Hydrodynamic friction of fakir-like super-hydrophobic surfaces

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, the series can be interpreted as Riemann sums, and the slip length can be obtained analytically. For posts on a square lattice, our analytical results are in excellent quantitative agreement with previous numerical computations

Low-Reynolds number swimming in gels

Many microorganisms swim through gels, materials with nonzero zero-frequency elastic shear modulus, such as mucus. Biological gels are typically heterogeneous, containing both a structural scaffold (network) and a fluid solvent. We analyze the swimming of an infinite sheet undergoing transverse traveling wave deformations in the "two-fluid" model of a gel, which treats the network and solvent as two coupled elastic and viscous continuum phases. We show that geometric nonlinearities must be incorporated to obtain physically meaningful results. We identify a transition between regimes where the network deforms to follow solvent flows and where the network is stationary. Swimming speeds can be enhanced relative to Newtonian fluids when the network is stationary. Compressibility effects can also enhance swimming velocities. Finally, microscopic details of sheet-network interactions influence the boundary conditions between the sheet and network. The nature of these boundary conditions significantly impacts swimming speeds.Comment: 6 pages, 5 figures, submitted to EP