Particles in Creeping Flow Near a Slip Wall

Abstract

The no‐slip boundary condition on a wall is the classical one for a viscous flow. But it has been recently recognized that a slip can take place at small scales, e.g., on hydrophobic surfaces. Slip is characterized by a slip length, that is the distance at which a fictitious no‐slip plane is moved away from the fluid to represent the slip plane in a shear flow. The slip length may be measured by following the motion of nanosized spheres close to such a surface. The motion of particles close to a slip surface also has applications in microfluidics, separation techniques in analytical chemistry, etc.With these applications in view, this paper considers a spherical solid particle in creeping flow close to a slip wall. The flow along the wall is locally approximated by a pure shear flow and the sphere is translating and rotating. The various perturbed flow fields around the sphere are solved analytically using the bispherical coordinates technique. The hydrodynamic force and torque on the sphere are obtained in terms of the slip length. Results are provided with a precision better than 10−7, even for a small gap down to 10−4 sphere radius.The translational and rotational velocities of a freely moving sphere in a pure shear flow near a slip wall are then calculated. The hindered Brownian diffusion of a freely rotating sphere close to a slip wall is characterized by a diffusion tensor, the coefficients of which are derived.Finally, the Aris‐Taylor dispersion of Brownian particles in a shear flow near a slip wall is calculated from the advection‐diffusion equation, using the expression for the particle velocity. For this purpose, the equation is first Fourier‐transformed in the direction of vorticity. The transformed two‐dimensional equation is solved by a finite elements package using a refined mesh.