  We investigate theoretically on the basis of the steady Stokes equations for
a viscous incompressible fluid the flow induced by a Stokeslet located on the
centre axis of two coaxially positioned rigid disks. The Stokeslet is directed
along the centre axis. No-slip boundary conditions are assumed to hold at the
surfaces of the disks. We perform the calculation of the associated Green's
function in large parts analytically, reducing the spatial evaluation of the
flow field to one-dimensional integrations amenable to numerical treatment. To
this end, we formulate the solution of the hydrodynamic problem for the viscous
flow surrounding the two disks as a mixed-boundary-value problem, which we then
reduce into a system of four dual integral equations. We show the existence of
viscous toroidal eddies arising in the fluid domain bounded by the two disks,
manifested in the plane containing the centre axis through adjacent
counterrotating eddies. Additionally, we probe the effect of the confining
disks on the slow dynamics of a point-like particle by evaluating the
hydrodynamic mobility function associated with axial motion. Thereupon, we
assess the appropriateness of the commonly-employed superposition approximation
and discuss its validity and applicability as a function of the geometrical
properties of the system. Additionally, we complement our semi-analytical
approach by finite-element computer simulations, which reveals a good
agreement. Our results may find applications in guiding the design of
microparticle-based sensing devices and electrokinetic transport in small scale
capacitors.
