Steady azimuthal flow field induced by a rotating sphere near a rigid disk or inside a gap between two coaxially positioned rigid disks

TL;DR

This paper provides a theoretical and computational analysis of azimuthal flow induced by a rotating sphere near rigid disks, revealing how geometric confinement influences flow and particle mobility at low Reynolds numbers.

Contribution

It introduces a novel analytical and numerical framework for solving the viscous flow around a rotating sphere near rigid disks, including explicit solutions and mobility expressions.

Findings

Analytical solution for flow near a stationary disk.

Numerical solution for flow between two disks.

Good agreement with finite-element simulations.

Abstract

Geometric confinements play an important role in many physical and biological processes and significantly affect the rheology and behavior of colloidal suspensions at low Reynolds numbers. On the basis of the linear Stokes equations, we investigate theoretically and computationally the viscous azimuthal flow induced by the slow rotation of a small spherical particle located in the vicinity of a rigid no-slip disk or inside a gap between two coaxially positioned rigid no-slip disks of the same radius. We formulate the solution of the hydrodynamic problem as a mixed-boundary-value problem in the whole fluid domain, which we subsequently transform into a system of dual integral equations. Near a stationary disk, we show that the resulting integral equation can be reduced into an elementary Abel integral equation that admits a unique analytical solution. Between two coaxially positioned stationary disks, we demonstrate that the flow problem can be transformed into a system of two Fredholm integral equations of the first kind. The latter are solved by means of numerical approaches. Using our solution, we further investigate the effect of the disks on the slow rotational motion of a colloidal particle and provide expressions of the hydrodynamic mobility as a function of the system geometry. We compare our results with corresponding finite-element simulations and observe very good agreement.