## A rigorous justification of the Reynolds equation.(English)Zbl 0661.76028

In a classical paper O. Reynolds formulated a differential equation which gives an asymptotic approximation of the Navier-Stokes equation. In the one-dimensional case this equation reads as follows: (1) $$(H^ 3P')'=6\mu \bar VH'$$. P is the pressure in a thin film of a viscous fluid of constant density, thickness H and viscosity $$\mu$$. $$\bar V$$ is the $$X_ 1$$-component of the moving surface. Equation (1) applies when the height of the fluid is very small compared to the span and the length. The justifications given in the literature for the transition from the full Navier-Stokes equations to (1) are mainly heuristic. A possible approach to the Reynolds equation, and thus to the theory of lubrication, is to suppose true the expressions of the Couette-Poiseuille flow, $$V_ 1=2\mu^{-1}(\partial P/\partial X_ 1)X_ 2(X_ 2-H)+\bar V(H-X_ 2)/H$$, even when H is not a constant. In the first part of this paper we examine in detail the asymptotic validity of this assumption of the theory.
We develop a formal expansion which gives in the first term the Reynolds equation. Various rigorus estimates of the remainder are proved. We limit ourselves to the two-dimensional linear case (i.e., we consider as “exact” the solution given by the linear Stokes equations). The transition from the Stokes equations to the Reynolds equation in the three-dimensional case (but only to the first order) is studied in a paper by G. Bayada and M. Chambat [(*) Appl. Math. Optimization 14, 73-93 (1986)], which recently appeared and which we saw after completing the present work. No use is made in (*) of the technique of the stream function on which this paper is largely based.

### MSC:

 76D08 Lubrication theory 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q99 Partial differential equations of mathematical physics and other areas of application
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