On the connection between evolution Stokes equation and Reynolds equation for thin-tilm flow

On the Connection Between Evolution Stokes Equation and Reynolds Equation for Thin-Film Flow ¹

J. Fabricius, Yu. Koroleva, P. Wall

Lule˚ a University of Technology

john.fabricius@ltu.se, yulia.koroleva@ltu.se, peter.wall@ltu.se

We study three-dimensional incompressible flow in a domain bounded by two moving surfaces. It is assumed that parameter ε > 0 characterizes the thickness of the gap between the surfaces. We rigorously derive the Reynolds equation which describes both the pressure and the velocity field in the limit as ε → 0.

Consider the incompressible time-dependent Stokes equation in domain Ω

, bounded by two moving surfaces x

= εh

(x

, x

, t), t ∈ [0, T ] :

D

U

− ν∆U

+ ∇P

= 0 in Ω

(0.1)

div U

= 0 in Ω

(0.2)

with initial-boundary values

U

= (

v

, v

, ε(D

h

+ v

D

h

+ v

D

h

) )

on Σ

(0.3)

U

= g

on Σ

(0.4)

U

= U

on Ω

(0), (0.5)

where g

and U

are prescribed, and Σ

, Σ

, Ω

(0) are boundaries of Ω

. For the convenience first we pass to the problem with homogeneous boundary conditions by introducing a lift function U

and then transform the original domain into fixed Ω = {(ξ

, ξ

) ∈ ω, ξ

∈ (0, 1)}, which depends neither on ε nor t, by a simple change of variables. We define a weak solution for the considered problem in terms of Ω. The solution of the problem in the fixed domain Ω is denoted by (u

, p

). We prove the existence and uniqueness of the limit (u

, p

) such that

u

= − ξ

(1 − ξ

)h

2ν ∇p

+ ξ

v

+ (1 − ξ

)v

− P U. (0.6)

1

The work was supported by Bernt J¨ armarks Foundation for scientific Research and in part by RFBR

Moreover p

∈ L

(0, T ; ˚ L

(ω) ∩H

(ω)) does not depend on the third variable and satisfies the Reynolds equation

 

 

 



D

h + div (

− h

12ν ∇p

+ h

2 (v

+ v

) )

= 0 in ω × (0, T ) (

− h

12ν ∇p

+ h

2 (v

+ v

) )

· ˆn = hg · ˆn on ∂ω × (0, T ).

(0.7)

Here P denotes the projection P y = (y

, y

, 0), y ∈ R

, h = h

− h

and g is the average of the prescribed function g in the ξ

direction.

The obtained results extend to three dimensions the analysis of the two-dimensional problem considered in [1]. In addition to previous studies, we allow both surfaces to be in motion. Moreover, the motions of the surfaces are assumed to originate from rigid body motion (i.e. both translational and rotational motion is allowed), whereas in [1]

the velocity was assumed constant.

References

[1] G. Bayada, M. Chambat and I. Ciuperca, Asymptotic Navier-Stokes equations in a

thin moving boundary domain, Asympt. Anal., 21 (1999), 117–132.