Research Article Existence of Weak Solution for a Free Boundary Lubrication Problem

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1 International Scholarly Research Network ISRN Mathematical Analysis Volume 11, Article ID 466, 1 pages doi:1.54/11/466 Research Article Existence of Weak Solution for a Free Boundary Lubrication Problem Abdelhamid Laouar Department of Mathematics, Faculty of Sciences, University of Annaba, P.O. Box. 1, 3 Annaba, Algeria Correspondence should be addressed to Abdelhamid Laouar, laouar.abdelhamid@univ-annaba.org Received 15 October 1; Accepted 9 November 1 Academic Editor: G. Schimperna Copyright q 11 Abdelhamid Laouar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is interested in a free boundary problem modelling a phenomenon of cavitation in hydrodynamic lubrication. We reformulate the problem see Boukrouche, 1993 in a large context by introducing two positive parameters, namely, N and a. We build a weak formulation and establish the existence of the solution to the problem. 1. Introduction The lubrication fields have many applications; one example is the study of the rotary mechanisms such as the bearing, joints. The study is concerned in looking for a moving free boundary problem related to the cavitation modelling in lubrication see 1 4. The experimental results make evidence of the occurrence of two distinct zones, one full of the fluid, namely, the saturated zone t ; the other c c \ t, where is the global domain, is the cavitated zone e.g., the mixture of fluid and air. Two approaches have been used to cope with phenomena. One of them 5 homogenizes the phenomena and considers it as a D phenomena, so introducing θ the saturation variable lubricant concentration ; the other one 6 takes full account of the three-dimensional character of this phenomena, with appearance of air bubbles and introduces in c the relative height as supplementary unknown for more details, see 1, 3. We use here the first approach but both approaches lead to the same mathematical problem. In this paper, we take the problem studied in 1 3 and rewrite it, here, in a large context, by introducing two positive parameters, namely, N and a. This formulation of the problem gives as advantages a proportionality relation

2 ISRN Mathematical Analysis ) Γ i Γ ex Axis t) Γt) c Figure 1: Aviewofdomains t and c. between N and the pressure p, and the parameter a which allows the control of a squeezing effects. The mathematical modelling is made according to the model of Jakobsson-Floberg see 1, 3, where the lubricant is not defined only by a pressure p but also by a saturation variable θ. This variable θ characterizes the cavitation phenomena, where θ 1in t and θ < 1in c. The interface between t and c constitutes the moving free boundary denoted by Γ t. The problem is a convection-diffusion problem type, and the Reynolds equation is elliptic in the saturated zone and hyperbolic in the other one. We note that the study of existence and uniqueness point of views to the problem has been established in 3 in the particular case of the N anda 1. For this, the author proved the existence of solution for this kind of problem by way of an approximation by an elliptic problem. In our work, we followed, exactly the same way, and the aim of this study is to construct a weak formulation and establish the existence of solution to the problem. The plan of the paper is as follows. Section proposes a state of the problem and a weak formulation. Section 3 introduces an elliptic nonlinear problem and gives the existence and uniqueness of the solution to this problem. Section 4 proposes an approximation of the elliptic nonlinear problem by a family of linear problems and proves a priori estimate. Last section gives a theorem of existence of the solution to the problem.. Notations and State of the Problem.1. Description of the Phenomenon We consider a global domain with border. The fluid is injected at a given rate w over the fixed internal boundary Γ I Γ ex \ Γ I. For each t ; T, the experimental results make evidence of the occurrence of two distinct zones: one full of the fluid is the saturated zone t, where the pressure p p > and the saturation variable θ θ 1, andthe other c c \ t is the cavitated zone, where the pressure is constant p and θ<1 e.g., the mixture of fluid and air. The free boundary of the region t containing fluid is Γ t and the region, withborderγ I, occupied by the fluid at t being given see Figure 1... State of the Problem The strong formulation of the problem described the phenomenon is written as follows.

3 ISRN Mathematical Analysis 3 1 θ Figure : Graph of saturation variable θ. For each t, find a pair p, θ L,T; H 1 L H 1,T; H 1 and Γ t such that K h, t, N p ) Vhθ a hθ, θ 1 in t,.1 Vhθ a hθ in c,. K h, t, N p h 1 θ V aυ n on Γ t,.3 n p on Γ t,.4 K h, t, N p n hθv n w on Γ I,.5 p on Γ ex,.6 p 1 θ in,.7 θ t θ,.8 where h 3 K h, t, N Φ h, t, N 1 N with N < 1, Φ h, t, N h 1 N ) 1 N ) coth N 4h 1 N h 1 N..9 h t, x is the thickness of the thin film supposed a regular and given function of the problem. V is the speed of the axis supposed being given. υ is the moving free boundary with υ on Γ I. n resp., n is the normal vector along Γ t resp., Γ I exterior to t resp.,. The saturation variable θ can be represented by a graph see, Figure. In.1.3, there are the diffusion term K h, t, N p, the shearing term hv, and the squeezing term h/.

4 4 ISRN Mathematical Analysis.3. Weak Formulation Before starting the construction of a weak formulation of the problem.1.8, we denote by,t, Σ I Γ I,T, Σ t Γ t,t, Σ ex Γ ex,t, { E H 1 : onσ ex and t..1 Indeed, multiplying.1 by E and integrating over,t t,weobtain T T T K P dxdt K Pn K Pn t Γ t Γ I T a { T t T Vh dxdt d dt t T h dx dt Γ t T Vhn Γ t T VhΦ ndt T Vhn Vh n ex Γ I Γ ex T Vhndt Γ I t h d dx dt, dt where n ex is the normal vector along exterior to. In the same way, we apply to. T T Vh dxdt c Γ t { T d a h dx dt dt c T Vh n T Γ t T Vh n Vh n ex Γ I Γ ex T Vhndt Γ I T Vh ndt c h d dt dx dt. By adding and in all t c,andusing.3.6 then the weak formulation can be written as follows. Find a pair p, θ L,T; H 1 xl H 1,T; H 1 θ<1, p onσ ex, p 1 θ, a.e. in,.11 { ) K p Vhθ a hθ hθ hθ d w, ɛe, dt Σ I.1 θ t θ H 1. As t implies that hθ.

5 ISRN Mathematical Analysis 5 1 β O Figure 3: Graph of the nonlinear function β. 3. An Elliptic Nonlinear Problem To solve the problem , we will approximate it by an elliptic problem in the same way as Boukrouche 3 and Gilardi 7. Let β be a real function see Figure 3 satisfying the following assumptions: β C R ξ R; β ξ 1, with β ξ, β ξ for ξ. 3.1 Put Σ ex. 3. consider now the problem, given >, find p H 1 such that p { ) K p Vhβ p a hβ p hβ p d dt I w, 3.3 p on, for all H 1 vanishing on. Introducing the operator τ is as follows: τ : H 1 H 1, p q τ p. 3.4 If p H 1, τ p is a unique solution q to the linear problem q H 1, q, on q K q hβ p V a d dt ) a hβ p w T I 3.5 for every H 1 vanishing on.

6 6 ISRN Mathematical Analysis Lemma 3.1. The operator τ is continuous from H 1 with the weak topology into H 1 with the strong topology. Moreover τ H 1 is bounded in H 1. Proof. Let p i H 1 with q i τ p i for i 1,. Taking q 1 q in 3.5, we have ) ) q1 q K ) q1 q h β p 1 β p ) V ) d )) q 1 q q 1 q a dt { a h β p 1 β p ) ) q 1 q. 3.6 Using Cauchy-Schwarz s inequality, we obtain α q1 q H 1 C 1{ q1 q H β p1 1 β p L { C q1 q L β p1 β p L, 3.7 where α is constant depending on h, N,and. C 1 and C are two constants depending on h, V,anda. As H 1/ L, H 1/ Σ I L Σ I, H 1 H 1/, H 1 H 1/ Σ I 3.8 and β is Lipschitz continuous function, there exists a constant C depending on h, V, N, a, and such that { q1 q H 1 C p 1 p L p 1 p L. 3.9 If p p converge weakly in H 1, then p p, L and p T p T, L. Thus τ p τ p, then the continuity of τ is shown. Taking q in 3.5 and using Cauchy-Schwarz s inequality, we obtain q H 1 k 1, where k 1 is a constant depending on h, V, N, β, T, a, and. Theorem 3.. If the function β satisfies hypothesis 3.1, then, for every >, there exists a solution to the problem 3.3. Proof. Use Lemma 3.1 and Schauder fixed-point theorem. Theorem 3.3 cf., 3. If the function β satisfies hypothesis 3.1, then for every >, the solution of the problem 3.3 is unique.

7 ISRN Mathematical Analysis 7 1 H O Figure 4: Graph of Heaviside function H. 4. Approximating Problems In order to solve the problem , we consider a new family of problems of type 3.3 in which the function β is an approximation of the Heaviside function see Figure 4. Therefore we consider a family of functions H :, R \ H C R, 4.1 H 1, H, 4. H, lim inf{ζ >:H ζ 1, 4.3 lim L, where L sup { H ζ : ζ>. 4.4 Consider now the following approximating problem. For fixed and a where, a ; 1, findp such that K p p H 1, p on, ) hh p V a d ) a dt hh p ) I w 4.5 for every H 1 vanishing on. From Theorems 3. and 3.3, we deduce the following theorem. Theorem 4.1 cf., 3. For every and a, 1, there exists at least one solution to the problem 4.5. Moreover if h and V are sufficiently regular, every solution belongs to H 1 L. Lemma 4.. If the function H verifies and dh/dt, then one has ) )dp hh p p hh p dt. 4.6 Proof. Putting H ζ ζ H τ dτ, we have then H ζ ζh ζ and hh p p hh p dp /dt hh p p h d/dt H p.

8 8 ISRN Mathematical Analysis Using the integration by parts, we obtain ) hh p p h d ) H ) [ ) ) p h H p p H ] dh p dt dt H ζ 4.7 as H p p H p, then Lemma 4. is shown. The following proposition gives some a priori estimates for pressure p. Proposition 4.3. There exists a constant C independent of, a, N, and p K { p dp a hh p hh C. dt 4.8 Proof. Taking p in 4.5, weobtain P K { p dp ) a hh p hh hv H p p wp. T dt Σ I 4.9 By using Lemma 4., we have K p mes β p L V L p L Σ I w L Σ I. 4.1 Applying Poincare s inequality, we have T p L p α1 p L, 4.11 where α 1 is constant depending on. As p L Σ I α p L Σ I, we finally deduce the result. Proposition 4.4. For every nonnegative and D, there exists a constant C such that,x C ) 1 L. 4.1

9 ISRN Mathematical Analysis 9 Proof. From 4.5, we have P ) )) )) K p VhH p a hh p K h, t, N p n hh Vn w on Σ I, p on, T on. in, 4.13 Multiplying 4.13 by / and integrating over,t, we have,x 1 K t p t { hh ) p t K p hh p ) V ) ah p ) h t 1 t K p. p t 4.14 Taking then we have A t 1 t { B t K p, K p hh p ) V ) ah p ) h,x A B, T,x T A B dt, T T Adt t T K p L t L K T L p t, p Using Proposition 4.3., we obtain T ) Adt c 1, 4.17

10 1 ISRN Mathematical Analysis where c 1 C max{ 1/ L, / K L L and C is constant of Proposition 4.3: T Bdt { T K p hv ) H p t T ) H p hv ah p ) h We obtain finally T,x T A B dt C ) 1 L An Existence Theorem of the Problem Theorem 5.1. There exists at least one solution to the problem Proof. If a, 1 and from Proposition 4.3, we can extract a subsequence of p, still denoted by p, such that p p in L ),T; H 1 weakly, 5.1 in L weakly. 5. Moreover H p, 1, for all >, then there exists θɛl such that H p ) θ in L weakly. 5.3 We can now proof that the p, θ given in 5.1 and 5.3 is solution of the problem We denote by k { ɛl ),T; H 1 : onσ ex and in, { k ɛl : 1, k k is convex space of L ),T; H 1 L, then k k is weakly closed. 5.4 Therefore pɛl,t; H 1, p a.e.in and θɛl L with θɛ, 1 a.e. in. Thus if in 4.5, wefindtheproblem In order to prove θ<1 and p 1 θ, a.e in, we give a brief demonstration for more details, see 7, pages

11 ISRN Mathematical Analysis 11 First, we need to prove p 1 θ. We notice that lim p 1 H p since lim p 1 H p )) mes sup { ζ :H p ) < Now we have to prove that )) p 1 θ lim p 1 H p. 5.6 We define w / p H p and use Then the couple p, θ is solution of the problem Theorem 5.. If w on Γ I,T and θ, thenp a.e. in,t. Proof. Following 7, we construct a sequence solution of the problem p p p T, on,t, 5.7 where p sup p,, p sup p, and p p p. From the classical Cauchy-Lipschitz-Picard theorem, there exists a unique solution p ɛc 1,T. Multiplying 5.7 by p resp., by / and integrating over, weobtain p L p L, 5.8 respectively, L p L. 5.9 Deriving 5.7 with respect to x, multiplying by p, and integrating over, weobtain p L p L. 5.1 We deduce that there exists pɛl,t; H 1 and ζɛl such that p p ζ weakly in L ),T; H 1, 5.11 weakly in L ),T; H

12 1 ISRN Mathematical Analysis From 5.11 we have p in L ),T; H 1, therefore ζ Passing to the limit in 5.7, we deduce that p p a.e. in. As E H 1,T; V is dense in L,T; H 1,.1 can be rewritten in the following form: T a < V θh, > V K p Vhθ w, ɛv, a.e. in,t Σ I Taking now p as test function in 5.14 and passing to the limit over, we deduce K p wp Σ 5.15 as w, therefore p a.e.in, thatis,p a.e.in. Next work will consist in finding some existence of relationship between the pressure p and parameter N and in completing numerical analysis study to the problem.1.8. Acknowledgment The author is grateful to Professor Boukrouche M. for a helpful discussion on this subject. References 1 G. Bayada and M. Chambat, Nonlinear variational formulation for a cavitation problem in lubrication, Mathematical Analysis and Applications, vol. 9, no., pp , 198. G. Bayada, M. Boukrouche, and M. El Alaoui Talibi, The transient lubrication problem as a generalized Hele-Shaw type problem, Journal for Analysis and Its Applications, vol. 14, no. 1, pp , M. Boukrouche, Contribution à la modélisation de problèmes à frontière libre, issus de la lubrification hydrodynamique: phénomènes de Hele-Shaw et de cavitation,thèse d état, Loyn, France, 1993, no M. El Alaoui Talibi and G. Bayada, Une méthode du type caractéristique pour la résolution d un problème de lubrification hydrodynamique en régime transitoire, Mathematical Modelling and Numerical Analysis, vol. 5, no. 4, pp , H. G. Elrod and M. L. Adams, A computer program for cavitation, in Cavitation and Related Phenomena in Lubrication, D. Dowson, M. Godet, and C. M. Taylor, Eds., Mech. Eng. Publ., G. Capris and G. Cimati, Partial lubrication of fullcylindrical bearing, C.N.U.C.E. Report, vol. C, pp. 4 79, G. Gilardi, A new approach to evolution free boundary problems, Communications in Partial Differential Equations, vol. 4, no. 1, pp , 1979.

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