From adherence to slip in nanofluidics: a mathematical justification based on a drop of viscosity—the scalar case

Abstract

The question of the origin of the unusually large slip length that may appear in some nanofluidic experimental settings is the object of much debate in the physics community. An idea suggested by Myers (Microfluid Nanofluid 10(5):1141–1145, 2011) is that in carbon nanotubes, this phenomenon may originate from a viscosity drop of the fluid near the wall of the tube, in a very thin region called ”depletion layer”. In order to investigate this claim mathematically, we introduce as a first step a scalar diffusion model whose coefficient of diffusion tends to zero in the depletion layer, as the thickness of this region goes to zero. Using the natural energy bound satisfied by weak solutions $$u_\varepsilon $$ of the system, we develop an adaptation of the unfolding method (Arbogast et al in SIAM J Math Anal 21:823–836, 1990; Casado-Díaz in Proc R Soc Edinb 130(A):249–276, 2000; Cioranescu et al in C R Acad Sci Paris Sér I 335:99–104, 2002), (which is related to the two-scale convergence method (Allaire in SIAM J Math Anal 23:1482–1518, 1992; Nguetseng in SIAM J Math Anal 20:608–623, 1989)), based on the introduction of a fast variable and on a compactness argument applied to the corresponding sequence of rescaled solutions in the depletion region. This fine analysis of $$u_\varepsilon $$ allows us to construct an adapted sequence of test functions and to pass to the limit in the variational formulation of the initial problem, in order to derive an effective model. We conclude that a viscosity drop may result in a change of boundary condition akin to the apparition of a slip length.