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Some effects of the boundary roughness in a thin film flow. (English) Zbl 0651.76013
Boundary control and boundary variations, Proc. IFIP WG 7.2 Conf., Nice/France 1987, Lect Notes Control Inf. Sci. 100, 96-115 (1988).
[For the entire collection see Zbl 0634.00013.]
The aim of this work is the mathematical study of an incompressible viscous fluid flow between two surfaces in motion as the thickness of the gap tends to zero. The authors pay attention to the asymptotic behaviour of the solutions of the Stokes equation with respect to the gap height and to the choice of the boundary conditions. The method of homogenisation is used for Reynolds equation (roughness parameter tends to zero) and for Stokes equation in an oscillating domain. The relation between gap height and roughness periodic parameter is considered essential. The authors point out some open questions but their study is a very rigorous one.
Reviewer: C.-I.Gheorghiu

76D08 Lubrication theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76D10 Boundary-layer theory, separation and reattachment, higher-order effects