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Stokes Flow in Thin Films

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Stokes Flow in Thin Films Stokes Flow in Thin Films Present understanding of the mechanisms of lubrication and the load carrying capacity of lubricant films mainly relies on models in which the Reynolds equation is used to describe the flow. The narrow gap assumption is a key element in its derivation from the Navier Stokes equations. However, the tendency in applications is that lubricated contacts have D. E. A. van Odyck to operate at smaller film thickness levels, and because engineering surfaces are never perfectly smooth, locally in the film this narrow gap assumption may violated. In addition C. H. Venner to this geometric limitation of the validity of the Reynolds equation may come a piezovis- cous and compressibility related limitation. In this paper the accuracy of the predictions University of Twente, of the Reynolds model in relation to the local geometry of the gap is investigated. A Faculty of Engineering Technology, numerical solution algorithm for the flow in a narrow gap has been developed based on Enschede, The Netherlands the Stokes equations. For a model problem the differences between the pressure and velocity fields according to the Stokes model and the Reynolds equation have been inves- tigated. The configuration entails a lower flat surface together with an upper surface (flat or parabolic) in which a local defect (single asperity) of known geometry has been embedded. It is investigated how the magnitude of the differences develops as a function of the geometric parameters of the film and the feature. Finally, it is discussed to what extend for these problems a perturbation approach can provide accurate corrections to be applied to the Reynolds solution. ͓DOI: 10.1115/1.1506317͔ 1 Introduction incorporated in a model ͑either Stokes or Reynolds͒. Two ap- proaches are distinguished: a deterministic approach and a statis- Technical surfaces are never perfectly smooth and an important tical approach. The statistical approach is based on the character- question is how to account for the effect of the surface roughness ization of a surface roughness profile by means of statistical on the film formation. This has been the subject of many studies in parameters. ‘‘Statistical’’ model equations are derived, e.g., a gen- the past decades. Clearly if the surfaces are so smooth that the eralized Reynolds equation, in which the coefficients are a func- computed minimum film thickness for the given operating condi- tions is large compared to the standard deviation of the surface tion of various statistical properties of the roughness. The most common quantities appearing in such models are the Root Mean roughness profile in the contact the effect of the roughness will be ͑ ͒ negligible. However, if the amplitude of the roughness is compa- Square RMS value and the auto-correlation length. In general it rable to the thickness of the film one has to take into account its has been observed that the rough surface analysis yields a larger effect on the film. load capacity then obtained for the smooth surface case. Examples ͑ ͒ of this approach based on the Reynolds equation are the works of For a Newtonian fluid the Navier-Stokes NS equations de- ͓ ͔ ͓ ͔ scribe the conservation of momentum. If the inertia terms are Tzeng and Saibel 3,4 , Christensen and co-workers 5–8 , Elrod ͓ ͔ ͓ ͔ ͓ ͔ small compared to the viscous terms, they reduce to the Stokes 9,10 , Cheng and Dyson 11 and also Chow and Cheng 12 , ͓ ͔ ͓ ͔ ͓ ͔ equations. Assuming next that H/LӶ1, where H is the local film Chow and Saibel 13 , Sun 14 , Phan Thien 15–17 , Phan-Thien ͓ ͔ ͓ ͔ thickness and L some typical length scale in the contact it then and Atkinson, 18 and Prakash 19 . A related approach referred follows from the Stokes equations that the pressure will be inde- to as ‘‘the flow factor method’’ is where the details of the rough- pendent of the cross-film coordinate which facilitates integration ness enter in coefficients of the Reynolds equation as so-called over the cross-film coordinate to obtain the velocity field and flow factors, see Patir and Cheng ͓20, 21͔, Teale and Lebeck ͓22͔, eventually leads to the Reynolds ͓1͔ equation from which the Elrod ͓2͔, and Tripp ͓23͔. This flow factor method is widely used pressure can be solved. However, for the case of non-smooth sur- in practice in both hydrodynamic lubrication as well as ElastoHy- faces the assumption H/LӶ1may͑locally͒ be violated. This is drodynamic Lubrication ͑EHL͒ problems. Examples of the latter already suggested by Elrod ͓2͔ where a table of values of a ratio are found in the works of Sadeghi and Sui ͓24͔, Patir and Cheng ⌳ ͓25͔, and Tripp and Hamrock ͓26͔. H/ c is presented as can appear in applications where H is the ⌳ film thickness and c a characteristic roughness wavelength. It is The deterministic approach differs from the statistical approach argued that this ratio could be as large as O(1). in that the model equations, Reynolds or Stokes, are solved taking Based on the anticipated validity of the Reynolds equation for the film thickness an exact description of the surface geometry roughness has been classified in two categories: Reynolds rough- as a function of the coordinates. An early example of deterministic ness and Stokes roughness, see Elrod ͓9͔: studies in the field of EHL are the papers of Goglia et al. ͓27,28͔. In general the model equations have to be solved numerically. An ⌳ Ӷ • Reynolds roughness H/ c 1. accurate description of small scale surface features requires many ⌳ ϭ • Stokes roughness H/ c O(1). gridpoints so, for a deterministic approach to be feasible, fast Reynolds roughness implies it is assumed that it is justified to use computers and, more important, efficient numerical algorithms, the Reynolds equation to describe the fluid flow in the lubricated are essential. This is clearly illustrated by the developments in the contact, otherwise the Stokes equations should be used. It is not field of EHL modeled using Reynolds’ equation. Fast and stable clear where the border between Reynolds and Stokes roughness algorithms based on multigrid/multilevel techniques aided by the should be drawn and how it is related to the problem parameters. developments in terms of faster computers have led to a situation A second classification of roughness effect studies that has been where at present the effects of surface features on film formation made is based on the way in which the roughness effects are in EHL point contacts can be simulated using a deterministic ap- proach, even taking into account the transient effects of the sur- face feature moving through the contact. Examples of such studies Contributed by the Tribology Division for publication in the ASME JOURNAL OF ͓ ͔ ͓ ͔ TRIBOLOGY. Manuscript received by the Tribology Division October 19, 2001 re- are the papers of Venner et al. 29,30 and Wijnant et al. 31 .At vised manuscript received June 26, 2002; Associate Editor: L. San Andre´s. the same time on the experimental side the optical interferometry Journal of Tribology Copyright © 2003 by ASME JANUARY 2003, Vol. 125 Õ 121 ! technique to measure the film thickness in model contacts im- u ucu, proved so much that it has become possible to validate predicted film thickness behavior even on a sub-contact scale. This is for H ͓ ͔ w!⑀u w, ⑀ϭ , example illustrated by the work of Kaneta 32 . The agreement c L between the computed film shapes and the measured profiles is good, as was for example shown by Venner and Lubrecht ͓33͔, for x!Lx, (1) the particular case of an artificial ridge moving through the con- tact. However, also some local discrepancies were observed that z!Hz, ͓ ͔ could not be explained. Venner and Lubrecht 33 suggest that ␩ ͑ cucL these differences could be an indication of limitations to the lo- p! p. cal͒ validity of the Reynolds equation. H2 Most likely because of the additional dimension and the asso- ciated complications related to the analysis and numerical solution In terms of these dimensionless variables the Stokes equations are only few studies exist in which the Stokes problem or Stokes given by: 2uץ 2uץ pץ roughness is considered. Some of the earliest examples are the ͓ ͔ Ϫ⑀2 Ϫ ϭ (2) ,0 2 ץ 2 ץ ץ works of Sun and Chen 34 and of a later date the paper of Phan-Thien ͓35͔ using the statistical approach. Stokes roughness x x z 2wץ 2wץ pץ -effects were also the subject of the works of Myllerup and Ham rock ͓36,37͔ but here a perturbation approach was used rather than Ϫ⑀4 Ϫ⑀2 ϭ (3) ,0 2 ץ 2 ץ ץ the full Stokes equations. Only a few studies have appeared in z x z which a lubrication ͑related͒ problem described with the Stokes wץ uץ equations is solved numerically. One example is the work of ϩ ϭ0. (4) zץ xץ Noordmans ͓38͔ where an algorithm was developed to solve the Stokes equations on a highly anisotropic-rectangular domain. The system formed by Eqs. ͑2͒, ͑3͒, and ͑4͒ is of elliptic type and ¨ ͓ ͔ Schafer et al. 39 solved the EHL problem but they used modi- will henceforth be referred to as the ‘‘Stokes model.’’ fied Stokes equations in which some viscous terms were omitted For ⑀ϭO(1) all terms in the equations are of equal importance to describe the fluid flow. They showed that especially when slid- for the solution so the full system must be solved. In lubrication ing occurs the Reynolds solution differs significantly from the ⑀ ͓ ͔ problems is usually small.
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