# On the Reynolds lubrication equation

Chipot, M (1988). On the Reynolds lubrication equation. Nonlinear Analysis: Theory, Methods & Applications, 12(7):699-718.

## Abstract

Let Ω be a bounded open set in R 2 with “smooth” boundary ∂Ω. Ω is the region where two solid bodies are in contact. These two solids are moving and the average of their velocities is denoted by V=(V 1 ,V 2 ). The pressure p=p(X) which develops in a fluid layer confined between these two bodies satisfies the so- called Reynolds lubrication equation:
(1)∇·(h 3 ρ∇p)=6μV·∇(ρh)inΩ
(2)p=p a on∂Ω·
Here ρ is the density of the fluid, μ>0 its dynamic viscosity, h=h(X) is the distance between the two bodies and p a ≥0 is the given ambient pressure.
We will assume all along that h is a Lipschitz continuous function such that
(3)0<h 1 ≤h(X)≤h 2 a·e·X=(x,y)∈Ω
(4)|∇h(X)|≤Ha·e·X=(x,y)∈Ω
where h 1 , h 2 , H are positive constants. When the fluid is incompressible, i.e. ρ is a positive constant, existence and uniqueness for a solution to (1), (2) is a trivial matter since the equation is linear. In the compressible case when (5) ρ=ρ(p) the problem becomes more challenging.
The paper is organized as follows: in Section 1 we investigate the problem of existence of a solution for (1), (2) when ρ is given by (5). In Section 2 we consider a case where (1), (2) reduces to a one dimensional problem and we study the question of uniqueness as well as the shape of the solution. Finally, in a third section we give some extensions of the results of Section 2 in higher dimension.

## Abstract

Let Ω be a bounded open set in R 2 with “smooth” boundary ∂Ω. Ω is the region where two solid bodies are in contact. These two solids are moving and the average of their velocities is denoted by V=(V 1 ,V 2 ). The pressure p=p(X) which develops in a fluid layer confined between these two bodies satisfies the so- called Reynolds lubrication equation:
(1)∇·(h 3 ρ∇p)=6μV·∇(ρh)inΩ
(2)p=p a on∂Ω·
Here ρ is the density of the fluid, μ>0 its dynamic viscosity, h=h(X) is the distance between the two bodies and p a ≥0 is the given ambient pressure.
We will assume all along that h is a Lipschitz continuous function such that
(3)0<h 1 ≤h(X)≤h 2 a·e·X=(x,y)∈Ω
(4)|∇h(X)|≤Ha·e·X=(x,y)∈Ω
where h 1 , h 2 , H are positive constants. When the fluid is incompressible, i.e. ρ is a positive constant, existence and uniqueness for a solution to (1), (2) is a trivial matter since the equation is linear. In the compressible case when (5) ρ=ρ(p) the problem becomes more challenging.
The paper is organized as follows: in Section 1 we investigate the problem of existence of a solution for (1), (2) when ρ is given by (5). In Section 2 we consider a case where (1), (2) reduces to a one dimensional problem and we study the question of uniqueness as well as the shape of the solution. Finally, in a third section we give some extensions of the results of Section 2 in higher dimension.

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