Influence of wall slip in dilute suspensions

Navier's (1823) slip condition, where the tangential velocity is proportional to the shear stress, may apply at micro-scales for a viscous liquid on walls with modified surfaces, e.g. hydrophobic ones in water. Experiments by Lumma et al (2003) show the difficulties involved in measuring such a slip. At small scales, particle-wall hydrodynamic interactions are important. They are modeled here for a dilute suspension of spherical solid particles near a slip wall. Consider a translating and rotating sphere (on which the no-slip condition applies) in an ambient parabolic flow. Analytical solutions of Stokes equations for the various elementary flow fields were obtained (Feuillebois et al 2009, 2011) as series in bispherical coordinates. The coefficients in the series are solutions of an infinite linear system, which is solved by an extension of Thomas' algorithm, allowing to calculate a large number of terms. Accurate results are available for the force, torque and stresslet on a sphere, velocity of a freely moving sphere and diffusion tensor. The Aris-Taylor dispersion of Brownian particles in a shear flow near a slip wall gives a large bias in the measurement of slip (Vinogradova et al, 2009). It is calculated here from the advection-diffusion equation, using the expressions for the particle velocity and diffusion tensor near a slip wall.