A Quantitative Determination of Minimum Film Thickness in Elastohydrodynamic Circular Contacts

The current work presents a quantitative approach for the prediction of minimum film thickness in elastohydrodynamic-lubricated (EHL) circular contacts. In contrast to central film thickness, minimum film thickness can be hard to accurately measure, and it is usually poorly estimated by classical film thickness formulae. For this, an advanced finite element-based numerical model is used to quantify variations of the central-to-minimum film thickness ratio with operating conditions, under isothermal Newtonian pure-rolling conditions. An ensuing analytical expression is then derived and compared to classical film thickness formulae and to more recent similar expressions. The comparisons confirmed the inability of the former to predict the minimum film thickness, and the limitations of the latter, which tend to overestimate the ratio of central-to-minimum film thickness. The proposed approach is validated against numerical results as well as experimental data from the literature, revealing an excellent agreement with both. This framework can be used to predict minimum film thickness in circular elastohydrodynamic contacts from knowledge of central film thickness, which can be either accurately measured or rather well estimated using classical film thickness formulae.


Abbreviations a
Dry or Hertzian contact radius (m) A 1 , A 2 Parameters in the modified Yasutomi-WLF model b 1 , b 2 Parameters in the modified Yasutomi-WLF model C 1  Glass transition temperature at ambient pressure (°C) u e Mean entrainment velocity (m/s) = (u 1 + u 2 )∕2 u 1 , u 2 Velocity in the x -direction of surfaces 1 and 2 (m/s) U Dimensionless speed parameter (

Introduction and background
Minimum film thickness in circular elastohydrodynamic (EHD) contacts has been significantly less studied than central film thickness, and the mechanisms behind the former remain certainly less understood. Experimentally, rational explanations can be found for this contrasting situation. The first techniques developed to measure film thickness in elastohydrodynamic lubrication or EHL (voltage discharge, electrical resistance, capacitance, X-rays, see [1] for more detail), have mostly allowed a global or mean measurement of the gap between moving solids, i.e. determined over the whole contact area, and then considered to be representative of the film thickness at its center.
It was not until the emergence of methods that are based on optical interferometry that it became possible to quantify the minimum film thickness more accurately. Availability of more sensitive CCD sensors with better spatial resolution enhanced minimum film thickness probing, leading to more quantitative measurements. This last point is important because, unlike central film thickness, which is representative of a relatively large surface, the minimum thickness area, which occurs on the so-called side lobes, is very small, sometimes reduced to a few µm 2 . This can make both its location and measurement uncertain, or subject to approximations. This situation can take place in the case of highly loaded contacts or in the very thin-film (VTF) regime when the thickness of the lubricating film reaches molecular dimensions [2][3][4][5]. In addition, operation in the VTF regime is becoming more and more frequent, due to the need for reducing the environmental impact of systems (e.g., less viscous lubricants leading to lower friction losses, but also to an overall reduction in film thickness) and due to the increasing severity of operating conditions (e.g., higher loads, increased working temperatures). Although interferometric techniques have been widely used for at least two decades now, it is clear that measurements reported in the literature are overwhelmingly concerned with film thicknesses at the contact center, and often ignore the minimum film thickness.
In terms of prediction, expressions have been available for more than 4 decades now. They are based on several operating parameters, usually bundled in dimensionless groups. They aim at predicting central and minimum film thicknesses in circular EHD contacts; sometimes in elliptical contacts as well. Surprisingly, it is not possible to confirm that all these expressions predict identical or very close predictions.
There is, however, relatively little work that has allowed an objective assessment of these EHD formulas. At most, one can cite Koye and Winer [6] who compared their experimental results obtained on point contacts to minimum film thickness predictions by Hamrock and Dowson [7], and who found a qualitative agreement only, with rather large deviations at high contact pressures. For highly loaded circular contacts, Smeeth and Spikes [8] have obtained quite substantial differences between measurements of central and minimum film thicknesses and predictions by Hamrock and Dowson [7], especially at high contact pressure, where their measurements agree with predictions by Venner [9]. On the basis of a wide set of measurements, Chaomleffel et al. [10] evidenced an overall poor agreement with the minimum thickness predictions of Hamrock and Dowson [7] and, on the other hand, a more satisfactory agreement with a model combining the approaches of Nijenbanning et al. [11] with that of Chevalier [12]. Finally, van Leeuwen [13,14] who, in reality, was trying to derive an appropriate value of the lubricant's pressure viscosity coefficient by combining film thickness measurements with different classical expressions, indirectly showed the relative inaccuracies that the latter could generate.
In all of the above-mentioned works, rather significant discrepancies between measurements and predictions have been noted, mainly concerning minimum film thicknesses. Wheeler et al. [15] highlighted some of these discrepancies by comparing their numerical predictions to the results obtained by five models: Hamrock and Dowson [7], Nijenbanning et al. and Chevalier [11,12], Evans and Snidle [16], Chittenden et al. [17] and Masjedi and Khonsari [18]. If on average Wheeler et al. found a relative deviation of about 10% on the central thickness, the deviation increases up to 40% for the minimum film thickness, with extreme values exceeding 90%. To make things worse, all these deviations are of a non-conservative nature, as they overestimate film thickness (both central and minimum values).
As for EHL numerical simulations, the solution of a contact case should be extracted after a full convergence of the corresponding numerical algorithms is attained, and grid-independence is verified. However, this full convergence is not easy to achieve, especially when the contact is subjected to a high normal load and/or in the presence of very thin films. Under these conditions, the deformation of the elastic bodies becomes very large compared to the thickness of the lubricating film, which implies that calculations must be carried out with the highest degree of caution to ensure accuracy and reliability of the results. In addition, the almost exponential increase of viscosity with pressure generates a very strong non-linearity in the Reynolds equation. All this can explain why, in the literature, there are very few papers only that have reported validated results found under these conditions, compared to the countless film thickness values obtained at moderate pressure, typically 0.5 GPa from steel/glass contacts. However, one can find works that have shown that controversies have erroneously arisen when the above precautions were not respected. For example, this is the case with the publications of Morales-Espejel et al.,  who collectively proved-experimentally and numerically-that film thickness decreases at a constant slope (on a log-log scale) when the entrainment speed is decreased. This has been verified even at high normal load or/and very low film thickness, any other tendency being only the consequence of errors, mainly due to inadequate mesh density in the case of numerical simulations.
Why minimum film thickness? Minimum film thickness is of primary interest to assess whether or not separation of the contacting surfaces is ensured. This has a crucial importance on the useful life of lubricated mechanisms. If separation is large enough, service life is likely to be long and only limited, for instance, by surface fatigue. If surfaces are not adequately separated, they interact by direct contacts, which result in shorter component life, or even failure. In summary, minimum film thickness is helpful in defining the frontier between full-film, long term lubrication and other regimes in which direct interactions (i.e. contacts) between surfaces may occur. In that case, direct contact between the surfaces is followed by the appearance of damages such as spalling fatigue, smearing, pitting, scuffing, wear, etc.
It is therefore of primary importance to be capable of predicting and/or measuring minimum film thickness, or the ratio of central-to-minimum film thickness with confidence. If validated over a wide range of operating conditions (including those that characterize heavily loaded EHD contacts or the VTF regime), this ratio will constitute a powerful mean, not only for correctly predicting the minimum film thickness (from knowledge of the central one), but also for determining the lubrication regime. Note that at the current stage of knowledge on this topic, it is imperative to carry out both approaches (i.e. prediction of the minimum film thickness directly, or from knowledge of the central one, using the central-to-minimum film thickness ratio) at the same time, or to compare them to each other, to ensure their coherence and the validity of the results.
The scope of this work is the quantitative prediction of the minimum film thickness in circular EHD contacts. This is the third part of a series (after [22,23]), which aims at clarifying or recalling some fundamental aspects of EHL, in order to stimulate the correct use of the tools, results or models that are available to date. This approach is intended to make EHL a rigorous and quantitative field of tribology, based on independently validated concepts and data, in opposition to the classical EHL approach, which most often leads to misinterpretations, or results that cannot be extended or generalized.

Brief Description of the Numerical Model
Quantitative minimum film thickness predictions are carried out in this work using the full-system finite element approach [24]. It consists of a simultaneous resolution of all governing equations of the isothermal Newtonian EHL circular contact problem. First, Reynolds equation which governs the hydrodynamic pressure build-up within the lubricating conjunction is given by: This equation is applied to a two-dimensional contact domain, with zero-pressure boundary conditions. The nonphysical negative pressures that arise in the outlet domain of the contact are forced towards zero through a mass-conservative penalty method [25]. Such negative pressures cannot be tolerated by the lubricant, which will cavitate, leading to a break-up of the lubricating film. Lubricant density and viscosity are highly dependent on pressure, according to mathematical models that are described in detail in Sect. 2.2 for the selected lubricant. The lubricant film thickness h for a circular contact, describing the geometry of the fluid conjunction separating the solid surfaces is given by: where h 0 corresponds to the rigid body separation, the middle term describes the geometry of the non-deformed conjunction, and is the combined normal surface deformation of the contacting solids, under the influence of the hydrodynamic pressure generated within the lubricant film. This deformation is obtained by applying the classical linear elasticity equations to a cubic three-dimensional The mechanical properties of this domain combine those of the two contacting solids, such that it would accommodate the total elastic deformation of both. The dimensions of the solid domain are taken to be sufficiently large to ensure a half-space configuration. All boundaries of this domain are subjected to a free displacement boundary condition, except for the contact domain on the surface of the solid, where a normal pressure boundary condition is applied (using the pressure field obtained from the solution of Reynolds equation) and the boundary within the depth of the solid (on the opposite side of the contact domain) which is subjected to a zero-displacement boundary condition. Finally, the value of the rigid body separation term h 0 is obtained by introducing a load balance equation, which balances the contact external applied load F with the integral of the pressure field generated over the contact domain Ω c : This equation is added (as a simple integral equation) to the overall system of algebraic equations formed by the discretized Reynolds and linear elasticity equations, while considering h 0 as an additional unknown. All equations are discretized using quadratic finite elements (2 nd order) and they are solved together in a monolithic system. Stabilized finite element formulations [26] are employed for the solution of highly loaded contacts. Given the highly non-linear nature of the arising system of equations, a damped-Newton [27] procedure is employed in its resolution. This iterative procedure is pursued until the normalized L-2 norm (with respect to the problem size, i.e. total number of unknowns) of the overall solution increment vector falls below 10 -6 . All numerical tests were carried out on a sufficiently fine mesh to guarantee grid-independent results, and symmetry of the problem with respect to the xz-plane passing through the contact center is taken into consideration (by applying symmetry boundary conditions to it, for both the Reynolds and linear elasticity equations) to reduce the computational overhead. The mesh employed throughout the numerical tests is the "Fine" mesh case of reference [28]. It consists of a nonregular tetrahedral meshing of the 3D solid domain, which triangular projection on the contact domain serves as the mesh for the latter. The mesh size is taken to be the smallest in the vicinity of the central part of the contact domain (i.e. Hertzian contact), and it is gradually increased when moving away from it in all directions. The total number of tetrahedral elements in the solid domain is 41,556 with a total of 67,743 nodes, which means that the total number of unknowns for the elastic part is 3 × 67,743 = 203,229 (since each node has 3 degrees of freedom, i.e. the x-, yand z-components of the elastic deformation field). The total number of triangular elements in the contact domain is 10,697 with a total of 21,546 nodes, which means that the total number of unknowns for the hydrodynamic part is 21,546 (since each node only has 1 pressure degree of freedom). Therefore, the overall number of unknowns for the entire model is 203,229 + 21,546 + 1 = 224,776 with the last additional unknown corresponding to h 0 . For further details about the FEM model, the interested reader is referred to [24] or [28].

Conditions
The considered configuration is that of a steel-steel ballon-disk contact under pure-rolling conditions, with a ball radius of 12.7 mm. An isothermal assumption was adopted here to provide a reliable basis for minimum film thickness predictions, under EHL conditions that are at the same time conventional and fairly easy to control experimentally. Thermal effects could be included in a future work, but experimental validation could be difficult to achieve as each setup has its own thermal characteristics. Perfectly smooth surfaces are considered and Newtonian lubricant response is assumed, with an ambient temperature T 0 = 30 • C = 303 K . The contact load and mean entrainment speed are varied to cover a wide range of the Moes [29] dimensionless parameters M and L , covering the gr id M = 10, 30, 50, 100, 300, 500, 1000, 2000 and 3000 andL = 1, 2, 3, 5, 7, 10, 15 and 20 . The lubricant selected for the analysis is a well-characterized mineral oil, known under the commercial name "Shell T9". Viscosity-pressure and density-pressure models had to be chosen to perform the simulations. Concerning the former, it should be reminded that film thickness is generated in a zone of rather mild pressures, which leads to a very low influence of the chosen model. The influence of the density-pressure model is certainly more pronounced at high contact pressure and/ or temperature. This has been known for a while now and under such extreme conditions, correction factors have been introduced [30,31]. The choice of the lubricant "Shell T9" allows to perform simulations under conditions that are representative of the standard compressibility of lubricants (i.e. neither very compressible, nor little compressible). Its density-pressure dependence may be accurately represented by the Murnaghan [32]

equation of state:
The values for the different parameters of the Murnaghan equation of state for this fluid are [33]: K � 0 = 10.545 , K 00 = 9.234 GPa , K = 6.09 × 10 −3 K −1 and 0 = 872 kg∕m 3 . As for its viscosity-pressure dependence, it is accurately represented by the modified Yasutomi-WLF model [34]: In the above relation, T g is the glass transition temperature at a given pressure, with T g0 being its ambientpressure value. The values for the different parameters of the modified Yasutomi-WLF model for Shell T9 are [33]: .47 • C and g = 10 12 Pa ⋅ s . This results in an ambient-pressure viscosity value 0 = 12.50 mPa ⋅ s and a reciprocal asymptotic isoviscous pressure coefficient [35], * = 21.21 GPa −1 . This latter value is used as the pressure-viscosity coefficient for evaluating the Moes parameter L.

Results
Since h c , the central film thickness can be accurately measured, or even estimated using several available models, and given the historic failure of film thickness classical formulae in accurately predicting minimum film thickness, most results are presented here in the form of a ratio of central-to-minimum film thickness h c ∕h m . Furthermore, since results are easier to present and discuss as a function of two dimensionless parameters (instead of three for those of Hamrock and Dowson), the two Moes parameters, M and L , are employed in the following to express and represent variations of the ratio h c ∕h m .
The dimensionless values (according to Moes [29]) of the central and minimum thicknesses, H c and H m , are plotted in Fig. 1  In addition, it is interesting to consider the variations of the central-to-minimum thickness ratio because (i) it has the advantage of being independent of the non-dimensioning convention ( h c ∕h m = H c ∕H m ) and (ii) it provides a quick way of correctly estimating the minimum thickness, from knowledge of a central thickness value that is easier to obtain experimentally or analytically, using film thickness formulae. Variations of this ratio for the considered values of M and L are given in Table 1 and plotted as a 3-D curve in Fig. 2.

Analytical Expression for h c ∕h m
To provide a reliable means of estimating h m from a knowledge of h c , the results in Table 1

Previous Works and Expressions
A large number of expressions predicting central and minimum film thicknesses in EHD circular contacts have been published in the literature; some more widely used than others; some more reliable than others, without any correlation between their usage and accuracy. The domains over which they have been established (described here by the parameters M and L ) are quite different from one fomula to another. They will be mentioned to point out the limitations of the formulas and to underline that, under many circumstances, these models have been used far beyond their domain of validity, leading to widely extrapolated film thickness values with no guarantee of their accuracy.
Three classical models are considered here: -Hamrock and Dowson [7], which are certainly the most used film thickness formulas in the literature, initially estab-  Note that the above-mentioned film thickness formulas have been originally derived for the general case of a point contact (circular or elliptical), and they have been adjusted here to only consider the particular case of a circular contact (i.e. unit ellipticity ratio).
In addition, three further contributions that directly report the variations of the h c ∕h m ratio are included: where film is defined in [37] as:  Chittenden et al. [17] Masjedi and Khonsari [18] There are still significant disparities between the above three models for h c ∕h m . First, in the central and minimum film thickness formulas of Chittenden et al., the dimensionless parameters U, G and W have the same exponents. Consequently, the ratio h c ∕h m remains constant and equal to 1.697 (for circular contacts). For the Hamrock and Dowson model, the exponents assigned to U, G and W for estimating h c ∕h m are very low, leading to weak variations of the film thickness ratio. It is somehow really surprising to face such trends, which do not reflect the countless film thickness experimental results published for at least 4 decades. The latter have clearly shown that, when the entrainment speed decreases over a sufficiently large range, the minimum film thickness reduction is significantly greater than the reduction (15) in central film thickness. Therefore, the ratio h c ∕h m should increase substantially. Numerical simulations of the EHD contact have obviously confirmed this observation. Figure 3 of Venner's publication [21] is one example among many. The author reported power law regressions that allow quantification (for the selected two fluids and operating conditions) of the variation of h c ∕h m versus u e (and thus, versus U ) by means of a power law with an exponent of − 0.20 ± 0.01.
Finally, the exponents in the Masjedi and Khonsari model are larger and should give larger and thus, more realistic variations in the h c ∕h m ratio than the two previous models.

Comparisons with Previous Models
The comparison of the six models and tables is presented in two formats. On the one hand, in Fig. 3 where M varies from 10 to 3000 and L takes the values 2, 5, 10 and 20, when these values do not exceed the domains over which the models were established. Otherwise, the curves are limited to the boundaries of the domains over which the models were originally based. On the other hand, Table 2  The four graphs in Fig. 3 show a rather wide dispersion of the values given by the three classical models and the three numerical ones. Apart from 40 ≤ M ≤ 100 , which is not a Fig. 3 Variations of the h c ∕h m ratio versus M for L = 2, L = 5 (top row), L = 10, L = 20 (bottom row) for 3 film thickness models (Hamrock and Dowson [7], Chittenden et al. [17], Masjedi and Khonsari [18]), the approaches of Chevalier [12], Sperka et al. [36] and this work (plotted with dotted lines) domain of great interest in this work, substantial differences are visible between the two approaches. Analytical expressions-whether recent or old, widely used (like Hamrock and Dowson) or not-are unable to capture the significant increase in the h c ∕h m ratio when M increases. Deviations are particularly large for M = 1000 (see Table 2) and they are even more pronounced for M = 3000 (see Fig. 3), where the approach adopted in this work indicates a ratio of the order of ≈3.8 for the most severe cases ( L = 2 or 5 ), while the classical models predict a ratio of the order of 1.7 to 2. This occurs in the case of highly loaded EHD contacts or very thin film thicknesses. These are typical cases where the determination of h m is most critical in determining whether or not a full lubricating film could be generated, and an over prediction could lead to significant surface damage.
An additional comparison with numerical results is shown in Fig. 4. Venner published results that were obtained over large M and L ranges, and down to very small film thickness values [21]. Furthermore, he showed that both h c and h m varied as power functions of u e , the entrainment speed; making it easy to express the h c ∕h m ratio as a power function of this parameter. However, in his simulations, Venner used the Roelands pressure-viscosity equation and the Dowson and Higginson compressibility formula. In theory, these two models differ from those described earlier in this work. However, the Hertzian pressure chosen by Venner is not too large (1.5 GPa) and the difference between Dowson and Higginson and Murnagham predictions is not really prohibitive. Moreover, Venner simulated two lubricants, one of which is a paraffinic base oil whose piezoviscosity is described in a rather close way by both the Roelands and modified Yasutomi-WLF models in the low-to-medium pressure domain, which is critical for the generation of EHD film thickness. For this latter lubricant, the film thickness ratio varies as follows according to Venner [21]: (18) h c ∕h m = 1.7711u e −0.1865 Although using different physical and rheological equations, and despite an extension of the analytical formula of Sect. 2.4 beyond its domain of application (for u e < 0.01m∕s ), the agreement with Venner's results is excellent. The average relative deviation between the two models for the cases plotted in Fig. 4 is slightly less than 1%. This comparison really covers cases with very high M and low L (both values are given in Fig. 4): M reaches almost 7500 and L = 5 for the lowest entrainment velocity, u e = 0.005 m/s.

Comparison and Validation Against Experiments
Experimental validation is certainly the most reliable way to validate results from numerical simulations. Experimental results from references [5] and [38] were utilized for this purpose. These were obtained from measurements that were performed twice for each test case. In addition, for each of the test cases, 3 to 5 interferograms were considered to enhance the reliability of the results. A first series of experiments was performed on a steel ball/sapphire disk contact [38], under pure-rolling conditions with smooth surfaces at T = 50 • C . The contact was lubricated with a well-characterized lubricant, 5P4E ( 0 = 160 mPa s and * = 28.4 GPa −1 at 50 °C). Tests were conducted at variable entrainment speed, for different applied normal loads, leading to Hertzian pressures ranging from 0.76 to 1.8 GPa. These operating conditions lead to values of L varying from 14 to 23 when u e varies from 0.1 to 0.63 m/s, regardless of the Hertzian pressure.
In   [17], Masjedi and Khonsari [18]), the numerical approaches of Chevalier [12], Sperka et al. [36] and this work.  These comparisons with experiments are instrumental because they validate the numerical approach developed in this work. Nevertheless, they remain limited to high, but not extreme, values of M and to rather large values of L . A second series of comparisons between experiments and simulations will allow for checking the validity of the model under more extreme and representative cases.
Experimental results were taken from Ref. [5], where a hydrocracked mineral base oil (Yubase 4) was proven to generate film thickness down to a few nanometers without deviating from a linear dependence on the entrainment speed (on a log-log scale); thus following a power law with a constant exponent. Tests were performed on a (Si0 2 -coated or uncoated, for h > 90nm only) glass disk in contact with a steel ball under pure-rolling conditions and smooth surfaces at p H = 0.5 GPa. They were conducted at T = 25 • C ( 0 = 28 mPa s and * = 16.6 GPa −1 at 25 °C) for different entrainment speeds, ranging from 1 m/s down to 0.024 m/s. As in Fig. 5, experimental results are plotted in Fig. 6 with black symbols. h c exp is compared with the Chittenden predictions, which gave the closest estimation of the central film thickness [5]. As for h m , different scenarios were considered: -The Chevalier h c ∕h m ratio was applied to the h c prediction by the Chittenden equation [17] (noted "hm Chitt.-Chev." in Fig. 6). Although this approach yields a good agreement in the middle-to-high entrainment speed domain, it leads to the largest underestimation of the experimental minimum film thickness for the lower u e values, i.e. for the more critical cases. -The Chevalier h c ∕h m ratio was applied to the h c prediction by the Nijenbanning, Venner and Moes [11] equation (noted "hm Moes-Chev." in Fig. 6). In agreement with previous works (see [10] for instance), this scenario yields, on average, a rather correct prediction of h m . But when looking into details, one can notice an overestimation of h m at high entrainment speeds and an underestimation at low speeds. Thus, the predicted slope of h m = f (u e ) does not quantitatively agree with the experimental one. -This work h c ∕h m ratio was applied to the h c prediction by the Chittenden expression (noted "hm Chitt.-This work" in Fig. 6). In this case, minimum film thicknesses at medium-to-high entrainment speeds are in very good agreement with experimental values, and those at low speeds are slightly underestimated. -Finally, the experimental central film thicknesses were divided by the h c ∕h m ratios from this work and noted "hm hc exp-This work" in Fig. 6. This hybrid combination of experimental results and numerical ones leads to the best agreement with h m measurements, regardless of the entrainment velocity.
This last scenario is of particular interest. Not only does the method allow a reliable prediction of h m , but it circumvents its measurement or direct estimation by means of an analytical expression, both of which can be relatively inaccurate. At low entrainment speeds, M becomes greater than 1000 (as highlighted in Fig. 6) and additionally, L takes very low values (here L < 2 when u e = 0.024 m/s). This reveals the robustness of the proposed approach, even in the very thin-film (VTF) regime.

Summary
This work presents a quantitative analysis of the prediction of minimum film thickness in elastohydrodynamic circular contacts. In contrast to central film thickness, minimum film thickness can be hard to accurately measure and it is poorly predicted by classical EHL formulas. A particular focus has been attributed to conditions that are nowadays very common in lubricated mechanisms, and which lead to highly loaded contacts or/and low or very low film thicknesses. An advanced numerical model was used to probe the variations of the central-to-minimum film thickness ratio as a function of the dimensionless parameters M and L , and an analytical expression was derived.
The latter was compared to the classical film thickness formulas and to more recent similar expressions. Comparisons confirmed the inability of the former to predict minimum film thickness, and the limitations of the latter, which tend to overestimate the h c ∕h m ratio, and thus to underestimate h m from knowledge of h c . In addition, an excellent agreement was found with simulations published by Venner.
The validation of this work was achieved by comparisons with experimental results, with a focus on highly loaded and very thin-film (VTF) cases. The analytical model from this work has proved its capability to accurately predict the h c ∕h m ratio under such conditions.
In conclusion, a reliable and validated method is proposed to predict the minimum film thickness in circular EHD contacts, from knowledge of central film thickness, which can be either accurately measured or rather well estimated using classical film thickness formulae. The proposed approach can be of particular interest to experimental researchers, or also design engineers wishing to assess the actual operating conditions of a lubricated contact from the macroscopic parameters.

Appendix A: Dimensionless Central and Minimum Film Thicknesses
See Tables 3 and 4. See Table 5.

Availability of Data and Material (Data Transparency)
The authors declare that all data supporting the findings of this study are available within the article.

Conflict of interest
The authors declare that they have no conflict of interests.