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Research Papers: Hydrodynamic Lubrication

Homogenization in Hydrodynamic Lubrication: Microscopic Regimes and Re-Entrant Textures

[+] Author and Article Information
İ. N. Yıldıran, B. Çetin

Department of Mechanical Engineering,
Bilkent University,
Ankara 06800, Turkey

İ. Temizer

Department of Mechanical Engineering,
Bilkent University,
Ankara 06800, Turkey
e-mail: temizer@bilkent.edu.tr

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received November 17, 2016; final manuscript received May 11, 2017; published online July 21, 2017. Assoc. Editor: Stephen Boedo.

J. Tribol 140(1), 011701 (Jul 21, 2017) (19 pages) Paper No: TRIB-16-1355; doi: 10.1115/1.4036770 History: Received November 17, 2016; Revised May 11, 2017

The form of the Reynolds-type equation which governs the macroscopic mechanics of hydrodynamic lubrication interfaces with a microscopic texture is well-accepted. The central role of the ratio of the mean film thickness to the texture period in determining the flow factor tensors that appear in this equation had been highlighted in a pioneering theoretical study through a rigorous two-scale derivation (Bayada and Chambat, 1988, “New Models in the Theory of the Hydrodynamic Lubrication of Rough Surfaces,” ASME J. Tribol., 110, pp. 402–407). However, the resulting homogenization theory still remains to be numerically investigated. For this purpose, after a comprehensive review of the literature, three microscopic regimes of lubrication will be outlined, and the transition between these three regimes for different texture types will be extensively demonstrated. In addition to conventional textures, representative re-entrant textures will also be addressed.

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References

Figures

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Fig. 1

The multiscale interface problem is depicted, where the numerical solution of the macroscopic problem requires the consideration of the microscale. Here, among representative quantities, L is a macroscopic dimension, λ is the wavelength for film thickness variations, h0 is the local average film thickness, and a is the amplitude of the oscillations.

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Fig. 2

Scale separation and different regimes are depicted qualitatively for three different textures. Cases 1a and 1b are two realizations of the same periodic texture which only differ by a phase. Here, it is assumed that {h0, L, a} are fixed and only λ is varied. The variation of the pressure with λ may depend on the particular problem. The depicted variation is not based on the numerical results but, where scale separation holds, is only qualitatively associated with the classical wedge problem in a one-dimensional setting with zero Dirichlet boundary conditions and a periodic texture.

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Fig. 3

The physical unit-cell geometry with relevant problem variables, the three-dimensional fluid domain for the solution of the Stokes problem, and its two-dimensional projection onto the in-plane coordinates for the solution of the Reynolds problem are depicted

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Fig. 4

Conventional/re-entrant texture unit-cells are depicted in two dimensions, and a modification of the original re-entrant texture toward a conventional one with a well-defined film thickness is proposed. In conventional textures, a vertical line (e.g., the solid blue line) across the fluid domain is unbroken by the solid domain, whereas it is broken in re-entrant textures (see color figure version online).

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Fig. 5

Unit-cell geometries for conventional textures. For all textures, the only free geometry variables are {λ, h0, a}. For the ellipsoidal and V-shaped textures, b is adjusted to obtain a desired h0 once {λ, a} are specified.

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Fig. 6

For the sinusoidal texture with λ = 1 μm, the variation of {AS, CS} with γ is shown at different {h0, a} combinations that all share the same value of ζ = a/h0 = 0.5. Here, and in several subsequent plots, the Reynolds/congested limit is evaluated explicitly and plotted as an empty/filled square with the same color as the corresponding curve: (a) AS variation at different {h0, a} combinations and (b) CS variation at different {h0, a} combinations (see color figure version online).

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Fig. 7

For the sinusoidal texture, normalization of the homogenized response with the homogeneous one collapses the three curves in Fig. 6 onto the black curves in (a-1) and (b-1). This normalization will be employed in all subsequent figures. This curve depends on the nondimensional parameter ζ. The boxes indicate the Reynolds and congested limits, as in Fig. 6. The variation of the response with varying ζ at fixed values of γ is shown explicitly in Eqs. (a-2) and (b-2): (a-1) AS variation at fixed a/h0 values, (b-1) CS variation at fixed a/h0 values, (a-2) AS variation at fixed h0/λ values, and (b-2) CS variation at fixed h0/λ values.

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Fig. 8

For different conventional textures, the variation of the macroscopic response with γ at a fixed value of ζ = 0.5 and the variation with ζ at a fixed value of γ = 1 are shown. The boxes in (a-1) and (b-1) indicate the Reynolds and congested limits, as in Fig. 6: (a-1) AS variation for conventional textures (ζ = 0.5), (b-1) CS variation for conventional textures (ζ = 0.5), (a-2) AS variation for conventional textures (γ = 1), and (b-2) CS variation for conventional textures (γ = 1).

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Fig. 9

Unit-cell geometries for two re-entrant textures. The case of b = c = λ/2 recovers the geometry of the square texture from Fig. 5. With respect to this reference configuration, to control the degree to which the texture is re-entrant, the angle of the vertical surfaces is changed in the trapezoidal surface at fixed values of {λ, h0, a} by varying {b, c} accordingly. Hence, the trapezoidal texture approaches the sawtooth texture in the limit as b/λ → 0. Similarly, for the T-shaped texture, the width of the top portion is changed so that c decreases as b increases. The trapezoidal texture is re-entrant only for b/λ > 0.5 but the whole range of 0–1 will be tested.

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Fig. 10

For the two re-entrant textures, the variation of the macroscopic response with an increasing degree ξ = b/λ of re-entrant features is shown at different γ = h0/λ values. Despite the value of γ=O(1), ξ→1 effectively leads to flow obstruction, the limit of which was evaluated explicitly via the congested limit and indicated by the dashed line as a lower bound: (a-1) AS variation for trapezoidal texture, (b-1) CS variation for trapezoidal texture, (a-2) AS variation for T-shaped texture, and (b-2) CS variation for T-shaped texture (see color figure version online).

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Fig. 11

The variation of the macroscopic response with γ is shown with different degrees ξ = b/λ of re-entrant features: (a-1) AS variation for trapezoidal texture, (b-1) CS variation for trapezoidal texture, (a-2) AS variation for T-shaped texture, and (b-2) CS variation for T-shaped texture

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Fig. 12

The macroscopic responses {A, C}={AS, CS} of the original re-entrant textures and the responses {A,C}={A¯S,C¯S} of the simplified textures are compared at two different re-entrant configurations (controlled by ξ = b/λ) from the Reynolds limit to the congested limit based on the formulation of the Stokes regime. The lines toward the Reynolds and congested limits can be further straightened by employing finer mesh discretizations—see Appendix B: (a-1) AS variation for trapezoidal texture, (b-1) CS variation for trapezoidal texture, (a-2) AS variation for T-shaped texture, and (b-2) CS variation for T-shaped texture.

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Fig. 13

Near the Reynolds limit (γ = h0/λ = 2 × 10−2), macroscopic predictions from different formulations are compared for different degrees ξ = b/λ of re-entrant features. For the trapezoidal texture, fixing h0 corresponds to the adjustment of the gap between the surfaces so as to keep the mean film thickness a constant during geometry simplification which, however, leads to an incorrect trend prediction: (a-1) A variation for trapezoidal texture, (b-1) C variation for trapezoidal texture, (a-2) A variation for T-shaped texture, and (b-2) C variation for T-shaped texture.

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Fig. 14

For the sinusoidal texture configuration with ζ = a/h0 = 0.5, the influence of regular edge refinement is demonstrated for an increasing total number of elements. The default regular mesh employs 3200 elements: (a) AS variation and (b) CS variation.

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Fig. 15

For re-entrant textures, presently for the trapezoidal texture, switching from a regular to a compatible mesh with the same number of elements (3200) delivers qualitatively better results: (a) AS variation and (b) CS variation

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Fig. 16

The compatible mesh results from Fig. 15 are improved by increasing the total number of elements employed from the default value of 3200–9600 in the range γ∈(10−2, 10−1). The region near the congested limit is excluded to highlight refinement effects more clearly: (a) AS variation and (b) CS variation.

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Fig. 17

The solution to the Poiseuille cell problem (2.9) of the Stokes regime is provided for representative texture configurations. The arrows indicate the magnitude and direction of ω while the background color represents π variation (red: high, blue: low) (see color figure version online).

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Fig. 18

The solution to the Couette cell problem (2.10) of the Stokes regime is provided for representative texture configurations. The arrows indicate the magnitude and direction of Ω, while the background color represents Π variation (red: high, blue: low) (see color figure version online).

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