Shearing of a stratified layer of amphiphilic (bio)molecules

With the goal to identify surface viscosity, this paper proposes the modelling of a shear flow within an annular channel whose floor is put in rotation while its two vertical cylindrical side walls are maintained stationary. The liquid surface at the top of the annular channel is covered by a layer of hydrophobic molecules. The flow is considered as permanent, axisymmetric and creeping. The ratio of the liquid depth to the outer radius is small enough (shallow flow) so that it is possible to develop a matched asymptotic model. In the rotating sub-phase, a core flow is therefore distinguished from the boundary layers along side walls. The modelling includes the possibility to take into account the impact of the radially-inwards molecular packing induced by centrifugation of the underlying bulk. More particularly, radial distribution of surface viscosity is taken into account via the transport equation for surface momentum. In this paper, the model by Mannheimer and Schechter [R.J. Mannheimer, R.S. Schechter, J. Colloid Interface Sci. 32 (2) (1970) 195] can be considered as revisited: a new integral formulation is made evident which enables a fair estimation of the Boussinesq number as well as a simple measurement of a stratified surface viscosity.