Knoxville, TN, United States
Knoxville, TN, United States

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Current industrial applications require a consideration of two-dimensional surface roughness effects in design and optimization of fluid bearings. Although the influence of striated surface roughness on fluid lubrication is now at a fairly mature level of understanding, the knowledge and understanding of two-dimensional roughness effects is not nearly at the same level as that achieved over the past several decades for one-dimensional striations. The subject of this paper includes the formulation of a practical roughness averaged lubrication equation that is appropriate for two-dimensional surface roughness and applicable over a wide range of Knudsen numbers. After derivation by multiple-scale analysis, the resulting lubrication equation is specialized to treat the patterned data islands located on a storage medium as a two-dimensional roughness pattern, and then used to determine the effect of this roughness on the air-bearing interface between recording head slider and disk. The roughness averaged lubrication equation is solved numerically by a variable-grid finite-difference algorithm, and computed results are included for several bearing geometries. © 2012 American Society of Mechanical Engineers.


Low clearance gas bearing applications require an understanding of surface roughness effects at increased levels of Knudsen number. Because very little information has been reported on the relative air-bearing influence of roughness location, this paper is focused on a comparison of the effects of moving and stationary striated surface roughness under high Knudsen number conditions. First, an appropriate lubrication equation will be derived based on multiple-scale analysis that extends the work of White (2010, A Gas Lubrication Equation for High Knudsen Number Flows and Striated Rough Surfaces, ASME J. Tribol., 132, p. 021701). The resulting roughness averaged equation, applicable for both moving and stationary roughness over a wide range of Knudsen numbers, allows an arbitrary striated roughness orientation with regard to both (1) the direction of surface translation and (2) the bearing coordinates. Next, the derived lubrication equation is used to analyze and compare the influences produced by a stepped transverse roughness pattern located on the moving and the stationary bearing surface of a wedge bearing geometry of variable inclination. Computed results are obtained for both incompressible and compressible lubricants, but with an emphasis on high Knudsen number flow. Significant differences in air-bearing performance are found to occur for moving versus stationary roughness. © 2012 American Society of Mechanical Engineers.


Numerical solution of heat conduction in a heterogeneous material with small spatial and time scales can lead to excessive compute times due to the dense computational grids required. This problem is avoided by averaging the energy equation over the smallscales, which removes the appearance of the short spatial and time scales while retaining their effect on the average temperature. Averaging does, however, increase the complexity of the resulting thermal energy equation by introducing mixed spatial derivatives and six different averaged conductivity terms for three-dimensional analysis. There is a need for a numerical method that efficiently and accurately handles these complexities as well as the other details of the averaged thermal energy equation. That is the topic of this paper as it describes a numerical solution for the averaged thermal energy equation based on Fourier conduction reported recently in the literature. The solution, based on finite difference techniques that are second-order time-accurate and noniterative, is appropriate for three-dimensional time-dependent and steady-state analysis. Speed of solution is obtained by spatially factoring the scheme into an alternating direction sequence at each time level. Numerical stability is enhanced by implicit algorithms that make use of the properties of tightly banded matrices. While accurately accounting for the nonlinearity introduced into the energy equation by temperature-dependent properties, the numerical solution algorithm requires only the consideration of linear systems of algebraic equations in advancing the solution from one time level to the next. Computed examples are included and compared with those for a homogeneous material. Copyright © 2016 by ASME.


White J.,6017 Glenmary Road
Journal of Heat Transfer | Year: 2015

In order to better manage computational requirements in the study of thermal conduction with short-scale heterogeneous materials, one is motivated to arrange the thermal energy equation into an accurate and efficient form with averaged properties. This should then allow an averaged temperature solution to be determined with a moderate computational effort. That is the topic of this paper as it describes the development using multiple-scale analysis of an averaged thermal energy equation based on Fourier heat conduction for a heterogeneous material with isotropic properties. The averaged energy equation to be reported is appropriate for a stationary or moving solid and three-dimensional heat flow. Restrictions are that the solid must display its heterogeneous properties over short spatial and time scales that allow averages of its properties to be determined. One distinction of the approach taken is that all short-scale effects, both moving and stationary, are combined into a single function during the analytical development. The result is a self-contained form of the averaged energy equation. By eliminating the need for coupling the averaged energy equation with external local problem solutions, numerical solutions are simplified and made more efficient. Also, as a result of the approach taken, nine effective averaged thermal conductivity terms are identified for three-dimensional conduction (and four effective terms for two-dimensional conduction). These conductivity terms are defined with two types of averaging for the component material conductivities over the short-scales and in terms of the relative proportions of the short-scales. Numerical results are included and discussed. © 2015 by ASME.


Discrete track recording (DTR) is a method for increasing the recording density of a data storage disk by use of a pattern arrangement of discrete tracks. The DTR track structure consists of a pattern of very narrow concentric raised areas and recessed areas underneath a magnetic recording layer. In order to design the air-bearing slider platform that houses the magnetic transducer for DTR application at very low fly heights, the influence of the disk surface topography as a surface roughness effect must be taken into account. This paper is focused on the numerical solution of the roughness averaged lubrication equation reported recently in the work of White (2010, "A Gas Lubrication Equation for High Knudsen Number Flows and Striated Rough Surfaces," ASME J. Tribol., 132, p. 021701) and is specialized for the influence of discrete disk data tracks on the recording head slider-disk air-bearing interface subject to a nonzero skew angle formed between the slider longitudinal axis and the direction of disk motion. The generalized lubrication equation for a smooth surface bearing and appropriate for high Knudsen number analysis is quite nonlinear. And including the averaging process required for treatment of a nonsmooth disk surface, as well as the rotational transformation required to allow for a nonzero skew angle, increases further the nonlinearity and general complexity of the lubrication equation. Emphasis is placed on development of a numerical algorithm that is fast, accurate, and robust for air-bearing analysis of complex slider surfaces. The numerical solution procedure developed utilizes a time integration of the lubrication equation for both steady-state and dynamic analyses. The factored-implicit scheme, a form of the more general alternating-direction-implicit numerical approach, was chosen to deal with the two-dimensional and highly nonlinear aspects of the problem. Factoring produces tightly banded coefficient matrices and results in an algorithm that is second-order accurate in time while requiring only the solution of tridiagonal systems of linear equations in advancing the computation from one time level to the next. Numerical solutions are presented that demonstrate the performance of the computational scheme and illustrate the influence of some discrete track parameters on skewed air-bearing performance as compared with a flat surface data storage disk. © 2011 American Society of Mechanical Engineers.


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. © 2010 by ASME.


White J.,6017 Glenmary Road
Journal of Tribology | Year: 2010

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. Copyright © 2010 by ASME.


The ability to predict surface roughness effects is now well established for gas bearings that satisfy the requirements for either high wave number-limited or high bearing number-limited conditions. However, depending on the parameters involved, a given bearing configuration may not satisfy either of these limited requirements for analysis of roughness effects. Well-established methods for the analysis of surface roughness effects on gas lubrication are not yet available outside of these two limited regions. With that as motivation, this paper then reports an analytical investigation of rough surface gas-bearing effects for the region bounded on one side by high wave number-limited conditions and on the other by high bearing number-limited effects. It emphasizes the gas-bearing region, where shear-driven flow rate and pressure-driven flow rate due to surface roughness are of the same order of magnitude. This paper makes use of the compressible continuum form of the Reynolds equation of lubrication together with multiplescale analysis to formulate a governing lubrication equation appropriate for the analysis of striated roughness effects collectively subject to high bearing number (Λ → ∞), high inverse roughness length scale (β → ∞), and unity order of magnitude-modified bearing number based on roughness length scale (Λ2 = Λ/β = O(1)). The resulting lubrication equation is applicable for both moving and stationary roughness and can be applied in either averaged or un-averaged form. Several numerical examples and comparisons are presented. Among them are results that illustrate an increased sensitivity of bearing force to modified bearing number for Λ2 = O(1). With Λ2 in this range, bearings with either moving or stationary roughness exhibit increased force sensitivities, but the effects act in opposite ways. That is, while an increase in modified bearing number causes a decrease in force for stationary roughness, the same increase in modified bearing number causes an increase in force for moving roughness. Copyright © 2013 by ASME.


Analytical methods and techniques are required for design and analysis of low clearance gas-bearings that account for the combined influence of surface roughness and Knudsen number. Analytical methods for the lubrication equation are currently available for bearings that are either high wave number-limited or high bearing number-limited. There are few useful analytical methods in the range between these limiting extremes that account for the combined effect of roughness and rarefaction. That is the focus of this paper as it extends the work reported by White (2013, "Surface Roughness Effects in the Region Between High Wave Number and High Bearing Number-Limited Lubricant Flows," ASME J. Tribol., 135(4), p. 041706) to include rarefaction effects. Results of an analytical study will be reported that investigates a wedge bearing geometry using perturbation methods and multiple-scale analysis over a wide range of Knudsen numbers for roughness on moving and stationary surfaces. The solution technique developed allows nonlinear aspects of the lubrication equation to be retained in the analysis. Solutions will be presented graphically and discussed. Results indicate that most of the bearing sensitivity to Knudsen number can be accounted for by a modified form of the bearing number. Copyright © 2015 by ASME.

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