Equilibrium analysis for a mass-conserving model in presence of cavitation

Transcription

1 Equilibrium analysis for a mass-conserving model in presence of cavitation Ionel S. Ciuperca CNRS-UMR 58 Université Lyon, MAPLY,, 696 Villeurbanne Cedex, France. ciuperca@maply.univ-lyon.fr Mohammed Jai CNRS-UMR 58, Insa de Lyon, Bat Léonard de Vinci, 696 Villeurbanne Cedex, France. Mohammed.Jai@insa-lyon.fr J.Ignacio Tello Matemática Aplicada, ETSISI. Universidad Politécnica de Madrid, 83 Madrid, Spain. jtello@eui.upm.es) Abstract We study the existence of equilibrium positions for the load problem in Lubrication Theory. The problem consists of two surfaces in relative motion separated by a small distance filled by a lubricant. The system is described by the modified Reynolds equation (Elrod-Adams model) which describes the behavior of the lubricant and an extra integral equation given the balance of forces. The balance of forces allows to obtain the unknown position of the surfaces, defined with one degree of freedom. Mathematics Subject Classification: Primary 35J5; Secondary 35J7, 34E5. Keywords: Reynolds lubrication equation; Inverse Problems; Weighted Sobolev Spaces; Singular perturbation analysis; Maximum Principle Introduction and problem setting Lubrication is the process used in mechanical systems to carry the load between two surfaces in relative motion and close proximity. The narrow space between the surfaces is filled by the lubricant, its characteristics allow to avoid the direct contact between the surfaces and reduce the wear. In that case, when distance is strictly positive we say that the system is in Hydrodynamic regime. The force induced by the pressure of the fluid is developed by the relative motion of the surfaces and it depends on the geometry of the space filled by the lubricant.

2 Equilibrium analysis for a mass-conserving model For simplicity, we assume that the bottom surface is planar and moves with a constant horizontal translation velocity. The lubricant is assumed incompressible and the distance between the surfaces belongs to the range of admissible distances satisfying the thin-film hypothesis, and therefore the pressure fluid does not depend on the vertical coordinate. Let us denote by the two-dimensional domain in which the hydrodynamical contact occurs. We suppose, for simplicity, that =], [ and the boundary is split into two parts: Γ (defined by {x = }) and the rest of the boundary, denoted by Γ. We also assume that the fluid flux at Γ is a given constant µ >. Without loss of generality we may assume that the velocity of the bottom surface is oriented in the direction of the x - axis and its normalized value is equal to. In order to take into account the cavitation in the fluid we introduce the so-called Elrod-Adams model in the stationary case. The problem consists of finding p (the pressure of the lubricant) and θ (the volume fraction occupied by the fluid), solution of the following system (see for example [9] and []): [h 3 (x) p] = (θh) x θ H(p), p p = x Γ hθ h 3 p x = µ on Γ x where h is the non-dimensional distance (the gap) between the surfaces and H is the Heaviside multivalued function defined by p > H(p) = [, ] p = p <. In many lubricated systems, the position of the surface is unknown and h may present some degrees of freedom. We reduce our study to the case where h will be given up to one degree of freedom which is the vertical translation, which results as a equilibrium position between the hydrodynamic force p(x) dx and the known exterior force F (assumed constant) applied upon the upper surface. Then, we assume that h is defined as follows (.) h(x) = h (x ) + a (.) where a > accounts for the vertical translation and h : [, ] [, [ is a given regular nonnegative function which represents the gap corresponding to a = and defines the geometry of the space. For simplicity we assume that the surface is rigid (i.e. h is independent of the forces applied), depends only on x and is C ([, ]) function. We also suppose that min x [,] h (x ) =, which allows to say that a represents the minimum distance between the two surfaces. We are interested in the equilibrium positions of the system, which are defined as the stationary solutions of the equation defined by the second Newton s Law. Then, for a given constant force F,

3 I. Ciuperca, M. Jai and J.I. Tello 3 the problem consists in finding a > such that, pdx = F (.3) with h is defined in (.) and (p, θ) is a solution of (.). The problem for a general h C () presents a high complexity and non-existence of solutions may occur for particular shapes h and some exterior forces. Therefore we should restrict the study to the case where the contact region in the limit case (a = ) satisfies the following assumptions: There exist γ ], [, α > and m, m two positive constants such that h (γ) = (.4) h (x ) < for x [, γ] (.5) h (x ) > for x [γ, ] (.6) h is convex on [, γ] (.7) m x γ α h (x ) m x γ α for x ], [ (.8) m x γ α h (x ) m x γ α for x ], [ (.9) The weak formulation of problem (.) consists of finding (p, θ) in V + L () such that { h3 p ϕdx = ϕ hθ x dx + Γ µϕdσ ϕ V (.) θ H(p) where V = { ϕ H () : ϕ = on Γ } and V + = {ϕ V : ϕ }. The main result of this paper is the following Theorem.. Under assumptions (.4)-(.9), there exists at least one solution (p, θ, a) V + L () ], + [ to (.),(.) and (.3). Remark. We notice that if a µ, a solution to (.) or (.) is p = and θ(x) = µ h. Then, if (x)+a the external force F is zero there exist infinity many solutions of the coupled problem (.), (.), µ (.3) defined by (, h, a) for any a µ. (x)+a In this paper we focus on the case F > which is more relevant from a physical point of view. Up to our knowledge the equilibrium problem in lubrication with the use of the Elrod-Adams model from a theoretical point of view was not considered before. The existing literature in numerical simulations and applications has been growing in the last forty years, nevertheless a deep mathematical study of the existence of equilibrium positions has not been accomplished. Uniqueness /

4 4 Equilibrium analysis for a mass-conserving model multiplicity of solutions are not studied in the paper, the existing techniques to prove uniqueness of inverse problems can not be applied directly and new ideas should be introduced to obtain such results. There exist in the literature other theoretical studies of equilibrium problems in lubrication using different models as Reynolds equation (see for example [], [4]) or Reynolds inequality (see [5] or [8]). The content of the paper is the following: in Section we consider a regularization of the problem and we prove the existence of at least a solution of this regularized problem, as well as appropriate estimations. In Section 3 we prove the main result by passing to the limit and in Section 4 we give some numerical simulations. The regularized problem. Setting of the regularized problem In this section we study a regularized problem of (.) with h given by (.), obtained by replacing in (.) the Heaviside function H by a regular approximation H ɛ defined by z ɛ z H ɛ (z) = z ɛ ɛ z. The regularized problem is as follows [h 3 (x) p ɛ,a ] = [hhɛ(pɛ,a)] x in, p ɛ,a = in Γ, hh ɛ (p ɛ,a ) h 3 p ɛ,a x = µ on Γ. The variational formulation of the problem consists of finding p ɛ,a V such that h 3 p ɛ,a ϕdx = hh ɛ (p ɛ,a ) ϕ dx + µϕdσ ϕ V. (.) x Γ For any given a > it is well known that the problem (.) has a unique solution (see Bayada- Vázquez []) where the main ideas of the proofs are summarized for the journal bearing problem (see Bayada-Martin-Vázquez [3] and the references therein for more details). Since the proof for the slide is similar to the journal-bearing system, we omit the details. We also have for ɛ (.) p ɛ,a p a weakly in V (.3) H ɛ (p ɛ,a ) θ a weakly star in L () (.4) where (p a, θ a ) is a solution of (.). The weak formulation of the regularized coupled problem (.), (.), (.3) consists of finding p ɛ solution of (.) and a ɛ > such that h 3 ɛ p ɛ ϕdx = h ɛ H ɛ (p ɛ ) ϕ dx + µϕdσ ϕ V (.5) x Γ

5 I. Ciuperca, M. Jai and J.I. Tello 5 p ɛ dx = F (.6) h ɛ (x) = h (x ) + a ɛ. (.7) In order to prove the existence of a solution of the coupled problem (.5), (.6), (.7) we re-write the problem as an scalar problem with only one unknown. To reduce it, we introduce the function g ɛ : a ], + [ g ɛ (a) = where p ɛ,a is the unique solution of (.) with h defined by (.). Then, (.5), (.6), (.7) is described as follows: Find a ɛ > such that p ɛ,a (x)dx g ɛ (a ɛ ) = F. (.8) Since g ɛ is a continuous function, we just need to obtain large and small values of g ɛ and apply the intermediate value theorem to prove the existence of at least one solution of (.8). The most difficult part of the proof is to obtain a large value of g ɛ and it can only be found for a small enough. So, the crucial step of the result is to prove that lim sup g ɛ(a) > F. (.9) a The idea is to bound from below the solution p ɛ by a subsolution of the problem which is large enough when a goes to zero. Definition.. We say that q ɛ V is a subsolution of (.) if it satisfies: h 3 q ɛ ϕdx hh ɛ (q ɛ ) ϕ dx + µϕdσ ϕ V +. (.) x Γ The next lemma gives a comparison principle for the solutions of (.). Lemma.. Let p ɛ be a solution of (.) and q ɛ a subsolution. Then p ɛ q ɛ a.e. on. Proof. We adapt a technique of Gilbarg and Trudinger [] to non-linear problems (see also [6]). We set w = q ɛ p ɛ and our goal is to prove that w + =. Substracting (.) from (.) we obtain h 3 w ϕdx Now for any δ > we take ϕ = w+ w + +δ []. h (H ɛ (q ɛ ) H ɛ (p ɛ )) ϕ dx c ɛ w x ϕ x ϕ V +. and the end of the proof is like in Gilbarg and Trudinger

6 6 Equilibrium analysis for a mass-conserving model. Construction of a subsolution In order to construct a subsolution we split the domain into several subdomains, we consider first the following subdomains: l =], γ[ ], [ and a = ] β, γ a /α[ ] 4, 3 [ 4 where β ], γ[ is a constant to be fixed later and a ], (γ β) α [. Let us denote by R ɛ,a : a R the solution of the Reynolds equation { [h 3 R ɛ,a ] = h x in a R ɛ,a = ɛ in a. (.) It is clear that R ɛ,a = R a + ɛ where R a is the unique solution of { [h 3 R a ] = h x in a R a = in a. (.) Since h x on a. on a and thanks to maximum principle we have that R a, which implies R ɛ,a ɛ In the following lemma we construct a subsolution to (.). Lemma.3. Let ξ ɛ H ( l a ) be such that ξ ɛ = R ɛ,a = ɛ on a (.3) ξ ɛ = on l Γ (.4) (hh ɛ (ξ ɛ )) (h 3 ξ ɛ ) x on l a (.5) hh ɛ (ξ ɛ ) h 3 ξ ɛ x µ on Γ (.6) h 3 R ɛ,a ν h 3 ξ ɛ ν on a (.7) h 3 ξ ɛ x on {x = γ} (.8) where ν denotes the unitary exterior normal to a. Then, q ɛ : R defined by R ɛ,a on a q ɛ = ξ ɛ on l a (.9) on l is a subsolution of (.).

7 I. Ciuperca, M. Jai and J.I. Tello 7 Proof. It is clear that q ɛ is an element of V. For any ϕ V + we have h 3 q ɛ ϕdx hh ɛ (q ɛ ) ϕ dx µϕdσ = h 3 R ɛ,a ϕ+ x Γ a h 3 ξ ɛ ϕ h ϕ hh ɛ (q ɛ ) ϕ µϕdσ l a a x l a x Γ since H ɛ (R ɛ,a ) =. Now we have h 3 R ɛ,a ϕ a a h ϕ x = a h 3 R ɛ,a ν ϕ a hν ϕ (.) and also h 3 ξ ɛ ϕ hh ɛ (ξ ɛ ) ϕ { l a l a } x = [hh ɛ (ξ ɛ )] [h 3 ξ ɛ ] ϕ h 3 ξ ɛ l a x a ν ϕ + hh ɛ (ξ ɛ )ν ϕ h 3 ξ ɛ ϕ + hh ɛ (ξ ɛ )ϕ a Γ x Γ + h 3 ξ ɛ ϕ hh ɛ (ξ ɛ )ϕ x {x =γ} {x =γ} Adding (.) and (.) and using the hypothesis of the lemma we obtain the result. (.) We precise the choice of β in the definition of a : we choose β ], γ[ such that h (β) βh (β) < µ (.) Such β clearly exists by continuity of h and h since h (γ) = h (γ) =. Then there exists a > small enough such that h(β) βh (β) < µ, for any a ], a [. (.3) Then, we may assume that { ( γ ) α } a < min (γ β) α,. (.4) Now we introduce the auxiliary function q a : [, γ] R defined by h(β)+h (β)(x β) h(x for x ) β q a (x ) = for β x γ a /α sin ( π (γ x a /α ) ) for γ a /α x γ. Since q a(β) = then q a H (, γ) with q a (γ) = (.5)

8 8 Equilibrium analysis for a mass-conserving model It is clear that the function q a is non-negative and from the convexity of h on ], γ[ we also deduce q a on [, γ]. (.6) We also consider q : [, ] R defined as follows sin(πx ) for x 4 q (x ) = sin(π( x )) for x for 3 x 4. It is clear that q H (, ) H (, ) and Let us now introduce the functions and q ɛ,a : l R defined by q on [, ] (.7) ξ ɛ,a : l R defined by ξ ɛ,a (x, x ) = ɛq a (x )q (x ) where R ɛ,a is the solution of (.). Observe that ξ ɛ,a = ɛ on a and ξ ɛ,a = on l Γ. R ɛ,a on a q ɛ,a = ξ ɛ,a on l a (.8) on l Lemma.4. Let a, β given by (.) (.4). Then for any a ], a ] there exists ɛ = ɛ (a) such that for any ɛ ], ɛ [ the function q ɛ,a given by (.8) is a subsolution of the problem (.). Proof. We prove that the hypothesis of Lemma.3 are verified with ξ ɛ = ξ ɛ,a. It is clear that (.3),(.4) and (.8) are satisfied. Since R ɛ,δ ɛ on a then R ɛ,δ on ν a which implies (.7) since ξɛ,a = on ν a. On the other hand, from (.6)-(.7) we deduce that H ɛ (ξ ɛ,a ) = q a (x )q (x ) on l a so the inequality (.6) is equivalent to [ h(β) βh (β) ɛh 3 ()q a() ] q (x ) µ x [, ] Since q a() is independent of ɛ, from (.3) and (.7) we deduce that (.6) is verified for ɛ > small enough depending on a. It remains to prove (.5) which is equivalent to q (x ) (hq a ) (x ) ɛq (x ) (h 3 q a) (x ) ɛ (h 3 q a ) (x )q (x ) in l a. (.9)

9 I. Ciuperca, M. Jai and J.I. Tello 9 Observe that Then the inequality (.9) is reduced to We consider three cases q = A q, with A = 4π ],/4[ ]3/4,[. (.3) (hq a ) (x ) ɛ ( h 3 q a) (x ) + ɛa (x ) ( h 3 q a ) (x ) in l a. (.3) Case. < x < β In this case (.3) becomes h (β) ɛ ( h 3 q a) (x ) + ɛa (x ) ( h 3 q a ) (x ). For ɛ small enough (depending on a) this inequality is satisfied since h (β) <. Case. β < x < γ a /α Then (.3) becomes h (x ) + ɛa (x ) ( h 3 q a ) (x ). (.3) From the fact that h is decreasing and convex on ], γ[ and using also (.9) we deduce h (x ) m a (α )/α for β < x < γ a /α. Then (.3) is verified for ɛ small enough (depending on a). Case 3. γ a /α < x < γ We have in this case q a(x ) = π ( π ) a cos /α a (γ x ) /α and q a = π 4a q a. /α Dividing (.3) by q a we deduce that (.3) is equivalent to h (x ) π h(x a /α ) cotan ( π (γ x a /α ) ) + 3π ɛ(h h )(x a /α ) cotan ( π (γ x a /α ) ) + π ɛh 3 (x 4a /α ) +ɛa (x )h 3 (x ) in l a. (.33) Observe that the three first terms in the above inequality are negative and the last two terms are non-negative. In the sub-case γ a /α < x < γ a/α, we have h (x ) m ( a /α / ) α and this gives (.33) for ɛ small enough. In the other sub-case γ a/α < x < γ we have π ( π ) a h(x ) cotan /α a (γ x ) π /α a a = π /α a /α and this gives (.33) for ɛ small enough.

10 Equilibrium analysis for a mass-conserving model This ends the proof of lemma.4. We now introduce a lower bound for a R a in the following lemma. Lemma.5. There exists a constant c > independent of a and a = ( µ )α such that a R a dx c a /α, a ], a ]. Proof. We proceed as in [7] and consider the variational formulation of (.) (h + a) 3 R a ϕ = a h ϕ, a ϕ H( a ) (.34) where R a H ( a ). From (.34) we deduce, thanks to Cauchy-Schwarz inequality a (h + a) 3 R a sup ϕ H (a),ϕ a h ϕ a (h + a) 3 ϕ. (.35) Now consider ψ D(R), ψ >, with support included in [, ] and ψ D(R), ψ > with support included in [/4, 3/4]. We take in (.34), ϕ(x, x ) = ψ ( γ x a /α ) ψ (x ) which is an element of H ( a ) with support included in [γ a /α, γ a /α ] [/4, 3/4]. Since h (x ) m (γ x ) α we have h ϕ a m γ a/α m a α α 3/4 γ a /α /4 3/4 /4 (γ x ) α ψ ( γ x ψ (x )dx γ a /α γ a /α ψ a /α ( γ x ) ψ (x )dx dx a /α ) dx and we easily see that there exists a positive constant c such that ( h ϕ γ ) α c a, a < (.36) a Using that h (x) m(γ x ) α we have that : (h + a) 3 ϕ [ a ( γ x a /α ψ a /α 3/4 γ a /α /4 ) ψ (x ) + (m (γ x ) α + a) 3 γ a /α ( γ ψ x a /α ) ψ (x ) ] This gives a (h + a) 3 ϕ c a 3 /α, a < ( γ ) α (.37)

11 I. Ciuperca, M. Jai and J.I. Tello We deduce from (.35), (.36) and (.37): a (h + a) 3 R a c c a /α, a < ( γ ) α. (.38) On the other hand taking ϕ = R a in (.34) and using the fact that h m on a and that R a > on a we deduce m R a h R a = (h + a) 3 R a a a a so we obtain the result using (.38), where a = ( ν ) α. The main result of this section is Theorem.6. For any F > there exists ɛ >, a > and a 3 > a depending possibly on F but independent of ɛ, such that for any ɛ ], ɛ ] there exists at least a solution (p ɛ, a ɛ ) V [a, a 3 ] of the coupled problem (.5)-(.6)-(.7) or (.8). Moreover we have p ɛ H () C (.39) where C is a constant independent of ɛ. Proof. For any fixed ɛ > the continuity of g ɛ is obvious as a consequence of the continuity of the solution of (.) with respect to a. On the other hand, for any a > take ϕ = p ɛ in (.). We have (h + a) 3 p ɛ (h + a) pɛ x + µ Γ p ɛ dx ( (h + a) 3 p ɛ ) ( / / dx h +a) + µc p ɛ H () (h + a) 3 p ɛ + + µc a p ɛ H () with C > a constant. We deduce using also the Poincaré inequality: C a 3 p ɛ H () a + µc p ɛ H () a + C a 3 p ɛ H () + C a 3 µ C. We then deduce the existence of a constant C 3 > such that p ɛ H () C 3 ( a + a 3 ), for any a > and ɛ >. (.4) By Poincaré inequality there exists C 4 > such that g ɛ (a) C 4 ( a + a 3 ), for any a > and ɛ >. (.4) Finally from Lemma.5, Lemma.4 and Lemma., we deduce that for any a ], min{a, a }], there exists ɛ = ɛ (a) such that for any ɛ ], ɛ ] we have g ɛ (a) > c a /α. (.4)

12 Equilibrium analysis for a mass-conserving model We now chose a > small enough such that c a /α F (.43) and a 3 > a large enough such that C 4 ( a 3 + ) F. (.44) a 3 3 It is clear from (.4)-(.4)-(.43)-(.44) and the continuity of g ɛ that there exists a ɛ [a, a 3 ] such that g ɛ (a ɛ ) = F, for any ɛ ], ɛ (a )] (.45) which ends the proof of the existence of the solution ( of (.8). ) From (.4) we deduce also (.39), with C = C 3. 3 Proof of the main theorem a + a 3 Now we can prove Theorem.. We can extract a subsequence of ɛ denoted also by ɛ and we have a [a, a 3 ] and p V such that a ɛ a and p ɛ p in V weakly. Passing to the limit ɛ in a classical manner in (.5), (.6) and (.7) we obtain the result. 4 Numerical results The goal of this section is the numerical illustration of the theoretical result of Theorem. as well as some extensions. We consider here h (x) = x / α corresponding to γ =, with different values of α > and we search for a numerical solution of the coupled problem (.)-(.)-(.3) with µ =.. To do this we represent graphically pdx as a function of a >, where p is the numerical solution of (.) with h = h + a. The simulation code is based on the Elrod-Adams algorithm [9], with the implementation detailed by Ausas et al. in []. In section 4. we take α = (which is included in the case studied theoretically in this work) and we observe numerically the existence of a unique solution for any given F >. An interesting question here is to see what happens in the case α, which is still an open theoretical question. Recall that the stationary problem with Reynolds variational inequality in the place of (.) was studied in [7]; in the case α = the authors still proved the existence of a solution for any F >, while in the case α < they proved the existence of a solution for F < F only, where F > is a threshold value. In Section 4. (α = ) and Section 4.3 (α = ) we observe numerically that the results in [7] remain valid with an Elrod-Adams model (system (.)-(.)- (.3)).

13 I. Ciuperca, M. Jai and J.I. Tello 3 4. Case. α = The left plot of Fig. represents the parabolic shape of h with α = while in the right plot we can see the profile of pdx (the load) with respect to a. We can conclude that for any F > the studied coupled problem has a unique solution a > (remark that the theoretical uniqueness result is still an open question). In Fig. we represent the 3d profile of the pressure p(x) and the contour plot of the corresponding θ field for, respectively, a =., a =. and a =.. One can observe that for a =. we have large values of the pressure which is concentrated around the middle of the domain (x = /), as predicted theoretically h Load x... a Figure : Case α =. The shape of the upper body h (left plot) and the load versus a (right plot)

14 4 Equilibrium analysis for a mass-conserving model Figure : Case α =. Pressure (left) and corresponding theta-field profiles for different values of a. From top to bottom : a =.,.,.. 4. Case. α = In this limit case we can expect that the result of Theorem. is still valid for any F >. Nevertheless, we expect (as in [7]) that pdx goes very slowly to + when a (as log ( a) ), which is confirmed by the numerical simulations (see Fig. 3) h.3. Load x.... a Figure 3: Case α =. The shape of the upper body h (left plot) and the load versus a (right plot)

15 I. Ciuperca, M. Jai and J.I. Tello Figure 4: Pressure (left) and corresponding theta field profiles for different values of a. Values of a from top to bottom: a =.,., Case 3. α = In this case we observe numerically the existence of a threshold F =.35 such that the result is valid only for F < F (see Fig. 5) h.4.3 Load x.... a Figure 5: Case α = /. The shape of the upper body h (left plot) and the load versus a (right plot).

16 6 Equilibrium analysis for a mass-conserving model Figure 6: Pressure (left) and corresponding theta-field profiles for different values of a. Values of a from top to bottom: a =.,.,.. References [] R. AUSAS, M. JAI AND G. BUSCAGLIA, A mass-conserving algorithm for dynamical lubrication problems with cavitation, ASME J. of Trib, 3 (3), 9. [] G. BAYADA, C. VÁZQUEZ, A survey on mathematical aspects of lubrication problems. Bol. Soc. Esp. Mat. Apl. S ema 39 (7), [3] G. BAYADA, S. MARTIN AND C. VÁZQUEZ, Two-scale homogenization of a hydrodynamic Elrod-Adams model. Asymptotic Analysis 44 (5) 75. [4] G. BUSCAGLIA, I. CIUPERCA, I. HAFIDI AND M. JAI, Existence of equilibria in articulated bearings, J. Math. Anal. Appl., 38 (7) [5] G. BUSCAGLIA, I. CIUPERCA, I. HAFIDI AND M. JAI, Existence of equilibria in articulated bearings in presence of cavity, J. Math. Anal. Appl., 335 (7)

17 I. Ciuperca, M. Jai and J.I. Tello 7 [6] I. CIUPERCA, M. EL ALAOUI TALIBI AND M. JAI, On the optimal control of coefficients in elliptic problems. Application to the optimization of the head slider, ESAIM Control Optim. Calc. Variat. (5) no., -. [7] I. CIUPERCA AND J.I. TELLO, Lack of contact in a lubricated system, Quarterly of Applied Mathematics, 69(), no., [8] I. CIUPERCA, M. JAI AND J.I. TELLO, On the existence of solutions of equilibria in lubricated journal bearings, SIAM J. Math. Anal., 4 (9) [9] H.G. ELROD AND M. ADAMS, A computer program for cavitation, st Leeds-Lyon Symposium on Cavitation and Related Phenomena in Lubrication, I.M.E. 3, 974. [] D. GILBARG AND N.S. TRUDINGER, Elliptic partial differential equations of second order. Second edition 983. Springer-Verlag, Berlin. [] L. LELOUP, Etude de la lubrification et Calculs de Paliers, Dunod, Paris, 96.