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Research Papers: Hydrodynamic Lubrication

# Surface Roughness Effects in the Region Between High Wave Number and High Bearing Number Limited Lubricant Flows

[+] Author and Article Information
James White

Knoxville, TN 37919

Contributed by the Tribology Division of ASME for publication in the Journal of Tribology. Manuscript received December 11, 2012; final manuscript received May 11, 2013; published online July 3, 2013. Assoc. Editor: Mihai Arghir.

J. Tribol 135(4), 041706 (Jul 03, 2013) (16 pages) Paper No: TRIB-12-1227; doi: 10.1115/1.4024709 History: Received December 11, 2012; Revised May 11, 2013

## Abstract

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 multiple-scale 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.

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## Figures

Fig. 1

Surface roughness configurations: (a) stationary roughness; (b) moving roughness

Fig. 5

Unaveraged pressure profile segments for stationary roughness as a function of the x coordinate with two different inlet conditions, 100 waves, and (h0 = 2,ɛ = 0.5,Λ2 = 1)

Fig. 2

Ensemble-averaged pressure profiles with stationary roughness for several values of Λ2 with (ɛ = 0.5): (a) h0 = 2; (b) h0 = 3

Fig. 3

Clearance profiles near the bearing inlet for stationary roughness

Fig. 4

Effect of modified bearing number and h0 on the unaveraged bearing performance for stationary roughness and two different inlet conditions (ɛ = 0.5): (a) net force ratio Fnet/Fnet,Λ2→∞; (b) mass flow rate ratio m·/m·Λ2→∞

Fig. 9

Ensemble-averaged solutions for stationary roughness with (ɛ = 0.5): (a) net force ratio Fnet/Fnet,Λ2→∞ as a function of (Λ2,h0); (b) mass flow rate ratio m·/m·Λ2→∞ as a function of (Λ2,h0)

Fig. 10

Time-averaged solutions for moving roughness with (ɛ = 0.5): (a) net force ratio Fnet/Fnet,Λ2→∞ as a function of (Λ2,h0); (b) mass flow rate ratio m·/m·Λ2→∞ as a function of (Λ2,h0)

Fig. 11

Mass flow rate function m·/(Λh0) as a function of ɛ and Λ2 for (h0 = 2)

Fig. 12

Force profiles as a function of roughness amplitude for several values of modified bearing number with (h0 = 2): (a) comparison of the force parameter Fnet/Fnet,ɛ = 0 for ensemble-averaged stationary roughness and time-averaged moving roughness; (b) ratio of ensemble-averaged stationary roughness net force to time-averaged moving roughness net force Fnet,stationary/Fnet,moving

Fig. 6

Unaveraged pressure profile segments for stationary roughness as a function of the x coordinate with two different inlet conditions, 100 waves, and (h0 = 2,ɛ = 0.5): (a) Λ2 = 0.1; (b) Λ2 = 10; (c) Λ2 = 100

Fig. 7

Time- and space-averaged profiles for moving surface roughness and (h0 = 3,ɛ = 0.5): (a) the product (p0h)AVG as a function of the x coordinate and Λ2; (b) the averaged pressure (p0)AVG as a function of the x coordinate and Λ2

Fig. 8

Moving and stationary roughness pressure profiles averaged after solution of the unaveraged lubrication equation with (h0 = 3,ɛ = 0.5,Λ2 = 1)

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