Effective leak conditions across a membrane

Abstract

The problem of effective boundary conditions for the flow of a viscous fluid across a type of permeable membranes is considered. Threshold leak conditions of subgradient type, introduced by [H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions, Res. Inst. Math. Sci. Kokyuroku 888 (1994), pp. 199–216; H. Fujita, A coherent analysis of Stokes flows under boundary conditions of friction type, J. Comput. Appl. Math. 149(1) (2002), pp. 56–79] are considered on the periodic solid part of the membrane: the normal velocity is zero unless the jump in the normal stress across the membrane reaches a yield limit. The effective conditions are of subgradient type with an effective yield limit, in the case of a densely distributed solid part, or of Navier type, in the case of dilute solid part; in the intermediate case the tangential slip cancels, whereas the normal velocity and stress are continuous. Unlike in the case of perforated walls [Sanchez-Palencia, Un problème de l'écoulement lent d'un fluide visqueux incompressible au travers d'une paroi finement perforée, in Les Méthodes de l'Homogénèisation: Théorie et Applications en Physique, Collection de la Direction des Études et Recherches d'Électricité de France, Vol. 57, Eyrolles, Paris, 1985, pp. 371–400; E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, Vol. 127, Springer-Verlag, Berlin, 1980], no stress concentrations are present.

Keywords:

• homogenization
• viscous fluids
• variational inequalities
• M-convergence
• leak conditions

AMS Subject Classifications::

• 35J86
• 74Q99
• 76M50

Acknowledgements

This research was partially supported by NSF Grant DMS-0807840.