Plastron induced drag reduction and increased slip on a superhydrophobic sphere

On low contact angle hysteresis superhydrophobic surfaces, droplets of water roll easily. It is intuitively appealing, but less obvious, that when such material is immersed in water, the liquid will flow more easily across its surface. In recent experiments it has been demonstrated that superhydrophobic surfaces with the same high contact angle and low contact angle hysteresis may not, in fact, have the same drag reducing properties. A key performance parameter is whether the surface is able to retain a layer of air (i.e. a plastron) when fully immersed. In this report, we consider an analytical model of Stokes flow (i.e. low Reynolds number, Re, creeping flow) across a surface retaining a continuous layer of air. The system is based on a compound droplet model consisting of a solid sphere encased in a sheathing layer of air and is the extreme limit of a solid sphere with a superhydrophobic surface. We demonstrate that an optimum thickness of air exists at which the drag on this compound object is minimized and that the level of drag reduction can approach 20 to 30%. Physically, drag reduction is caused by the ability of the external flow to transfer momentum across the water-air interface generating an internal circulation of air within the plastron. We also show that the drag experienced by the plastron-retaining sphere can be viewed as equivalent to the drag on a non-plastron retaining sphere, but with the no-slip boundary condition replaced by a slip boundary condition. If the plastron layer becomes too thin, or the liquid-gas interface is rigidified, circulation is no longer possible and drag increases to the value expected for a solid object in direct contact with water. We discuss the implications of this physical understanding in terms of its general applicability to the intelligent design of drag reducing superhydrophobic surfaces at low Re. We emphasize that the length scales and connectivity of surface topography generating superhydrophobicity are also likely to determine whether a plastron is of a suitable size to reduce drag.