On a coupled free boundary problem for a piezoviscous fluid in thin film

Transcription

1 On a coupled free boundary problem for a piezoviscous fluid in thin film G. Bayada, L. Chupin, B. Grec 1 Institut Camille Jordan, Lyon, France 1 berenice.grec@insa-lyon.fr International Conference on Mathematical Fluid Mechanics May 21st, 2007

2 Outline of the talk 1 Introduction 2 The model of the flow 3 Known results 4 Existence result 5 Conclusion and prospects

3 Outline of the talk 1 Introduction 2 The model of the flow 3 Known results 4 Existence result 5 Conclusion and prospects

4 Introduction Considered features of the flow: Non-newtonian behavior: no constant viscosity. Diphasic behavior (cavitation). Fluid-structure interaction. Deformation of the domain Full film flow Cavitation h Goal of the work: Knowledge of the pressure in a lubricated device. General result: no smallness assumption on the data.

5 Outline of the talk Anisotropy Deformation Non-newtonian fluid Cavitation Weak form 1 Introduction 2 The model of the flow 3 Known results 4 Existence result 5 Conclusion and prospects

6 Anisotropic flow Anisotropy Deformation Non-newtonian fluid Cavitation Weak form Thin lubricated film: Navier-Stokes equations (3-D) reduced to Reynolds equation (2-D). ([Bayada, Chambat, 86], [Nazarov, 90]) Equation on the pressure p in : ( ) 1 div x,y µ h3 p 6s h x = 0, z p = 0. s given velocity. y Γ 0 z s h x The input flow is imposed on Γ 0.

7 Anisotropic flow Anisotropy Deformation Non-newtonian fluid Cavitation Weak form Thin lubricated film: Navier-Stokes equations (3-D) reduced to Reynolds equation (2-D). ([Bayada, Chambat, 86], [Nazarov, 90]) Equation on the pressure p in : ( ) 1 div x,y µ h3 p 6s h x = 0, z p = 0. s given velocity. y Γ 0 z s h x The input flow is imposed on Γ 0.

8 Anisotropy Deformation Non-newtonian fluid Cavitation Weak form Deformation of the domain High pressures = deformation of the upper surface. h(p) = h 0 + k p, with the kernel k(x, y) = some constants. k 0 x2 + y 2, h 0 and k 0 h h 0 ( h 3 ) (p) div µ p = 6s h(p) x,

9 Anisotropy Deformation Non-newtonian fluid Cavitation Weak form Deformation of the domain High pressures = deformation of the upper surface. h(p) = h 0 + k p, with the kernel k(x, y) = some constants. k 0 x2 + y 2, h 0 and k 0 h h 0 ( h 3 ) (p) div µ p = 6s h(p) x,

10 Anisotropy Deformation Non-newtonian fluid Cavitation Weak form Non-newtonian behavior High pressures = viscosity µ non-constant. ([Szeri, 98], [Blair 01] for physical considerations, [Hron, Málek, Nečas, Rajagopal, 03] for mathematical approach) Roelands formula: µ(p) = exp (C [ 1 + (1 + cp) z ]) Barus relation: µ(p) = µ 0 e αp ( ) 1 div e αp h 3 (p) p 6s h(p) = 0 µ 0 x

11 Cavitation Anisotropy Deformation Non-newtonian fluid Cavitation Weak form y p = 0 Cavitation: Appearance of a non-saturated zone. Two models: p > 0 a variational inequality ([Cimatti], [Rodrigues]): minimization of the functional J(p) = ( ) 1 µ h3 p 2 6s h x under the condition on the pressure p 0. x

12 Cavitation Anisotropy Deformation Non-newtonian fluid Cavitation Weak form y p = 0 Cavitation: Appearance of a non-saturated zone. x p > 0 a pressure-saturation model (Elrod-Adams) ([Bayada, Chambat], [Alvarez]). The cavitation zone is characterized: by a constant pressure (p = 0), by a blend of air and fluid. It introduces a function θ H(p) (Heaviside function) corresponding to the local proportion of the fluid. ( h 3 ) (p) div µ(p) h3 p 6s θ(p) h(p) = 0 x

13 Weak formulation Anisotropy Deformation Non-newtonian fluid Cavitation Weak form Function spaces V = { φ H 1 (), φ Γ\Γ0 = 0 }, V + = {φ V, φ 0}. Find p V + and θ L () such that: h 3 (p) p ϕ = 6s h(p)θ(p) x ϕ + G 0 ϕ, ϕ V, (P) µ(p) Γ 0 θ H(p). y p = 0 G 0 is the input flow, given on Γ 0. G 0 p > 0 x

14 Outline of the talk 1 Introduction 2 The model of the flow 3 Known results 4 Existence result 5 Conclusion and prospects

15 Known results 1 Conditional results Small data assumption: sµ 0 α C. (1) Variational inequality model ([Oden, Wu], [Rodrigues]) Theorem Under hypothesis (1), the problem admits a solution satisfying p H 1 0 () L (). Elrod-Adams model ([Bayada, El Alaoui, Vasquez]) Theorem Under hypothesis (1), the problem admits a solution satisfying p H 1 () L ().

16 Known results 2 Unconditional results Variational inequality model ([Bayada, Bellout]) Theorem Under a realistic assumption on the asymptotic behavior of µ(p), the problem admits a solution satisfying p H 1 0 (). Elrod-Adams model : goal of this work. Difficulties: deal with two parameters of regularization (for p and θ). deal with the boundary terms (no zero Dirichlet boundary condition, input flow imposed).

17 Outline of the talk Change of functions Regularization Regularized problem Limit Estimates 1 Introduction 2 The model of the flow 3 Known results 4 Existence result 5 Conclusion and prospects

18 Change of functions Regularization Regularized problem Limit Estimates Reduced pressure Reduce the problem to one close to an isoviscous one. Reduced pressure P: p ds P = µ(s), p = γ(p), 0 A = + ds µ(s) <. 0 p = γ(p) A P h 3 (p) p ϕ = 6s (P) µ(p) θ H(p). h(p)θ(p) x ϕ + Γ 0 G 0 ϕ, ϕ V,

19 Change of functions Regularization Regularized problem Limit Estimates Reduced pressure Reduce the problem to one close to an isoviscous one. Reduced pressure P: p ds P = µ(s), p = γ(p), 0 A = + ds µ(s) <. 0 p = γ(p) A P H 3 (P) P ϕ = 6s (P ) H(P) = h 0 + k γ(p). H(P)θ(P) x ϕ + Γ 0 G 0 ϕ, ϕ V,

20 Heaviside function θ(p) regularized by θ η (P). Regularization Change of functions Regularization Regularized problem Limit Estimates Truncation γ ε (P) of the function γ(p). γ ε(p) θ η(t) γ(a ε) ε η t A P Goal : let η, ε tend to zero. H 3 (P ηε ) P ηε ϕ = 6s H(P ηε ) = h 0 + k γ ε (P ηε ). H(P ηε )θ η (P ηε ) x ϕ + G 0 ϕ, Γ 0

21 Change of functions Regularization Regularized problem Limit Estimates Existence result for the regularized problem Theorem For fixed η and ε, there exists a solution of the regularized problem satisfying P ηε V +. P also satisfies P ηε L R. Proof: Fixpoint theorem in V +. Proving that the set A k = {(x, y), P ηε k} is of measure zero for k big enough. In the case of conditional results: suppose that R < A. γε(p) ε γ(a ε) A P

22 Passing to the limit Change of functions Regularization Regularized problem Limit Estimates There exists P such that P ηε η,ε 0 It remains to: P in H1 (). prove by contradiction P L A γ(p) is well-defined. prove that γ(p) L 1 C. prove more regularity on P in order to pass to the limit in the non-linear term H 3 (P ηε ) P ηε. Since Pηε converges weakly in L 2, it suffices to prove that H(P ηε) converges strongly in L 6. H(P) = h0 + k γ ε(p), with k a regularizing kernel. It suffices to prove that γ ε(p ηε) is bounded in L 6 ().

23 Additional estimates Change of functions Regularization Regularized problem Limit Estimates We want to show γ σ ε (P ηε ) H 1 C for some σ. We compute γε σ 2 = C γ ε γ 2(σ 1) ε P ηε 2. Using the weak formulation with any test function δ(p ηε ) leads to H 3 P ηε 2 δ δ. Choose δ = γ ε γ2(σ 1) ε, i.e. δ = γε 2σ 1. Thus Hmin 3 γ ε γε 2(σ 1) P ηε 2 γ ε γ 2(σ 1) ε. We know γ ε C. We choose γ ε such that γ ε = γβ ε. (corresponds to the asymptotic behavior µ(p) = p β for p big enough). Choice of σ in function of β leads the needed estimate.

24 Additional estimates Change of functions Regularization Regularized problem Limit Estimates We want to show γ σ ε (P ηε ) H 1 C for some σ. We compute γε σ 2 = C γ ε γ 2(σ 1) ε P ηε 2. Using the weak formulation with any test function δ(p ηε ) leads to H 3 P ηε 2 δ δ. Choose δ = γ ε γ2(σ 1) ε, i.e. δ = γε 2σ 1. Thus Hmin 3 γ ε γε 2(σ 1) P ηε 2 γ ε γ 2(σ 1) ε. We know γ ε C. We choose γ ε such that γ ε = γβ ε. (corresponds to the asymptotic behavior µ(p) = p β for p big enough). Choice of σ in function of β leads the needed estimate.

25 Additional estimates Change of functions Regularization Regularized problem Limit Estimates We want to show γ σ ε (P ηε ) H 1 C for some σ. We compute γε σ 2 = C γ ε γ 2(σ 1) ε P ηε 2. Using the weak formulation with any test function δ(p ηε ) leads to H 3 P ηε 2 δ δ. Choose δ = γ ε γ2(σ 1) ε, i.e. δ = γε 2σ 1. Thus Hmin 3 γ ε γε 2(σ 1) P ηε 2 γ ε γ 2(σ 1) ε. We know γ ε C. We choose γ ε such that γ ε = γβ ε. (corresponds to the asymptotic behavior µ(p) = p β for p big enough). Choice of σ in function of β leads the needed estimate.

26 Additional estimates Change of functions Regularization Regularized problem Limit Estimates We want to show γ σ ε (P ηε ) H 1 C for some σ. We compute γε σ 2 = C γ ε γ 2(σ 1) ε P ηε 2. Using the weak formulation with any test function δ(p ηε ) leads to H 3 P ηε 2 δ δ. Choose δ = γ ε γ2(σ 1) ε, i.e. δ = γε 2σ 1. Thus Hmin 3 γ ε γε 2(σ 1) P ηε 2 γ ε γ 2(σ 1) ε. We know γ ε C. We choose γ ε such that γ ε = γβ ε. (corresponds to the asymptotic behavior µ(p) = p β for p big enough). Choice of σ in function of β leads the needed estimate.

27 Additional estimates Change of functions Regularization Regularized problem Limit Estimates We want to show γ σ ε (P ηε ) H 1 C for some σ. We compute γε σ 2 = C γ ε γ 2(σ 1) ε P ηε 2. Using the weak formulation with any test function δ(p ηε ) leads to H 3 P ηε 2 δ δ. Choose δ = γ ε γ2(σ 1) ε, i.e. δ = γε 2σ 1. Thus Hmin 3 γ ε γε 2(σ 1) P ηε 2 γ ε γ 2(σ 1) ε. We know γ ε C. We choose γ ε such that γ ε = γβ ε. (corresponds to the asymptotic behavior µ(p) = p β for p big enough). Choice of σ in function of β leads the needed estimate.

28 Additional estimates Change of functions Regularization Regularized problem Limit Estimates We want to show γ σ ε (P ηε ) H 1 C for some σ. We compute γε σ 2 = C γ ε γ 2(σ 1) ε P ηε 2. Using the weak formulation with any test function δ(p ηε ) leads to H 3 P ηε 2 δ δ. Choose δ = γ ε γ2(σ 1) ε, i.e. δ = γε 2σ 1. Thus Hmin 3 γ ε γε 2(σ 1) P ηε 2 γ ε γ 2(σ 1) ε. We know γ ε C. We choose γ ε such that γ ε = γβ ε. (corresponds to the asymptotic behavior µ(p) = p β for p big enough). Choice of σ in function of β leads the needed estimate.

29 Outline of the talk 1 Introduction 2 The model of the flow 3 Known results 4 Existence result 5 Conclusion and prospects

30 Conclusion and prospects We obtained an unconditional existence result for the problem with the reduced pressure. Non restrictive assumption on the asymptotic behavior of µ. 1D-problem : possible to come back to an existence result for the usual pressure. 2D-problem : not clear if the two problems are equivalent. Is it possible to make a direct proof without the reduced pressure? Uniqueness? Consider other types of deformation of the domain, for example another kernel k modelling roll bearings. plane-sphere plane-cylinder

31 Bibliography I G. Bayada and H. Bellout, An unconditional existence result for the quasi-variational elastohydrodynamic free boundary value problem, G. Bayada, M. El Alaoui Talibi, and C. Vázquez, Existence of solutions for elastohydrodynamic piezoviscous lubrication problems with a new model of cavitation, J. T. Oden and S. R. Wu. Existence of solutions to the Reynolds equation of elastohydrodynamic lubrication, J.-F. Rodrigues, Remarks on the Reynolds problem of elastohydrodynamic lubrication, 1993.

32 Proof of the estimate P L A Assume τ > 0 such that τ = {(x, y), P A + τ} has a non-zero measure. Let ε = {(x, y), P ηε A}. Then γ ε (P ηε γ(a ε) ε. γε(p) ε A γ(a ε) P But for P ηε P, we can prove that ε 1 2 τ. A P P ηε This implies γ ε (P ηε 1 2 γ(a ε) τ ε 0 +. This is a contradiction with γ ε (P ηε ) bounded uniformly in L 1 (). Back