Axisymmetric Stokes flow due to a point-force singularity acting between two coaxially positioned rigid no-slip disks

TL;DR

This paper provides a semi-analytical solution for the flow caused by a point-force between two coaxial no-slip disks, revealing flow structures, assessing approximation methods, and validating results with simulations.

Contribution

It introduces a semi-analytical method for solving Stokes flow between two disks, including flow structures and mobility effects, validated by finite-element simulations.

Findings

Viscous toroidal eddies are present between the disks.

The superposition approximation's validity depends on system geometry.

Good agreement between semi-analytical results and finite-element simulations.

Abstract

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.