The lift on a strongly non-spherical vesicle in a bounded shear flow, is studied in the case the membrane moves with a velocity, which is a linear function of the coordinates. The magnitude of the induced drift is calculated as a function of the axes lengths, of the distance from the wall, and of the ratio of the cell to the solvent viscosity. It appears that the main mechanism for lift, in the presence of tank-treading motions, is the fixed orientation of the vesicle with respect to the flow. Tank-treading vesicles in suspensions, flowing through narrow gaps under small Reynolds number conditions: Re = Lv/ν < 1 (with L the gap width, v the flow velocity and ν the viscosity) migrate away from the walls, with a velocity that is O(Re −1 ) larger than predicted by inertia.

Thin films can be effectively described by the lubrication approximation, in which the equation of motion is h, +(h h ")"=0.Here h is a necessarily positive quantity which represents the height or thickness of the film. Different values of n, especially 1, 2, and 3 correspond to different physical situations. This equation permits solutions in the form of traveling disturbances with a fixed form. If u is the propagation velocity, the resulting equation for the disturbance is uh =(h h ""). Here, quantitative and qualitative solutions to the equation are presented. The study has been limited to the intervals in x where the solutions are positive. It is found that transitions between different qualitative behaviors occur at n =3, 2,~, and 2. For example, if u is not zero, solitonlike solutions defined on a finite interval are only possible for n & 3. More specific results can be obtained. In the case in which the velocity is zero, solitons occur for n & 2. For n = 1, the region-& n is characterized by the presence of advancingfront solutions, with support on (-~, t). For n) 2, single-minimum solutions diverging at~are possible. The generic solution, present for all positive values of n, is a receding front, which diverges at finite x for n &0.

The behavior of a spheroidal vesicle, in a plane shear flow bounded from one side by a wall, is analysed when the distance from the wall is much larger than the spheroid radius. It is found that tank treading motions produce a transverse drift away from the wall, proportional to the spheroid eccentricity and the inverse square of the distance from the wall. This drift is independent of inertia, and is completely determined by the characteristics of the vesicle membrane. The relative strength of the contribution to drift from tank-treading motions and from the presence of inertial corrections, is discussed. J. Phys. A (Math. Gen.)In press

A dispersion relation for gravity waves in water covered by disk-like impurities embedded in a viscous matrix is derived. The macroscopic equations are obtained by ensemble-averaging the fluid equations at the disk scale in the asymptotic limit of long waves and low disk surface fraction. Various regimes are identified depending on the disk radii and the thickness and viscosity of the top layer. Semi-quantitative analysis in the close-packing regime suggests dramatic modification of the dynamics, with orders of magnitude increase in wave damping and wave dispersion. A simplified model working in this regime is proposed. Possible applications to wave propagation in ice-covered ocean is discussed and comparison with field data is provided.

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