About the Inverse Theory and the Non-Unicity of the Film Thickness: Novel Approach Generating Equivalent Micro-Grooves and Roughness


 Since the 1960s, all studies have assumed that a film thickness “h” provides a unique pressure field “p” by resolving the Reynolds equation. However, it is relevant to investigate the film thickness unicity under a given hydrodynamic pressure within the inverse theory.
This paper presents a new approach to deduce from an initial film thickness a widespread number of thicknesses providing the same hydrodynamic pressure under a specific condition of gradient pressure. For this purpose, three steps were presented: 1) computing the hydrodynamic pressure from an initial film thickness by resolving the Reynolds equation with Gümbel’s cavitation model, 2) using a new algorithm to generate a second film thickness, 3) comparing and validating the hydrodynamic pressure produced by both thicknesses with the modified Reynolds equation.
Throughout three surface finishes: the macro-shaped, micro-textured, and rough surfaces, it has been demonstrated that under a specific hydrodynamic pressure gradient, several film thicknesses could generate the same pressure field with a slight difference by considering cavitation. Besides, this paper confirms also that different ratios of the averaged film thickness by the root mean square (RMS) similar hydrodynamic pressure could be generated, thereby the deficiency of this ratio to define the lubrication regime as commonly known with Patir and Cheng theory.


February 09th, 2021
Dear Editor, I wish to submit a new manuscript entitled "About the Inverse theory and the nonunicity of the film thickness: Novel approach generating equivalent micro-grooves and roughness" for consideration by the Tribology Letters.
I confirm that this work is original and has not been published elsewhere nor is it currently under consideration for publication elsewhere.
In this paper, throughout three surface finishes: the macro-shaped, micro-textured, and rough surfaces, it has been demonstrated that under a specific hydrodynamic pressure gradient, several film thicknesses could generate the same pressure field with a slight difference by considering cavitation. This is significant to confirm that different ratios of the averaged film thickness by the root mean square (RMS) similar hydrodynamic pressure could be generated, thereby the deficiency of this ratio to define the lubrication regime as commonly known with Patir and Cheng theory. The paper should be of interest to readers in the areas of lubricated contact based on the Inverse approach.
Please address all correspondence concerning this manuscript to me at "m.elgadari@ensam-umi.ac.ma." Thank you for your consideration of this manuscript.

Introduction
The differential equation that governs the generation of pressure in lubricating films is known as the Reynolds equation. In steady-state condition, this equation is given for one dimensional, isoviscous, and incompressible hydrodynamic lubrication problem by: Where: "x" the axial direction, "U" the velocity of the lower surface, and "µ" the lubricant viscosity.
To design the lubricated components (bearings, thrust, seals ...) the equation (1) is solved by using two theories: -The first is performed to dimension the supporting and/or guiding devices [1,2,3], so-called the direct approach. For a given film thickness, "h(x)" the pressure distribution, "p(x)" within the fluid film is obtained by solving Reynolds equation (1).
-The second is the inverse method that is widely used for elastomeric seals. Indeed, according to this theory, the static contact pressure "ps(x)" is assumed equal to the hydrodynamic pressure "p(x)" and the film thickness "h(x)" is computed by using the first and second derivatives of "p(x)" [4,5,6] under non-cavitation conditions. This second method is based on the Reynolds equation and two computational procedures are possible (2) and (3): Usually, equation (2) is the most used in the inverse hydrodynamic lubrication [6] to determine "h(x)" where "h * " is the film thickness at the location of the maximum pressure. The second approach is rarely used and is given by equation (3) with first and second derivative pressures "dp/dx" and "d 2 p/dx 2 ".
Additionally, when the surface roughness is excessive, two approaches are used to resolve the Reynolds equation: -Stochastic method, so-called Patir and Cheng flow factors method [7] by solving the transformed (or averaged) Reynolds equation for micro bearings having a realistic surface roughness.
-Deterministic method by assuming the rough surface as an analytical function.
In most cases, it is given by sinusoidal form [8].
The proposed method describes a new approach to find different geometries, microgrooves, and roughnesses of the lubricated contact that keep the same operational parameters: friction force F flow rate Q and lifting force W.
To consider the cavitation effect, the most used methods are: Method 1: Since 1921, Gümbel [11] has proposed simply to neglect pressures less than atmospheric pressure. Thus, to compute the hydrodynamic pressure in a full film lubricated contact, Reynolds equation (1) is resolved with the cavitation condition: -Method 2: The following modified Reynolds equation [12] is resolved: where ϕ the cavitation index equal to 0 in the cavitation zone when D≤ 0 and equal to 1 in the active zone when D>0.
By considering the replenishment r, where  and 0 are the density of cavitated zone and lubricant respectively: We admit also: Before beginning the numerical analysis, the reader must keep in mind the processing steps: -First analysis: With an initial film thickness "H1(x)", the Reynolds equation (1) is resolved, and the hydrodynamic pressure "p(x)" with cavitation effect is searched according to Gümbel [11] condition.
-Second analysis: Based on the hydrodynamic pressure "p(x)" of the first analysis an original approach is proposed to compute all the different film thicknesses "H2(x)".
-Third analysis: By replacing the two film thicknesses "H1(x)" and "H2(x)" in the modified Reynolds equation (6), the hydrodynamic pressure for each film thickness is computed to compare and discuss the accuracy of the approach.
2 Numerical approach

Computing the hydrodynamic pressure
The proposed approach consists of two depending on steps: -Step 1: By using an appropriate numerical method (finite differences, finite volumes, and finite elements, analytic ...), the Reynolds equation (1) corresponding to the first analysis and modified Reynolds equation (6) for the third analysis is resolved. Thus, according to the computed pressure p(x), the lifting force W the friction force F, and the flow rate Q are deduced from equations (4).
In this initial step, the film thickness h(x) is assumed known and equal to the guessed initial thickness H1(x).
Equations (1) and (6) are discretized with the classical finite volumes method briefly described in figure 1. Thus the discretized equations are given by: Where for the Reynolds equation (1) with Gümbel [11] condition for cavitation effect: And for the modified Reynolds equation (6): To validate the numerical model used in our Hydrodynamic lubrication simulations, we refer to the work of Fowell et al. [9,10] that studied the effects of textured surfaces. The analytical formulas proposed in this work allow simple but at the same time precise calculation of the pressure distribution across a textured surface. Cavitation zones are also located in the contact using a formulation based on mass conservation. The geometry proposed by Fowell [9] in figure 2 consists of a slot located on the stationary surface of a parallel faced pad. Thereby, in the divergent-convergent zone with the operating conditions (  Table 1. Geometry and operating conditions of the textured pad [9]. The boundary conditions are given by: at "x=0", "p(0)=p0" and at "x=B", "p(B)= p0". Comparing simulations of the present model and the analytical results [9], figure 3 confirms a good agreement between the pressure distributions. However, the figure shows that the difference is mainly caused by the discontinuous domain of the film thickness h(x). Indeed, the large mesh size of the geometry impacts the derivative accuracy of the function h(x).

Computing different film thickness distributions
A single and unique film thickness investigation can now be initiated to generate the same hydrodynamic pressure given by the Reynolds equation solution. Let us assume at the beginning, the existence of two roots for the equation (2), H1 and H2. Thus the equation (2) becomes: where: H1(x) is equal to the film thickness h(x) initially given.
In this section, we aim to check if H2(x) is equal to H1(x) whatever x between 0 and L.
By subtracting equations (10.1) and (10.2), we find: Thus the equivalent equation to resolve with H2 as unknown thickness is given by: To find the roots of equation (12), the algorithm below is used for each x between 0 and L: else: Equation (13) leads to the following condition regarding the unicity of the film thickness: dp dx ≥ 8μU H 1 2 or dp dx ≤ 0 for each "x" between "0" and "L" It is important to note that, when the condition (15) is not verified, all the roots film thickness H2 could be combined to the initial film thickness H1, and generate a new film thickness h(x) with the equation: where: α is an entire coefficient of geometry depending on the coordinate x and is equal to 1 or 0, and H1(x) the initial film thickness and H2(x) the roots film thickness.
Engineers could choose the parameter ( ) to define a new surface generating the same hydrodynamic pressure. Later in this paper, an investigation is performed to demonstrate how far the pressure is similar by considering the cavitation effect.

Different bearing geometry: without cavitation effect
To explain carefully this new theory, an example of a commonly known bearing "Sloping surface" is studied as described in figure 4. The guessed initial film thickness, in this case, is given by the following equation: By assuming the following data: L=20mm, hb=0.008mm, ha=0.002mm, μ=0.01 Pa.s, U=200mm/s. With those operational conditions, the current bearing geometry generates the hydrodynamic pressure shown in figure 5. A comparison between the finite difference method used in the previous section and the analytical result was performed and the numerical algorithm was easily validated. Figure 6 proposes the different configurations that engineers could make among two solutions: H1 and H2 to design an appropriate geometry for the load support, keeping  (16). This approach is a very convenient tool that allows engineers to choose the optimal surface to guide machinery components.  (13) and (14) Zone where the film thickness is not unique according to equation (15) It is relevant to confirm that with all the three bearing surface shape #1, #2, and #3, the same hydrodynamic pressure is generated as shown in figure 8, and the same flow rate as demonstrated in figure 9. It is also important to note that the frictional force difference between the three bearing surfaces is slightly regular. Thus with shape#1, the friction force per length is about 12.98N/m, shape#2 is 13.16N/m, and 13.17N/m for shape#3.
Thus the designers need to perform several iterations by changing geometry to maintain the same lifting force and flow rate and minimizing the viscous friction effect.

Micro-textured surfaces
After confirming the existence of different film thicknesses generating similar hydrodynamic pressure in the case of bearing with macro shaped geometry, this section presents a second application in the lubricated contact by considering micro-textured surfaces.
This part aims to find all micro-textured surfaces providing the same lifting force by considering the phenomena resulting from the convergent-divergent domain, especially the cavitation effect. Figure 10. The micro-textured bearing geometry.  A micro-textured surface is considered with an initial film thickness H1(x) given in figure 10.
This study is performed throughout three steps: 1) computing the hydrodynamic pressure "p(x)" with the guessed initial film thickness "H1(x)" and resolving the Reynolds equation (1), 2) using the equations (13) and (14) to find the roots film thickness H2(x), 3) comparing hydrodynamic pressures "p(x)" corresponding to the film thickness H1(x) and H2(x) with modified Reynolds equation.
Step 1: The hydrodynamic pressure corresponding to the initial film thickness H1(x): According to the discretized equation (6), the Reynolds equation is resolved. Figure 11 shows the hydrodynamic pressure "p(x)" corresponding to the initial film thickness H1(x).
Step 2: The second film thickness roots H2(x): By using the equations (14) and (15), the second film thickness H2(x) is computed. Figure 12 shows the big difference between the first film thickness H1(x) and H2(x). It is important to underline that both film thicknesses generate the same hydrodynamic pressure p(x).

Step 3: Comparing the hydrodynamic pressure for each film thickness
According to numerical results of step 2, figure 13 confirms the similar hydrodynamic pressure provided by differents film thicknesses H1(x) and H2(x). To investigate the cavitation effect on the second film thickness H2(x), this section aims to study microtextured surface generating cavitated zone with L=76mm and d=0mm as described in figure 10. This analysis is performed throughout three steps: 1) resolving the Reynolds equation (1) to find the hydrodynamic pressure "p(x)" with the initial film thickness "H1(x)", 2) using the equations (13) and (14) to compute the second film thickness H2(x), 3) comparing hydrodynamic pressures "p(x)" corresponding to the film thickness H1(x) and H2(x).
Step 1: The hydrodynamic pressure corresponding to the initial film thickness H1(x), with Gümbel [11]

cavitation condition.
Reynolds equation is resolved by considering the Gümbel conditions [11] given in the equation (5). Figure 11 shows the hydrodynamic pressure "p(x)" corresponding to the initial film thickness H1(x), where the existence of the cavitated zones is highlighted.

Figure 13. Hydrodynamic pressure with the different micro-textured surface without cavitation
Step 2: The film thickness roots H2(x): The second film thickness H2(x) is computed from equations (14) and (15) by using the hydrodynamic pressure corresponding to the initial film thickness H1(x). Figure 15 demonstrates the impact of the cavitation on the second film thickness H2(x). Indeed, the initial film thickness H1(x) is nearly equal to the second H2(x) except in the vicinity of crossing cavitated to the non-cavitated zone.
Step 3: Comparing the hydrodynamic pressure for each film thickness (with modified Renolds equation) To compare the hydrodynamic pressure generated with the film thicknesses: the initial H1(x) and the second H2(x), the modified Reynolds equation was resolved of both thicknesses by considering the mass conservation instead of the Gümbel model [11] according to equation (5). Figure 16 confirms a slight difference in the hydrodynamic pressure corresponding to the film thicknesses H1(x) and H2(x), despite the big difference with the hydrodynamic pressure given by the initial film thickness H1(x) computed according to the Gümbel model [11]. Therefore, it is also relevant to underline that the same initial film thickness H1(x) gives two differents hydrodynamic pressures with mass conservation according to modified Reynolds equation (6) and with the Gümbel cavitation model (5).

Effect of separating space film thickness: h0.
It is well known that the gap that separates lubricated surfaces affects significantly the hydrodynamic pressure. Thus, this section aims to prove the effectiveness of such an operational parameter on the second film thickness by considering the cavitation effect. To perform this investigation, the similar micro-textured surface is considered with L=76mm and d=0mm as described in figure 10 and three differents separating gap are designed as follow: Case#1 with h0=5microns, Case#2 with h0=1microns and Case#3 with h0=10micons. Figure 17 shows the impact of the gap between lubricated surfaces. As long as surfaces are close, as the hydrodynamic pressure is high and, the deviation between the hydrodynamic pressure produced by the second thickness is large compared to that generated by the initial thickness. Although the differences do not exceed 2% between pressure, the deviation maximum is located at the cavitation zone crossing to the noncavitated zone. Consequently, the proposed method for generating a second film thickness is applicable for any spacing gap between lubricated surfaces.

Rough surfaces
In general, the roughness surface is the relevant parameter that significantly affects the lubricated contact. Previous works have demonstrated that the arithmetic roughness Ra, the standard deviation σ, the Kurtosis and Skewness parameters Ku, and Sk respectively define the topography surface capacity to characterize tribological behavior of surfaces.
One of the pillar methods to find the hydrodynamic pressure for each rough surfaces is based on the stochastical approach with Patir and Cheng flow factors [7]. This stochastic method defines pressure as a function of the averaged film thicknesses ratio by the standard deviation roughness h0/σ.
The main subject of this section is to prove with different surface topography parameters: Ra, Ku, and Sk could generate the same hydrodynamic pressure. It is also very relevant to demonstrate that with different standard deviation σ the hydrodynamic pressure could be maintained, thus the adimensional film thickness h0/σ is not sufficient to define the Hydrodynamic or Elasto-hydrodynamic regime. Table 3 gives the initial film thickness (respectively the initial surface roughness), and the second film thickness is given by equations (13) and (14), with operational parameters as described in table 1  where: R is the roughness function represented in the realistic case with a randomized signal, and N is the total node number.  Table 3. Initial and second roughnesses corresponding to film thickness H1(x) and H2(x) respectively.  (1) and using Gümbel cavitation model [11].
To compare the accuracy of simulations, the hydrodynamic pressure produced by both film thicknesses H1(x) and H2(x) is computed by the Modified Reynolds equation (6). Figure 18.b confirms that the pressure has slight differences although the significant differences between surface topography, especially the Root Mean Square (RMS) σ.
Thereby, it is important to note that the ratio "h0/σ" is not sufficient to determine the lubrication regime, thereby the deficiency of this ratio to define the lubrication regime as commonly known with Patir and Cheng theory.

Conclusions
In this paper the existence of several film thicknesses that provide the same hydrodynamic pressure by considering the cavitation effect was investigated throughout three depending steps: Step 1: With the operational conditions and an initial film thickness H1(x), the hydrodynamic pressure is computed from the Reynolds equation and using the Gümbel cavitation condition.
Step 2: By using the hydrodynamic pressure of step 1 and under a specific pressure gradient condition, a second film thicknesses H2(x) is found with the presented new algorithm.
Step 3: Comparing the hydrodynamic pressure accuracy of both thicknesses H1(x) and H2(x) computed from the mass conservation equation with the Modified Reynolds equation.
This new approach was applied for different lubricated surfaces: -Macro-shaped surfaces: By considering initial bearing surfaces, several surfaces have been proposed that give the same hydrodynamic pressure. It was also demonstrated that with different bearing macro-shapes the frictional force varies slightly and the flow rate is regular.
-Micro-textured surfaces without cavitation: This case concerns a specific geometry condition of the surfaces that avoid the cavitation effect. By using the presented method, the initial and the second film thicknesses generate the same hydrodynamic pressure computed from the Modified Reynolds equation.
-Micro-textured surfaces with cavitation effect: A specific geometry condition was considered to produce cavitation zones within the lubricated contact. 0.1% is a difference gap between the pressure obtained by resolving the Modified Reynolds equation with both film thicknesses.
-Rough surface: A randomized surface was considered and a second film thickness H2(x) corresponding to a new roughness generating the same hydrodynamic pressure was computed. It was confirmed also that the pressure has a slight difference although the significant differences between the Root Mean Square (RMS) σ.
The effect of the separating gap between the lubricated surfaces was also investigated and the second film thickness was found. The impact of this separation is about 2% by comparing the hydrodynamic pressure for both thicknesses computed from the Modified Reynolds equation.
This paper underlines that the ratio "h0/σ" is not sufficient to determine the lubrication regime as commonly known with Patir and Cheng theory. Additionally, this novel approach gives to engineers a numerical tool for changing geometry by maintaining the same lifting force and flow rate and minimizing the viscous friction effect.
It must be recalled that this method is validated without considering the dry contact phenomenon. This work opens new tracks to be investigated to be able to design lubricated contact with any pressure distribution coupled with the surface flexibility. Tables: Table 1. Parameters adopted for the parametric study Table 2. Geometry and operating conditions of the microtextured surface. Table 3. Initial and second roughnesses corresponding to film thickness H1(x) and H2(x) respectively.