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Laminar flow over a bubble mattress is expected to experience a significant reduction in friction since the individual surfaces of the bubbles are shear-free. However, if the bubbles are sufficiently curved, their protrusion into the fluid and along the flow direction can lead to an increase in friction as was recently demonstrated experimentally and computationally. We provide in this paper a simple model for this result. We consider a shear flow at low Reynolds number past a two-dimensional array of bubbles, and calculate analytically the effective slip length of the surface as function of the bubble geometry in the dilute limit. Our model is able to reproduce quantitatively the relationship between effective friction and bubble geometry obtained in numerical computations, and in particular: (a) The asymmetry in friction between convex and concave bubbles, and (b) the existence of a geometric transition from reduced to enhanced friction at a critical bubble protrusion angle

Planar flow in the interfacial region of an open porous medium is investigated by finding solutions for Stokes flow in a channel partially filled with an array of circular cylinders beside one wall. The cylinders are in a square array oriented across the flow and are widely spaced, so that the solid volume fraction ϕ is 0.1 or less. For this spacing, singularity methods are appropriate and so they are used to find solutions for both planar Couette flow and Poiseuille flow in the open portion of the channel. The solutions, accurate to O(ϕ), are used to calculate the apparent slip velocity at the interface, Us, and results obtained for Us are presented in terms of a dimensionless slip velocity. For shear-driven flow, this dimensionless quantity is found to depend only weakly on ϕ and to be independent of the height of the array relative to the height of the channel and independent of the cylinder size relative to the height of the channel. For pressure-driven flow, Us is found to be less than that under comparable shear-flow conditions, and dependent on cylinder size and filling fraction in this case. Calculations also show that the external flow penetrates the porous medium very little, even for sparse arrays, and that Us is about one quarter of the velocity predicted by the Brinkman model.

A fluid droplet located on a super-hydrophobic surface makes contact with the surface only at small isolated regions, and is mostly in contact with the surrounding air. As a result, a fluid in motion near such a surface experiences very low friction, and super-hydrophobic surfaces display strong drag-reduction in the laminar regime. Here we consider theoretically a super-hydrophobic surface composed of circular posts (so called fakir geometry) located on a planar rectangular lattice. Using a superposition of point forces with suitably spatially-dependent strength, we derive the effective surface slip length for a planar shear flow on such a fakir surface as the solution to an infinite series of linear equations. In the asymptotic limit of small surface coverage by the posts, φs, the series can be interpreted as Riemann sums, and the slip length can be obtained analytically. For posts on a square lattice of periodicity L, our analytical results predict that in the low φs limit, the surface slip length, λ, scales aswhich is in excellent quantitative agreement with previous numerical computations.

Marangoni propulsion is a form of locomotion wherein an asymmetric release of surfactant by a body located at the surface of a liquid leads to its directed motion. We present in this paper a mathematical model for Marangoni propulsion in the viscous regime. We consider the case of a thin rigid circular disk placed at the surface of a viscous fluid and whose perimeter has a prescribed concentration of an insoluble surfactant, to which the rest of its surface is impenetrable. Assuming a linearized equation of state between surface tension and surfactant concentration, we derive analytically the surfactant, velocity and pressure fields in the asymptotic limit of low Capillary, Peclet and Reynolds numbers. We then exploit these results to calculate the Marangoni propulsion speed of the disk. Neglecting the stress contribution from Marangoni flows is seen to over-predict the propulsion speed by 50%