LACK OF CONTACT IN A LUBRICATED SYSTEM

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LACK OF CONTACT IN A LUBRICATED SYSTEM IONEL CIUPERCA AND J. IGNACIO TELLO Abstract. We consider the problem of a rigid surface moving over a flat plane. The surfaces are separated by a small gap filled by a lubricant fluid. The relative position of the surfaces is unknown except for the initial time t = 0. The total load applied over the upper surface is a know constant for t > 0. The mathematical model consists in a coupled system formed by Reynolds variational inequality for incompressible fluids and Newton?s second Law. We study the steady states of the problem, the global existence on time and uniqueness of solutions. We assume one degree of freedom for the position of the surface. We consider different cases depending on the geometry of the upper surface. 1. Introduction Lubricated contacts are widely used in mechanical systems to connect solid bo- dies that are in relative motion. A lubricant fluid is introduced in the narrow space between the bodies with the purpose of avoiding direct solid-to-solid contact. This contact is said to be in the hydrodynamic regime, and the forces transmitted between the bodies result from the shear and pressure forces developed in the lubricant film. We consider one of the simplest lubricated systems which consists of two rigid surfaces in hydrodynamic contact. The bottom surface, assumed planar and ho- rizontal moves with a constant horizontal translation velocity and a vertical given force F > 0 is applied vertically on the upper body.

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Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 36
• Langue: English

Extrait

LACK OF CONTACT IN A LUBRICATED SYSTEM

IONEL CIUPERCA AND J. IGNACIO TELLO

Abstract.problem of a rigid surface moving over a flatWe consider the plane. The surfaces are separated by a small gap filled by a lubricant fluid. The relative position of the surfaces is unknown except for the initial time t total load applied over the upper surface is a know constant= 0. The fort > mathematical model consists in a coupled system formed by0. The Reynolds variational inequality for incompressible fluids and Newton 0 s second Law. We study the steady states of the problem, the global existence on time and uniqueness of solutions. We assume one degree of freedom for the position of the surface. We consider different cases depending on the geometry of the upper surface.

1.Introduction

Lubricated contacts are widely used in mechanical systems to connect solid bo- dies that are in relative motion. A lubricant fluid is introduced in the narrow space between the bodies with the purpose of avoiding direct solid-to-solid contact. This contact is said to be in the hydrodynamic regime, and the forces transmitted between the bodies result from the shear and pressure forces developed in the lubricant film. We consider one of the simplest lubricated systems which consists of two rigid surfaces in hydrodynamic contact. The bottom surface, assumed planar and ho- rizontal moves with a constant horizontal translation velocity and a vertical given forceF >0 is applied vertically on the upper body. The wedge between the two surfaces is filled with an incompressible fluid. We suppose that the wedge satisfies the thin-film hypothesis, so that a Reynolds-type model can be used to describe the problem. We denote by Ω the two-dimensional domain in which the hydrodynamic con- tact occurs. We assume that Ω is open, bounded and with regular boundary∂Ω. Without loss of generality we consider 0∈ assume that the upper body,Ω. We theslider normalized distanceis allowed to move only by vertical translation. The, between the surfaces is given by

h(x, t) =h 0 (x) +η(t)

Date: November 5th, 2009. 2000Mathematics Subject Classification.Primary 35J20, 47H11, 49J10; Secondary: 76D08. Key words and phrases.lubricated systems, Reynolds variational inequality, global solutions, stationary solutions. The second author was partially supported by project MTM2009-13655 Ministerio de Ciencia eInnovaci´on(Spain). 1

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IONEL CIUPERCA AND J. IGNACIO TELLO

whereη(t)>0 represents the vertical translation of the slider and h 0 : Ω−→[0,∞[ describes the shape of the slider and is a given function satisfying (1.1)h 0 ∈C 0 (Ω), x m ∈ i Ω nh 0 (x) =h 0 (0) = 0. The mathematical model we study considers the possible cavitation in the thin film, so the (normalized) pressure “p” of the fluid satisfies the Reynolds variational inequality (see [8]): (1.2) Z Ω h 3 rp∙ r(ϕ−p)≥ Z Ω x∂ h ∂ (ϕ−p)−η 0 (t) Z Ω (ϕ−p),∀ϕ∈K 1 where K=  ϕ∈H 01 (Ω) :ϕ≥0  and “r” denotes the gradient with respect to the variablesx∈Ω. Without loss of generality we assume the velocity of the bottom surface is oriented in the direction of thex 1 - axis and its normalized value is equal to 1. The equation of motion of the slider is (1.3)η 00 = Z Ω pdx−F(second Newton Law) completed with the initial conditions: (1.4)η(0) =η 10 (1.5)η 0 (0) =η 0 2 ,

whereη 10 >0,η 20 ∈IRare given data. The unknowns of the problem are the pressurep(x, t) and the vertical displacement of the sliderη(t is known that for any given). ItC 1 functionη(t) the problem (1.2) is well posed (see for instance [12]). The system (1.2)-(1.5) is equivalent to the following Cauchy problem for a second order ordinary differential equation inη:  η 00 =G(η, η 0 ) (1.6)   η(0) =η 0 1 ,   η 0 (0) =η 0 2 , whereG:]0,∞[×IR−→IRis given by β, γ) := Z Ω , G(q(x)dx−F andq∈K(depending onβandγ) is the unique solution to    Z Ω (h 0 +β) 3 rq∙ r(ϕ−q)≥ Z Ω h 0 ∂ ∂x 1 (ϕ−q)−γ Z Ω (ϕ−q) (1.7)   ∀ϕ∈K. The main goal of the paper is to give sufficient conditions on the shapeh 0 of the slider to obtain global existence on time to (1.6), i.e. there is no contact solid- to-solid fort <∞. We also study the existence of steady states of the problem. Another interesting physical question which we adress here is to see if there exists a “barrier” valueη b >0 such thatη(t)≥η b for allt >0. We prove the existence ofη b  case (the so Thirdfor two of the three cases studied.