◽  
Wide Range ◽  
Averaging Techniques

Design of a near contact air bearing interface such as that created by a recording head slider and data storage disk requires consideration of a lubrication equation that is appropriate for high Knudsen number flows. The Poiseuille flow database reported by Fukui and Kaneko, 1990 [“A Database for Interpolation of Poiseuille Flow Rates for High Knudsen Number Lubrication Problems,” ASME J. Tribol., 112, pp. 78–83] is appropriate over a wide range of Knudsen numbers and is used throughout the data storage industry for analysis of the low flying recording head slider air bearing. However, at such low clearances, the topography of the air bearing surfaces also comes into question, making it important to consider both rarefaction and surface roughness effects in the air bearing design. In order to simplify the air bearing analysis of rough surfaces, averaging techniques for the lubrication equation have been developed for situations where the number of roughness elements (or waves) is either much greater or much less than the gas bearing number. Between these two extremes there are currently no roughness averaging methods available. Although some analytical and numerical studies have been reported for continuum and first-order slip conditions with simple geometries, little or no results have appeared that include both surface roughness and high Knudsen number flows outside the limited ranges where surface averaging techniques are used. In order to better understand the influence of transverse surface roughness over a wide range of Knudsen numbers and the relationship of key parameters involved, this paper describes a primarily analytical air bearing study of a wide, rough surface slider bearing using the Poiseuille flow database reported by Fukui and Kaneko. The work is focused outside the limited ranges where current surface averaging methods for the lubrication equation are expected to be valid.

Thin Spacing Analysis for Head-Tape Interface

1996 ◽  
Vol 118 (4) ◽  
pp. 800-806 ◽  
Author(s):  
Kazuo Sakai ◽  
Yasumasa Nagawa ◽  
Koetsu Okuyama ◽  
Takao Terayama
Keyword(s):  
Surface Roughness ◽  
Boltzmann Equation ◽  
Flow Model ◽  
Slip Flow ◽  
First Order ◽  
Lubrication Equation ◽  
Linearized Boltzmann Equation ◽  
Deformation Equation ◽  
High Density Magnetic Recording ◽  
Average Flow Model

Very thin head-tape spacing, combining contact and floating conditions, is investigated for high density magnetic recording. A generalized lubrication equation, based on a linearized Boltzmann equation, is coupled with the tape deformation equation for analysis. Tape-surface roughness is also taken into account in the lubrication equation. The average flow model is adopted to analyzing tape-surface roughness. For very thin spacing conditions, it is found that the spacing based on the linearized Boltzmann equation is smaller than that based on first-order slip flow, and larger than that based on second-order slip flow. It is also found that considering tape-surface roughness reduces the calculated minimum spacing. Analytical results agreed with the experimental ones.

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