  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.
