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

Abdallah Daddi-Moussa-Ider<EMAIL_ADDRESS>Institut für Theoretische Physik
II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße
1, D–40225 Düsseldorf, Germany  Abteilung Physik lebender Materie, Max-Planck-
Institut für Dynamik und Selbstorganisation, Am Faßberg 17, D–37077 Göttingen,
Germany  Alexander R. Sprenger  Institut für Theoretische Physik II: Weiche
Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D–40225
Düsseldorf, Germany  Thomas Richter  Institut für Analysis und Numerik, Otto-
von-Guericke-Universität Magdeburg, Universitätsplatz 2, D–39106
Magdeburg,Germany  Hartmut Löwen  Institut für Theoretische Physik II: Weiche
Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D–40225
Düsseldorf, Germany  Andreas M. Menzel  Institut für Physik, Otto-von-
Guericke-Universität Magdeburg, Universitätsplatz 2, D–39106 Magdeburg,
Germany

(August 27, 2024)

###### 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.

##  I Introduction

Hydrodynamic interactions in viscous flows are ubiquitous in nature and find
numerous applications in various industrial and environmental processes.
Simultaneously, confinements play a pivotal role in a wide range of biological
and biotechnological processes, including the dynamics of polymer solutions
and melts in microfluidic devices Zhou  _et al._ (2019); Müller, Pastorino,
and Servantie (2009); Piette, Saengow, and Giacomin (2019); Kanso  _et al._
(2019), DNA translocation through pores Storm  _et al._ (2005); Izmitli  _et
al._ (2008), transport and rheology of red blood cell suspensions in
microcirculation Secomb  _et al._ (1986); Pozrikidis (2005); McWhirter,
Noguchi, and Gompper (2009); Freund (2014); Secomb (2017); Barthès-Biesel
(2016), colloidal gelation Furukawa and Tanaka (2010), biofilm formation in
microchannels Rusconi  _et al._ (2010, 2011); Drescher  _et al._ (2013); Kim
_et al._ (2014), and swimming behavior of active self-propelled agents in
viscous media Berke  _et al._ (2008); Yazdi and Borhan (2017); Bianchi,
Saglimbeni, and Di Leonardo (2017); Reigh  _et al._ (2017); Bianchi,
Saglimbeni, and Di Leonardo (2017); Daddi-Moussa-Ider  _et al._ (2018a); Dhar,
Burada, and Sekhar (2020).

Fluid flows at small length scales are characterized by low Reynolds numbers,
where viscous forces typically dominate inertial forces. Under such
conditions, the fluid dynamics can well be described by the linear Stokes
equations Kim and Karrila (2013). Over the past few decades, there has been
mounting interest in the theoretical and experimental characterization of the
behavior of hydrodynamically interacting particles near confining interfaces
Diamant (2009). These include, for instance, a flat rigid wall MacKay and
Mason (1961); Gotoh and Kaneda (1982); Cichocki and Jones (1998); Lauga and
Squires (2005); Swan and Brady (2007); Franosch and Jeney (2009); Felderhof
(2012); De Corato  _et al._ (2015); Huang and Szlufarska (2015); Rallabandi,
Hilgenfeldt, and Stone (2017), a planar surface with partial slip Lauga and
Squires (2005), a flat interface separating two immiscible fluids Lee,
Chadwick, and Leal (1979); Berdan II and Leal (1982); Bławzdziewicz, Ekiel-
Jeżewska, and Wajnryb (2010a, b), an interface covered with a surfactant
Bławzdziewicz, Cristini, and Loewenberg (1999); Blawzdziewicz, Wajnryb, and
Loewenberg (1999), a rough boundary characterized by random surface textures
Kurzthaler  _et al._ (2020), or a soft deformable membrane possessing elastic
and bending properties Felderhof (2006); Bickel (2006, 2007); Boatwright  _et
al._ (2014); Daddi-Moussa-Ider, Guckenberger, and Gekle (2016a); Daddi-Moussa-
Ider and Gekle (2016); Jünger  _et al._ (2015); Salez and Mahadevan (2015);
Daddi-Moussa-Ider, Lisicki, and Gekle (2017a, b); Daddi-Moussa-Ider and Gekle
(2018); Rallabandi  _et al._ (2018); Daddi-Moussa-Ider  _et al._ (2018b, c);
Hoell  _et al._ (2019). Thanks to the advent of new particle tracking and
measurement techniques, the field has benefited from important recent advances
in the characterization of the behavior of colloidal particles near
confinement at small scales Lobry and Ostrowsky (1996); Graham (2011);
Faucheux and Libchaber (1994); Lin, Yu, and Rice (2000); Tränkle, Ruh, and
Rohrbach (2016).

From a chronological standpoint, one of the first attempts to address the
creeping flow induced by a spherical particle confined between two infinitely
extended planar walls dates back to Faxén Faxén (1921). One century ago, Faxén
provided in his doctoral dissertation a few approximate analytical expressions
of the hydrodynamic mobility function for parallel translational motion in a
channel bounded by two flat plates. Later, using the method of images, Liron
and Mochon Liron and Mochon (1976) obtained in a pioneering work an exact
solution of the Stokes flow induced by a point-force singularity acting
between two parallel no-slip walls. The problem of fluid motion in a channel
bounded by two no-slip walls has further been addressed using the multipole
expansion technique Bhattacharya and Bławzdziewicz (2002); Swan and Brady
(2010) and a strong interaction theory Ganatos, Weinbaum, and Pfeffer (1980);
Ganatos, Pfeffer, and Weinbaum (1980).

In the present article, we proceed a step further by examining the low-
Reynolds-number flow induced by a point-like particle rotating near a rigid
finite-sized no-slip disk or between two coaxially positioned rigid finite-
sized no-slip disks of the same radius. Mathematically, we model the situation
using a rotlet (also called a point-torque or point-couple) singularity acting
on the surrounding fluid medium. We formulate the flow problem at hand as a
mixed-boundary-value problem which we subsequently transform into a system of
dual integral equations on the domain boundary. For their solution, we employ
conventional procedures outlined by Sneddon and Copson so as to express the
solutions of the flow problems in terms of convergent definite integrals.
Moreover, we quantify the effect of the confining finite-sized disks on the
rotational motion by calculating the effect on the corresponding hydrodynamic
mobility function.

The remainder of this article is organized as follows. In Sec. II we derive
the solution of the hydrodynamic equations for a rotlet singularity acting
near a fixed no-slip disk. We show that the induced velocity field can be
presented in a compact analytical form in terms of a definite one-dimensional
integral. Afterwards, we obtain in Sec. III a semi-analytical solution of the
flow problem inside a gap between two coaxially positioned rigid no-slip
disks. We demonstrate that the solution can be reduced into a system of two
Fredholm equations of the first kind that can be solved by means of standard
numerical approaches. Finally, concluding remarks are contained in Sec. IV.
Technical aspects and simulation details regarding the finite-element method
we employ to compare our theory with are shifted to the appendices.

##  II Solution near a single disk

###  II.1 Problem formulation

Figure 1: (Color online) Graphical illustration of the system setup. A point-
like particle undergoing slow rotational motion near a rigid no-slip disk of
radius $R$ sets the fluid into motion. The center of the particle is located
at a distance $h$ above the center of the disk, while the surrounding viscous
fluid medium is characterized by a dynamic shear viscosity $\eta$.
$\bm{L}=L\bm{\hat{e}}_{z}$ sets the torque acting via the particle at the
particle position on the fluid.

First, we examine the low-Reynolds-number dynamics of a point-like particle
undergoing rotational motion near one fixed finite-sized disk of radius $R$.
We assume a no-slip boundary condition to hold at the surface of the disk. In
addition, we suppose that the disk is located within the plane $z=0$ and that
the center of the disk coincides with the origin of our coordinate frame; see
Fig. 1 for an illustration of the system setup. In addition, we assume that
the fluid is incompressible and Newtonian with constant shear viscosity
$\eta$.

At low Reynolds numbers, the fluid dynamics is thus governed by the steady
Stokes equations Happel and Brenner (2012)

$\eta\bm{\nabla}^{2}\bm{v}-\bm{\nabla}p+\bm{F}_{\mathrm{B}}=\bm{0}\,,\qquad\bm{\nabla}\cdot\bm{v}=0\,,$
(1)

wherein $\bm{v}$ and $p$ denote the hydrodynamic velocity and pressure fields,
respectively. In addition, $\bm{F}_{\mathrm{B}}$ represents an arbitrary bulk
force density acting on the fluid at position $\bm{r}_{0}=h\bm{e}_{z}$ with
$\bm{e}_{z}$ denoting the unit vector directed along the $z$ direction. The
torque $\bm{L}$ on the particle is transmitted to the fluid and linked to the
surface force density $\bm{F}$ acting on the fluid via the surface of the
spherical particle

$\bm{L}=\oint_{A}\left(\bm{r}-\bm{r}_{0}\right)\times\bm{F}\,\mathrm{d}S\,,$
(2)

with $A$ denoting the surface area of the tiny particle. In the point-particle
approximation, the asymmetric dipolar term in the multipole expansion is
associated with the flow field induced by a rotlet singularity of strength
$\bm{L}$ acting above the disk at position $\bm{r}_{0}$. Here we consider the
case in which the point torque is directed along the axis of symmetry of the
disk and set $\bm{L}=L\bm{e}_{z}$.

´

In an unbounded (infinite) fluid medium, the flow field induced by a rotlet
singularity is given by

$\bm{v}^{\infty}(\bm{r})=\frac{1}{8\pi\eta}\frac{\bm{L}\times\bm{s}}{s^{3}}\,,$
(3)

wherein $\bm{s}=\bm{r}-\bm{r}_{0}$ and $s=|\bm{s}|$ is the distance from the
singularity position. Using cylindrical coordinates $(r,\phi,z)$, the
azimuthal component of the flow velocity field induced by a free-space rotlet
oriented along the $z$ direction reads

$v_{\phi}^{\infty}(r,z)=\frac{Kr}{\left(r^{2}+\left(z-h\right)^{2}\right)^{\frac{3}{2}}}\,,$
(4)

where we have defined, for convenience, the abbreviation
$K=L/\left(8\pi\eta\right)$ of dimension (length)3(time)-1.

The solution of the flow problem in the presence of the confining disk can
generally be expressed as a linear superposition of the solution in an
unbounded fluid medium, given by Eq. (4), and a complementary solution that is
required to satisfy the underlying regularity and boundary conditions for the
total induced flow field. Specifically,

$v_{\phi}=v_{\phi}^{\infty}+v_{\phi}^{*}\,,$ (5)

with $v_{\phi}^{*}$ standing for the complementary solution for the azimuthal
flow velocity, also sometimes called the image solution Blake (1971).

The solution of the homogeneous equations governing the fluid motion can be
expressed in our case in terms of a single harmonic function as
$v_{\phi}^{*}=-\partial\Omega/\partial r$ (c.f. Ref. Happel and Brenner, 2012,
Eq. (3–3.58), or Ref. Shail and Packham, 1987, Eq. (5), and references therein
for the expression of the complete solution) with $\Omega$ satisfying the
Laplace equation $\bm{\nabla}^{2}\Omega=0$. Then, the harmonic function
$\Omega$ can be written in the general form in terms of a Fourier-Bessel
integral of the form

$\Omega(r,z)=K\int_{0}^{\infty}\omega(\lambda)J_{0}(\lambda
r)e^{-\lambda|z|}\,\mathrm{d}\lambda\,,$ (6)

wherein $\omega(\lambda)$ is an unknown wavenumber-dependent function to be
subsequently determined from the prescribed boundary conditions. In addition,
$J_{\nu}$ denotes the Bessel function Abramowitz and Stegun (1972) of the
first kind of order $\nu$. The image solution for the azimuthal component of
the velocity field is then obtained as

$v_{\phi}^{*}(r,z)=K\int_{0}^{\infty}\lambda\omega(\lambda)J_{1}(\lambda
r)e^{-\lambda|z|}\,\mathrm{d}\lambda\,.$ (7)

Evidently, the solution form given by Eq. (7) satisfies the natural continuity
of the azimuthal velocity field at the plane $z=0$. We note that the rotlet
singularity does not induce a pressure gradient and that the radial and axial
components of the fluid velocity vanish. Therefore, the solution of the flow
problem reduces to the search for the azimuthal component of the velocity
field only.

###  II.2 Boundary conditions and dual integral equations

We require no-slip boundary conditions on the surface of the disk and assume
the continuity of the azimuthal component of the normal stress vector on the
plane $z=0$ outside the disk. Specifically,

$\displaystyle\left.v_{\phi}^{\infty}+v_{\phi}^{*}\right|_{z=0}$
$\displaystyle=0\quad\text{for}\quad r<R\,,$ (8a)
$\displaystyle\left.\eta\,\frac{\partial v_{\phi}^{*}}{\partial
z}\right|_{z=0^{+}}-\left.\eta\,\frac{\partial v_{\phi}^{*}}{\partial
z}\right|_{z=0^{-}}$ $\displaystyle=0\quad\text{for}\quad r>R\,.$ (8b)

By inserting the expressions of the free-space and image fields given by Eqs.
(4) and (7), respectively, into Eqs. (8), we obtain the mixed-boundary-value
problem on the inner and outer domain boundaries. Specifically,

$\displaystyle\int_{0}^{\infty}\lambda\omega(\lambda)J_{1}(\lambda
r)\,\mathrm{d}\lambda$ $\displaystyle=f(r)\quad\quad(r<R)\,,$ (9a)
$\displaystyle\int_{0}^{\infty}\lambda^{2}\omega(\lambda)J_{1}(\lambda
r)\,\mathrm{d}\lambda$ $\displaystyle=0\qquad\quad\,\,(r>R)\,,$ (9b)

with the radial function

$f(r)=-\frac{r}{\left(r^{2}+h^{2}\right)^{\frac{3}{2}}}\,,$ (10)

stemming from the free-space rotlet field.

The solution of the type of dual integral equations stated by Eqs. (9) can
generally be obtained using the theory of Mellin transforms Tranter (1951);
Titchmarsh (1948). We will follow in the present article a different route
based on the analytical approach outlined by Sneddon Sneddon (1960) and Copson
Copson (1961). In the sequel, we will show that the present dual integral
equations problem with Bessel function kernels can conveniently be reduced to
an elementary Abel integral equation that may readily be inverted. This
solution strategy has previously been employed to examine the low-Reynolds-
number flow induced by nonrotational force singularities near a finite-sized
elastic disk possessing shear and bending properties Daddi-Moussa-Ider, Kaoui,
and Löwen (2019); Daddi-Moussa-Ider (2020), the flow field near a no-slip disk
Daddi-Moussa-Ider  _et al._ (2020a); Kim (1983), or the axisymmetric flow due
to a Stokeslet acting between two coaxially positioned rigid no-slip disks
Daddi-Moussa-Ider  _et al._ (2020b).

We search a solution for the unknown wavenumber-dependent function
$\omega(\lambda)$ of the integral form

$\omega(\lambda)=\lambda^{-\frac{1}{2}}\int_{0}^{R}\hat{\omega}(t)J_{\frac{1}{2}}(\lambda
t)\,\mathrm{d}t\,,$ (11)

wherein $\hat{\omega}(t)$, with $t\in[0,R]$, is an unknown function later to
be determined. We will show in the sequel that the equation for the outer
problem (9b) is indeed satisfied using this form of solution.

First, it can readily be checked that Eq. (11) can further be expressed in the
form

$\omega(\lambda)=\lambda^{-\frac{3}{2}}\int_{0}^{R}\hat{\omega}(t)\,t^{-\frac{3}{2}}\frac{\mathrm{d}}{\mathrm{d}t}\left(t^{\frac{3}{2}}J_{\frac{3}{2}}(\lambda
t)\right)\mathrm{d}t\,.$ (12)

By defining

$\hat{F}(t)=t^{\frac{3}{2}}\,\frac{\mathrm{d}}{\mathrm{d}t}\left(t^{-\frac{3}{2}}\hat{\omega}(t)\right)$
(13)

and assuming that $t^{\frac{3}{2}}\,\hat{\omega}(t)\to 0$ as $t\to 0^{+}$, Eq.
(12) can be rewritten upon integration by parts as

$\omega(\lambda)=\lambda^{-\frac{3}{2}}\left(\hat{\omega}(R)J_{\frac{3}{2}}(\lambda
R)-\int_{0}^{R}\hat{F}(t)J_{\frac{3}{2}}(\lambda t)\,\mathrm{d}t\right).$ (14)

Then, by substituting the modified form of solution given by Eq. (14) into Eq.
(9b), the integral equation for the outer problem can be expressed as

$\mathcal{K}_{+}(r,R)\,\hat{\omega}(R)-\int_{0}^{R}\mathcal{K}_{+}(r,t)\hat{F}(t)\,\mathrm{d}t=0\quad(r>R)\,.$
(15)

In this context, we define the kernel functions

$\mathcal{K}_{\pm}(r,t)=\int_{0}^{\infty}\lambda^{\frac{1}{2}}J_{1\pm\frac{1}{2}}(\lambda
t)J_{1}(\lambda r)\,\mathrm{d}\lambda\,.$ (16)

It turned out that the latter improper (infinite) integral can be evaluated
analytically as

$\mathcal{K}_{+}(r,t)=\left(\frac{2}{\pi
t}\right)^{\frac{1}{2}}\frac{r}{t}\frac{\Theta(t-r)}{\left(t^{2}-r^{2}\right)^{\frac{1}{2}}}\,,$
(17)

with $\Theta(\cdot)$ denoting the Heaviside step function (or the unit step
function). Since $r>R$, it can readily be perceived that the transformed
integral equation for the outer problem stated by Eq. (15) is trivially
satisfied.

Thereafter, substituting Eq. (11) into the integral equation for the inner
problem given by Eq. (9a) yields

$\int_{0}^{R}\mathcal{K}_{-}(r,t)\,\hat{\omega}(t)\,\mathrm{d}t=f(r)\qquad(r<R)\,,$
(18)

By noting that

$\mathcal{K}_{-}(r,t)=\left(\frac{2t}{\pi}\right)^{\frac{1}{2}}\frac{1}{r}\frac{\Theta(r-t)}{\left(r^{2}-t^{2}\right)^{\frac{1}{2}}}\,,$
(19)

Eq. (18) can subsequently be rewritten in a much simplified form as

$\int_{0}^{r}\frac{t^{\frac{1}{2}}\hat{\omega}(t)}{\left(r^{2}-t^{2}\right)^{\frac{1}{2}}}\,\mathrm{d}t=\left(\frac{\pi}{2}\right)^{\frac{1}{2}}rf(r)\qquad(r<R)\,.$
(20)

Equation (20) is a classical Abel integral equation which constitutes a
special form of the linear Volterra equation of the first kind having a weakly
singular kernelCarleman (1921); Smithies (1958); Anderssen, De Hoog, and Lukas
(1980). It admits a unique solution if and only if $f(r)$ is a continuously
differentiable function Whittaker and Watson (1996); Carleman (1922); Tamarkin
(1930). Its solution is formally given in an integral form as (c.f. Appendix A
for further details)

$\hat{\omega}(t)=\left(\frac{2}{\pi
t}\right)^{\frac{1}{2}}\frac{\mathrm{d}}{\mathrm{d}t}\int_{0}^{t}\frac{v^{2}f(v)\,\mathrm{d}v}{\left(t^{2}-v^{2}\right)^{\frac{1}{2}}}\,.$
(21)

Inserting the expression of $f(r)$ stated by Eq. (10) into Eq. (21) and
performing the resulting integration yields

$\hat{\omega}(t)=-2\left(\frac{2t}{\pi}\right)^{\frac{1}{2}}\frac{ht}{\left(t^{2}+h^{2}\right)^{2}}\,.$
(22)

Evidently, the condition $t^{\frac{3}{2}}\,\hat{\omega}(t)\to 0$ as $t\to
0^{+}$ assumed above upon integrating by parts is well satisfied.

Next, by substituting Eq. (22) into Eq. (11) upon noting that

$J_{\frac{1}{2}}\left(\lambda
t\right)=\left(\frac{2}{\pi}\right)^{\frac{1}{2}}\left(\lambda
t\right)^{-\frac{1}{2}}\sin\left(\lambda t\right)\,,$ (23)

the unknown wavenumber-dependent function $\omega(\lambda)$ can be written in
a compact integral form as

$\omega(\lambda)=-\frac{4h}{\pi\lambda}\int_{0}^{R}\frac{t\sin(\lambda
t)\,\mathrm{d}t}{\left(t^{2}+h^{2}\right)^{2}}\,.$ (24)

The latter integral can be expressed in a general way in terms of hyperbolic
functions and trigonometric integrals. However, choosing the integral form is
more convenient for later treatment. In particular, it follows that
$\omega(\lambda)=-e^{-\lambda h}$ as $R\to\infty$. Finally, by inserting Eq.
(24) into Eq. (7) and interchanging the order of integration with respect to
the variables $\lambda$ and $t$, the solution for the image azimuthal velocity
field follows as

$v_{\phi}^{*}(r,z)=-\frac{4Kh}{\pi}\int_{0}^{R}\frac{t\,\mathcal{Q}(r,z,t)\,\mathrm{d}t}{\left(t^{2}+h^{2}\right)^{2}}\,,$
(25)

where we have defined the kernel function

$\mathcal{Q}(r,z,t)=\int_{0}^{\infty}J_{1}(\lambda r)\sin(\lambda
t)\,e^{-\lambda|z|}\,\mathrm{d}\lambda\,.$ (26)

To obtain an analytical expression for $\mathcal{Q}$, we proceed by making use
of Euler’s formula in complex analysis Ahlfors (1966) and write $\sin(\lambda
t)=\operatorname{Im}\left\\{e^{i\lambda t}\right\\}$, with Im denoting the
imaginary part of the argument. By using the change of variable $u=\lambda r$,
Eq. (26) can be rewritten in the form

$\mathcal{Q}(r,z,t)=\frac{1}{r}\operatorname{Im}\left\\{\int_{0}^{\infty}J_{1}(u)\,e^{-su}\,\mathrm{d}u\right\\},$
(27)

with $s=\left(|z|-it\right)/r$. By invoking the Laplace transform Widder
(2015) of $J_{1}(u)$ given as $1-s\left(1+s^{2}\right)^{-\frac{1}{2}}$, Eq.
(27) can then be evaluated as

$\mathcal{Q}(r,z,t)=-\frac{1}{r}\operatorname{Im}\left\\{\frac{|z|-it}{\left(r^{2}+\left(|z|-it\right)^{2}\right)^{\frac{1}{2}}}\right\\}.$
(28)

The latter can further be cast in the final form as

$\mathcal{Q}(r,z,t)=\frac{2^{\frac{1}{2}}}{2rU}\left(t\left(U+V\right)^{\frac{1}{2}}-|z|\left(U-V\right)^{\frac{1}{2}}\right),$
(29)

where we have defined

$U=\left(\left(\rho^{2}+t^{2}\right)^{2}-4r^{2}t^{2}\right)^{\frac{1}{2}}\,,\qquad
V=\rho^{2}-t^{2}\,,$ (30)

where $\rho^{2}=r^{2}+z^{2}$. It can be shown that $\mathcal{Q}$ is always
well defined except when $U=0$, for which $z=0$ and $t=r$. In this case,
$\mathcal{Q}(r,z,t)\sim\left(r-t\right)^{-\frac{1}{2}}\Theta\left(r-t\right)$.
We note that $Q(r,z,t)\to 0$ as $r\to 0$.

An analytical integration of Eq. (25) is delicate, if not downright
impossible. Therefore, recourse to numerical procedures is necessary. To this
end, we approximate the integral by a standard middle Riemann sum using the
partition $t_{1},\dots,t_{N}$, where $t_{i}=\left(i-1/2\right)\delta$, with
$i=1,\dots,N$ and $\delta=R/N$. Here, $N$ denotes the number of discretization
points. Throughout this work, we consistently set $N=10\,000$.

Figure 2: (Color online) Contour plot of the amplitude of the scaled azimuthal
flow velocity as obtained theoretically [first row, $(a)$, $(b)$, and $(c)$]
and using finite-element simulations [second row, $(d)$, $(e)$, and $(f)$].
The flows are induced by a rotlet singularity positioned at $h/R=4$ [$(a)$ and
$(d)$], $h/R=1$ [$(b)$ and $(e)$] and $h/R=0.25$ [$(c)$ and $(f)$] on the axis
of a no-slip disk of radius $R$ (red). The scaled azimuthal velocity is
defined as $V_{\phi}=v_{\phi}/\left(L/\left(8\pi\eta R^{2}\right)\right)$.
Results are presented on a decimal logarithmic scale.

In Fig. 2, we show contour plots of the scaled azimuthal flow velocity induced
by a rotlet singularity positioned at different distances along the axis of
the disk. The analytical predictions [$(a)$, $(b)$, and $(c)$] are in good
agreement with the results from finite-element simulations [$(d)$, $(e)$, and
$(f)$; see Appendix B for technical details regarding the simulation method].

The magnitude of the scaled velocity field is shown on a logarithmic scale to
emphasize the difference between the different regions. Similarly, as in the
bulk, the flow velocity is decaying faster along the $z$ direction (or axis of
rotation) compared to the radial direction. However, near the disk, the
azimuthal flow velocity becomes asymmetric with respect to the rotlet
position, and for $z<0$ the total flow field almost vanishes completely. Last,
we remark that the overall magnitude of the flow field reduces as the rotlet
gets closer to the disk.

###  II.3 Exact solution for $R\to\infty$

The solution for a rotlet oriented normal to a hard wall can be obtained using
the image system technique noted by Blake Blake (1971). For completeness, we
here derive this solution in a different way using our formalism. For an
infinitely large disk located at $z=0$, the integral equations (9a) for the
inner domain hold for all positive $r$, and thus we can directly apply a
Hankel transform Piessens (2000) on both sides of this equation. Here, we make
use of the orthogonality property of Bessel functions Abramowitz and Stegun
(1972)

$\int_{0}^{\infty}rJ_{\nu}(\lambda
r)J_{\nu}(\lambda^{\prime}r)\,\mathrm{d}r=\lambda^{-1}\delta(\lambda-\lambda^{\prime})\,,$
(31)

and obtain the solution for

$\omega(\lambda)=\int_{0}^{\infty}rf(r)J_{1}(\lambda
r)\,\mathrm{d}r=-e^{-\lambda h}\,.$ (32)

Inserting the latter result into Eq. (7) yields

${v_{\phi}^{*}}_{\text{Blake}}(r,z)=-\frac{Kr}{\left(r^{2}+\left(|z|+h\right)^{2}\right)^{\frac{3}{2}}}\,.$
(33)

The image solution for the azimuthal component in the limit $R\to\infty$ can
alternatively be calculated analytically from Eq. (25). Consequently, the
total flow field vanishes underneath the disk, i.e. for $z<0$.

Figure 3:  (Color online) Percentage relative error in the flow field as
obtained using Blake’s solution for an infinitely extended disk in comparison
to the exact solution for a disk of finite size, as a function of increasing
vertical distance $h/R$ of the rotlet from the disk. The flow field is
evaluated at five different values of $z/h$ while keeping $r/R=0.2$.

Figure 3 shows the variation of the percentage relative error in the flow
velocity field as obtained using Blake’s solution given by Eq. (33) in
comparison to the exact solution derived in the present work for a finite-
sized disk given in an integral form by Eq. (25). Here, the flow field is
evaluated at five different axial distances $z/h$, while keeping the radial
position $r/R=0.2$. We observe that the error is vanishingly small for $h\ll
R$ and increases monotonically as the ratio $h/R$ gets larger. In addition,
the error amounts to small values in the fluid domain close to the singularity
position in which the flow velocity is primarily determined by the infinite-
space rotlet. Upon increasing $z/h$, the maximum percentage relative error
(MPRE) increases and it was found to be as high as approximately 55% for
$z/h=10$ and $h/R=10^{2}$. We have systematically checked that the MPRE is, in
general, less sensitive to variations in the radial position.

###  II.4 Hydrodynamic rotational mobility

Having derived the solution of the flow problem for a point-torque singularity
acting near a finite-sized disk, we next investigate how the presence of the
nearby disk affects the rotational mobility. For this purpose, we think of the
rotlet generated by a small colloidal particle of radius $a$. By restricting
ourselves to the situation in which $a\ll h$, the leading-order correction to
the particle rotational mobility can be obtained by evaluating the image
angular velocity of the fluid, $\tfrac{1}{2}\,\bm{\nabla}\times\bm{v}^{*}$, at
the singularity position Swan and Brady (2007). In a scaled form, it can be
presented as

$\frac{\Delta\mu}{\mu_{0}}=\frac{a^{3}}{K}\lim\limits_{(r,z)\to(0,h)}\frac{1}{2r}\frac{\partial}{\partial
r}\left(rv_{\phi}^{*}\right)\,,$ (34)

wherein $\mu_{0}=\left(8\pi\eta a^{3}\right)^{-1}$ is the bulk rotational
mobility, i.e., in the absence of the disk.

Following the notation employed in our previous considerations Daddi-Moussa-
Ider, Kaoui, and Löwen (2019); Daddi-Moussa-Ider (2020); Daddi-Moussa-Ider
_et al._ (2020a), we define the positive dimensionless number $k_{1}$ to be
the scaled correction factor to the mobility near a no-slip disk as

$k_{1}=-\left.\frac{\Delta\mu}{\mu_{0}}\middle/\left(\frac{a}{h}\right)^{3}\right.,$
(35)

and substituting Eq. (7) expressing the image azimuthal velocity into Eq.
(34), we obtain

$k_{1}=-\frac{h^{3}}{2}\int_{0}^{\infty}\lambda^{2}\omega(\lambda)e^{-\lambda
h}\,\mathrm{d}\lambda\,.$ (36)

Next, by inserting the expression of $\omega(\lambda)$ stated by Eq. (24) into
Eq. (36) and using the changes of variables $u=\lambda h$ and $v=t/R$, the
scaled correction factor can be presented as an integral over the interval
$[0,1]$ as

$k_{1}(\xi)=\frac{2}{\pi}\,\xi^{2}\int_{0}^{1}\frac{v\,G(v,\xi)\,\mathrm{d}v}{\left(v^{2}+\xi^{2}\right)^{2}}\,.$
(37)

with the dimensionless number $\xi=h/R$. Here,

$G(v,\xi)=\int_{0}^{\infty}u\sin\left(\frac{uv}{\xi}\right)e^{-u}\,\mathrm{d}u=\frac{2v\xi^{3}}{\left(v^{2}+\xi^{2}\right)^{2}}\,.$
(38)

Finally, evaluating the definite integral in Eq. (37) yields the expression of
the scaled correction factor as

$k_{1}(\xi)=\frac{1}{8}-\frac{1}{4\pi}\left(\arctan\xi+\frac{\xi\left(\xi^{2}-3\right)\left(1+3\xi^{2}\right)}{3\left(1+\xi^{2}\right)^{3}}\right).$
(39)

In particular, for $\xi\ll 1$ (or $h\ll R$), we obtain

$k_{1}(\xi)=\frac{1}{8}-\frac{4}{5\pi}\,\xi^{5}+\mathcal{O}\left(\xi^{7}\right)\,.$
(40)

Notably, the familiar correction factor $k_{1}=1/8$ near an infinitely
extended hard wall is recovered in the limit $\xi\to 0$. For $\xi\gg 1$, we
get

$k_{1}(\xi)=\frac{4}{3\pi}\,\xi^{-3}+\mathcal{O}\left(\xi^{-5}\right)\,.$ (41)

Interestingly, the correction factor takes a particularly simple expression
when $\xi=\sqrt{3}$, for which $k_{1}=1/24$.

Figure 4: Scaled correction factor $k_{1}$ to the rotational hydrodynamic
mobility near a rigid no-slip disk, Eq. (39), versus the dimensionless number
$\xi=h/R$ plotted on a semilogarithmic scale. The same curve is shown in the
inset on a log-log scale, where the scaling law $\xi^{-3}$ is displayed in the
range $\xi\gg 1$ (gray).

In Fig. 4, we present the variation of the scaled correction factor given by
Eq. (39) as a function of the dimensionless number $\xi=h/R$. We observe that
the scaled correction factor is a monotonically decaying function of $\xi$. On
a semilogarithmic scale (main plot), the curve exhibits an inverse logistic-
like (sigmoid) evolution between two plateau values. The scaled correction
factor undergoes a cubic decay with $\xi$ while vanishing in the limit
$\xi\to\infty$.

Recapitulating, we have presented a dual integral equation approach to
determine the solution of the hydrodynamic equations for a point-torque
singularity acting near a rigid disk. In the following, we will employ a
similar technique to obtain the corresponding solution of the flow problem in
the presence of two coaxially positioned rigid disks.

##  III Solution for two coaxially positioned disks

Figure 5: (Color online) Slow rotational motion of a point-like particle
confined between two coaxially positioned rigid no-slip disks of identical
radius $R$. The confining disks are located within the planes $z=\pm H/2$ with
$H$ denoting the separation distance between the parallel disks. The particle
is located at a distance $h$ from the origin of the system of coordinates on
the axis of the disks.

###  III.1 Problem formulation

We now assume that two parallel coaxially positioned rigid disks are located
within the planes at $z=-H/2$ and $z=H/2$, where $H$ represents the distance
separating the two disks. The $z$ axis passes through the centers of the
coaxially positioned disks. We suppose that the rotlet is acting between the
disks at position $\bm{r}_{0}=h\bm{e}_{z}$, where $-H/2<h<H/2$; see Fig. 5 for
a graphical illustration of the setup.

To find a solution to the flow problem, we partition the fluid medium into
three distinct parts. We label by the superscript 1 the flow velocity field in
the fluid domain beneath the plane $z=-H/2$ containing the lower disk,
subscript 2 the fluid region delimited by the planes at $z=-H/2$ and $z=H/2$,
and we designate by the subscript 3 the fluid domain above the plane $z=H/2$
containing the top disk. In the remainder of this article, we choose, for
convenience, to scale all lengths by the gap width $H.$

We now express the solution for the azimuthal velocity field in each region of
the fluid domain as

$\displaystyle{v_{\phi}^{*}}_{1}$
$\displaystyle=K\int_{0}^{\infty}A(\lambda)e^{\lambda z}J_{1}(\lambda
r)\,\mathrm{d}\lambda\,,$ (42a) $\displaystyle{v_{\phi}^{*}}_{2}$
$\displaystyle=K\int_{0}^{\infty}\left(B(\lambda)e^{-\lambda
z}+C(\lambda)e^{\lambda z}\right)J_{1}(\lambda r)\,\mathrm{d}\lambda\,,$ (42b)
$\displaystyle{v_{\phi}^{*}}_{3}$
$\displaystyle=K\int_{0}^{\infty}D(\lambda)e^{-\lambda z}J_{1}(\lambda
r)\,\mathrm{d}\lambda\,,$ (42c)

where $A(\lambda)$, $B(\lambda)$, $C(\lambda)$, and $D(\lambda)$ are unknown
functions to be determined from the underlying boundary conditions. It can
readily be checked that the regularity condition of a finite velocity field is
inherently satisfied in the whole domain.

###  III.2 Boundary conditions and dual integral equations

Requiring the natural continuity of the velocity field at the planes $z=\pm
1/2$ yields the expressions of the functions associated with the intermediate
fluid domain in terms of those related to the lower and upper domains.
Specifically,

$\displaystyle B(\lambda)$
$\displaystyle=\frac{1}{2}\left(A(\lambda)-D(\lambda)e^{-\lambda}\right)\operatorname{csch}(\lambda)\,,$
(43a) $\displaystyle C(\lambda)$
$\displaystyle=\frac{1}{2}\left(D(\lambda)-A(\lambda)e^{-\lambda}\right)\operatorname{csch}(\lambda)\,,$
(43b)

with csch denoting the hyperbolic cosecant function, defined as
$\operatorname{csch}(\lambda)=1/\sinh\lambda=2/\left(e^{\lambda}-e^{-\lambda}\right)$.

By imposing the no-slip boundary condition at the surfaces of the two disks,
we obtain the equations for the inner problem for $r<R$ as

$\displaystyle\int_{0}^{\infty}A(\lambda)e^{-\frac{\lambda}{2}}J_{1}(\lambda
r)\,\mathrm{d}\lambda$ $\displaystyle=\psi_{+}(r)\,,$ (44a)
$\displaystyle\int_{0}^{\infty}D(\lambda)e^{-\frac{\lambda}{2}}J_{1}(\lambda
r)\,\mathrm{d}\lambda$ $\displaystyle=\psi_{-}(r)\,,$ (44b)

where we have defined the radially symmetric functions

$\psi_{\pm}(r)=-\frac{r}{\left(r^{2}+\left(h\pm\frac{1}{2}\right)^{2}\right)^{\frac{3}{2}}}\,.$
(45)

In addition, the continuity of the azimuthal stress vector outside the regions
containing the disk yields the equations for the outer problem for $r>R$.
Specifically,

$\displaystyle\int_{0}^{\infty}\lambda\left(A(\lambda)e^{\frac{\lambda}{2}}-D(\lambda)e^{-\frac{\lambda}{2}}\right)\operatorname{csch}(\lambda)J_{1}(\lambda
r)\,\mathrm{d}\lambda$ $\displaystyle=0,$ (46a)
$\displaystyle\int_{0}^{\infty}\lambda\left(A(\lambda)e^{-\frac{\lambda}{2}}-D(\lambda)e^{\frac{\lambda}{2}}\right)\operatorname{csch}(\lambda)J_{1}(\lambda
r)\,\mathrm{d}\lambda$ $\displaystyle=0.$ (46b)

Equations (44) and (46) constitute a system of dual integral equations for the
unknown functions $A(\lambda)$ and $D(\lambda)$. For its solution, we employ
the standard solution approach outlined by Sneddon Sneddon (1960) and Copson
Copson (1961) and set

$\displaystyle\frac{1}{2}\left(A(\lambda)e^{\frac{\lambda}{2}}-D(\lambda)e^{-\frac{\lambda}{2}}\right)\operatorname{csch}(\lambda)$
$\displaystyle=\lambda^{\frac{1}{2}}f_{1}(\lambda)\,,$ (47a)
$\displaystyle\frac{1}{2}\left(A(\lambda)e^{-\frac{\lambda}{2}}-D(\lambda)e^{\frac{\lambda}{2}}\right)\operatorname{csch}(\lambda)$
$\displaystyle=\lambda^{\frac{1}{2}}f_{2}(\lambda)\,,$ (47b)

where

$f_{i}(\lambda)=\int_{0}^{R}\hat{f}_{i}(t)J_{\frac{1}{2}}(\lambda
t)\,\mathrm{d}t\,,$ (48)

with $\hat{f}_{i}(t),i\in\\{1,2\\}$, are two unknown functions defined on the
interval $[0,R]$ to be subsequently determined. In this way, the equations for
the outer problem are automatically satisfied following the same reasoning in
Sec. II. Solving Eqs. (47) for $A(\lambda)$ and $D(\lambda)$ yields

$\displaystyle
A(\lambda)=\lambda^{\frac{1}{2}}\left(f_{1}(\lambda)e^{\frac{\lambda}{2}}-f_{2}(\lambda)e^{-\frac{\lambda}{2}}\right),$
(49a) $\displaystyle
D(\lambda)=\lambda^{\frac{1}{2}}\left(f_{1}(\lambda)e^{-\frac{\lambda}{2}}-f_{2}(\lambda)e^{\frac{\lambda}{2}}\right).$
(49b)

Upon substitution of Eqs. (49) into Eqs. (44), the inner problem can be
expressed in the form

$\displaystyle\int_{0}^{R}\left(\mathcal{K}_{-}(r,t)\hat{f}_{1}(t)-\mathcal{S}(r,t)\hat{f}_{2}(t)\right)\mathrm{d}t$
$\displaystyle=\psi_{+}(r)\,,$ (50a)
$\displaystyle\int_{0}^{R}\left(\mathcal{S}(r,t)\hat{f}_{1}(t)-\mathcal{K}_{-}(r,t)\hat{f}_{2}(t)\right)\mathrm{d}t$
$\displaystyle=\psi_{-}(r)\,.$ (50b)

Here, we have defined the kernel function

$\mathcal{S}(r,t)=\left(\frac{2}{\pi
t}\right)^{\frac{1}{2}}\mathcal{Q}(r,1,t)\,,$ (51)

where $\mathcal{Q}$ has been defined earlier by Eq. (27). We note that the
expression of the kernel function $\mathcal{K}_{-}$ has previously been given
by Eq. (19) and can further be expressed in term of $\mathcal{Q}$ as

$\mathcal{K}_{-}(r,t)=\left(\frac{2}{\pi
t}\right)^{\frac{1}{2}}\mathcal{Q}(r,0,t)\,.$ (52)

Equations (50) represent a system of Fredholm integral equations of the first
kind Tricomi (1985). Due to the complicated expressions of their kernel
functions, exact analytical expressions for the unknown functions
$\hat{f}_{1}(t)$ and $\hat{f}_{2}(t)$ are far from trivial. Following a
computational approach, we partition the integration intervals $[0,R]$ into
$N$ subintervals, approximating the integrals by the standard middle Riemann
sum. We then evaluate the two resulting equations at $N$ discrete values of
$r$ that are distributed uniformly over the interval $[0,R]$. Inverting the
resulting linear system of $2N$ independent equations, we obtain accurate
values of $\hat{f}_{1}(t)$ and $\hat{f}_{2}(t)$ at each discretization point.

Inserting the expressions of $A(\lambda)$ and $D(\lambda)$ stated by Eqs. (49)
into Eqs. (42), the image solution for the azimuthal flow field everywhere in
the fluid domain can be cast in the final compact form

$v_{\phi}^{*}(r,z)=K\int_{0}^{R}\left(\frac{2}{\pi
t}\right)^{\frac{1}{2}}\hat{g}(r,z,t)\,\mathrm{d}t\,,$ (53)

wherein

$\hat{g}(r,z,t)=\mathcal{Q}\left(r,z+\frac{1}{2},t\right)\hat{f}_{1}(t)-\mathcal{Q}\left(r,z-\frac{1}{2},t\right)\hat{f}_{2}(t)\,.$

Notably, the system of Fredholm integral equations stated by Eqs. (50) is
recovered when enforcing the no-slip condition at $z=\pm 1/2$.

Likewise, the definite integral given by Eq. (53) can be discretized via the
middle Riemann sum to yield an approximate solution for the image velocity
field at any point $(r,z)$ in the entire fluid domain.

Figure 6: (Color online) Contour plots of the amplitude of the scaled
azimuthal velocity as obtained semi-analytically [first row, $(a)$ through
$(d)$] and by means of finite-element simulations [second row, $(e)$ through
$(h)$]. Results are shown for a rotlet acting at $h/H=0$ [$(a)$ and $(e)$],
$h/H=0.1$ [$(b)$ and $(f)$], $h/H=0.2$ [$(c)$ and $(g)$] and $h/H=0.3$ [$(d)$
and $(h)$] on the axis of two coaxially positioned no-slip disks of radius
$R=H$ (red). Here, the scaled azimuthal velocity is defined as
$V_{\phi}=v_{\phi}/\left(L/\left(8\pi\eta R^{2}\right)\right)$ and results are
presented on a decimal logarithmic scale.

Figure 6 shows contour plots of the amplitude of the scaled azimuthal flow
velocity for different positions on the axis of two coaxial disks, as obtained
analytically and by means of finite-element simulations. Again, the flow
velocity is the highest in the near vicinity of the rotlet and decays faster
along the $z$ direction compared to the radial direction. Moreover, the
magnitude is substantially smaller above and below the disks. As the rotlet
approaches one side of the confining disks, the overall magnitude of the flow
field becomes reduced and the structure becomes more asymmetric. Good
agreement is obtained between the semi-analytical theory and the finite-
element simulations.

###  III.3 Solution for $R\to\infty$

For completeness, we additionally address by our approach the solution for the
flow field in a gap bounded by two infinitely extended planar walls located at
$z=\pm 1/2$. To this end, a Hankel transform is applied on both sides of Eqs.
(44). We obtain

$\displaystyle A(\lambda)$ $\displaystyle=\lambda
e^{\frac{\lambda}{2}}\bar{\psi}^{+}(\lambda)=-\lambda e^{-\lambda h},$ (54a)
$\displaystyle D(\lambda)$ $\displaystyle=\lambda
e^{\frac{\lambda}{2}}\bar{\psi}^{-}(\lambda)=-\lambda e^{\lambda h},$ (54b)

with

$\bar{\psi}^{\pm}(\lambda)=\int_{0}^{\infty}r\psi^{\pm}(r)J_{1}(\lambda
r)\,\mathrm{d}r=-e^{-\lambda(\frac{1}{2}\pm h)},$ (55)

for $|h|<1/2$. Then, it follows from Eqs. (43) that the wavenumber-dependent
function associated with the fluid domain bounded by the planes $z=\pm 1/2$
are given by

$\displaystyle B(\lambda)$ $\displaystyle=-\frac{\lambda}{2}\left(e^{-\lambda
h}-e^{-\lambda\left(1-h\right)}\right)\operatorname{csch}(\lambda)\,,$ (56a)
$\displaystyle C(\lambda)$ $\displaystyle=-\frac{\lambda}{2}\left(e^{\lambda
h}-e^{-\lambda\left(1+h\right)}\right)\operatorname{csch}(\lambda)\,.$ (56b)

Figure 7:  (Color online) Contour plots of the amplitude of the scaled
azimuthal velocity induced by a rotlet singularity acting between two
infinitely extended no-slip walls. Again, the scaled azimuthal velocity is
defined as $V_{\phi}=v_{\phi}/\left(L/\left(8\pi\eta R^{2}\right)\right)$.

Finally, the corresponding solution of the image flow field can be obtained by
inserting Eqs. (54) and (56) into Eqs. (42). As expected, the total velocity
in the lower and upper regions vanishes in the limit $R\to\infty$. Figure 7
illustrates exemplary contour plots of the azimuthal flow field induced by a
rotlet acting between two infinitely extended no-slip walls for three
different positions of the singularity.

###  III.4 Hydrodynamic rotational mobility

The scaled correction to the particle rotational mobility in the point-
particle approximation can be obtained from the image velocity field via Eq.
(34). Between two coaxially positioned disks, we alternatively choose to
define the scaled correction factor as

$k_{2}=-\left.\frac{\Delta\mu}{\mu_{0}}\middle/a^{3}\right.\,.$ (57)

Then, it follows from Eq. (53) that the scaled correction to leading order can
conveniently be expressed as

$k_{2}=\frac{1}{2}\left(\frac{2}{\pi}\right)^{\frac{1}{2}}\int_{0}^{R}t^{-\frac{1}{2}}\big{(}\chi_{-}(t)\hat{f}_{2}(t)-\chi_{+}(t)\hat{f}_{1}(t)\big{)}\mathrm{d}t\,,$
(58)

where we have defined

$\chi_{\pm}(t)=\operatorname{Im}\left\\{\left(\frac{1}{2}\pm
h-it\right)^{-2}\right\\}.$ (59)

Again, an approximate evaluation of the definite integral given by Eq. (58)
can be performed by numerical discretization via the standard middle Riemann
sum.

In the limit of an infinitely extended channel $R\to\infty$, we obtain

$k_{2}=\frac{1}{2}\int_{0}^{\infty}\lambda^{2}\operatorname{csch}(\lambda)\left(\cosh\left(2\lambda
h\right)-e^{-\lambda}\right)\mathrm{d}\lambda\,.$ (60)

Using computer algebra systems such as Mathematica Wolfram (1991), the latter
can further be expressed as

$k_{2}=\frac{1}{8}\left(\zeta\left(3,\frac{1}{2}+h\right)+\zeta\left(3,\frac{1}{2}-h\right)-2\zeta(3)\right),$
(61)

wherein

$\zeta\left(s,t\right)=\sum_{n>m\geq 1}n^{-s}m^{-t}$ (62)

denotes the double zeta functions with $s>1$ and $t\geq 0$. Moreover,
$\zeta(s)$ is the Riemann zeta function with $s>1$ defined as
$\zeta(s)=\sum_{n\geq 1}n^{-s}$. In particular, $\zeta(3)$ is an irrational
number known as Apéry’s constant Van der Poorten and Apéry (1979). We quote
the famous identity derived by Euler $\zeta(3)=\zeta(2,1)$. In the mid-plane
of the channel, the correction factor reaches its minimum value
$k_{2}(h=0)=\frac{3}{2}\,\zeta(3)\approx 1.8031$. Performing a Taylor
expansion near the upper wall around $h=1/2$, we obtain

$k_{2}=\frac{1}{8}\,\epsilon^{-3}+\frac{3}{2}\,\zeta(5)\epsilon^{2}+\mathcal{O}\left(\epsilon^{4}\right),$
(63)

where $\epsilon=\frac{1}{2}-h$.

Figure 8: (Color online) Scaled correction factor stated by Eq. (61) versus
$R/H$ for various singularity positions along the axis of two coaxially
positioned rigid no-slip disks. Horizontal dashed lines correspond to the
scaled correction factors inside an infinitely extended channel. The lines
shown in gray displays the scaling law $k_{2}\sim\left(R/H\right)^{3}$ in the
range of small values of $R\ll H$ (c.f. Appendix C for the derivation of this
scaling relation.)

In Fig. 8, we present on a log-log scale the variation of the scaled
correction factor to the rotational hydrodynamic mobility given by Eq. (58)
versus the scaled radius of the coaxially positioned rigid disks for various
values of the singularity position within the gap between the two disks. The
correction factor increases monotonically upon increasing the size of the
disks because the rotational motion of the confined particle becomes more
restricted. In the limit of infinitely large disks, the correction factor
asymptotically tends to the value given by Eq. (61).

####  III.4.1 Superposition approximation

In the presence of two sufficiently separated confining boundaries, the
correction to the hydrodynamic mobility can sometimes be approximated by
superimposing the individual contributions arising from each boundary
Dufresne, Altman, and Grier (2001); Benesch, Yiacoumi, and Tsouris (2003);
Polin, Grier, and Quake (2006); Daddi-Moussa-Ider, Guckenberger, and Gekle
(2016b); Mathijssen  _et al._ (2016); Daddi-Moussa-Ider  _et al._ (2018d);
Daddi-Moussa-Ider, Löwen, and Liebchen (2021). This superposition
approximation has originally been introduced by Oseen Oseen (1927) to estimate
the translational hydrodynamic mobility in a channel bounded by two plates. In
this approach, the scaled correction factor can be approximated by

$k_{2}^{\mathrm{Sup}}=R^{-3}\left(\xi_{-}^{-3}k_{1}\left(\xi_{-}\right)+\xi_{+}^{-3}k_{1}\left(\xi_{+}\right)\right),$
(64)

wherein $\xi_{\pm}=\left(\frac{1}{2}\pm h\right)/R$ with $k_{1}$ representing
the scaled correction factor near a single disk given by Eq. (39). In
particular, near the upper wall, a Taylor expansion around $h=1/2$ leads to

$k_{2}^{\mathrm{Sup}}=\frac{1}{8}\,\epsilon^{-3}+k_{1}\left(R^{-1}\right)+\mathcal{O}\left(\epsilon\right)\,.$
(65)

Figure 9: (Color online) Percentage relative error of the scaled correction
factor of the rotational hydrodynamic mobility between two coaxially
positioned rigid no-slip disks as obtained using the simplistic superposition
approximation stated by Eq. (64) in comparison to the exact formula given by
Eq. (58).

Figure 9 displays the evolution of the percentage relative error committed by
using Oseen’s superposition approximation. We observe that the error increases
monotonically with the system size and reaches its maximum value in the limit
$R\to\infty$. In particular, the error attains its extreme value in the mid-
plane of the channel for $h=0$, where the MPRE amounts to about 11%. The
latter value is remarkably lower than the one previously obtained for
axisymmetric translational motion along the axis of two coaxially positioned
rigid no-slip disks Daddi-Moussa-Ider  _et al._ (2020b) for which the MPRE was
found to be as high as 55% in the mid-plane. Consequently, the superposition
approximation can be employed to estimate the rotational mobility between two
confining disks without significantly compromising the accuracy of the
prediction.

##  IV Conclusions

To summarize, we have presented an analytical and semi-analytical theory to
quantify the low-Reynolds-number flow induced by a point-torque singularity
acting near a single disk or between two coaxially positioned rigid disks,
respectively, satisfying no-slip boundary conditions on the surfaces of the
disks. The rotlet is assumed to be located on the axis of symmetry of the
disks with the torque directed along that axis. We have formulated the
solution of the hydrodynamic equations as mixed-boundary-value problems, which
we subsequently reduced into systems of dual integral equations with Bessel-
function kernels. On the one hand, we have demonstrated that, near a single
disk, the resulting integral equation can appropriately be transformed into a
classical Abel integral equation that admits a unique solution. On the other
hand, we have shown that, between two coaxially positioned disks, a system of
two Fredholm integral equations of the first kind arises. For its solution, we
have approximated the integral by standard middle Riemann sums reducing the
dual integral equations into a linear system of equations amenable to
immediate inversion using standard numerical approaches.

Moreover, we have made use of the derived solution of the flow problems to
probe the effect of confinement on the rotational hydrodynamic mobility of a
small colloidal particle through which the torque is exerted on the fluid.
More importantly, we have assessed the accuracy and reliability of Oseen’s
superposition approximation, which is commonly employed to predict the
rotational mobility in confined geometries. We have found that the maximum
percentage relative error of Oseen’s approximation is only about 11% in the
mid-plane of the channel, suggesting that this simplistic approximation could
generally be employed to estimate the rotational mobility between two finite-
sized disks.

The systems addressed in the present study may find useful applications in
various biologically and technologically relevant processes. On the one hand,
the solution of the flow problem for a rotlet singularity acting near a
finite-sized disk may be employed in the context of micromixing, as a small-
scale analog to a magnetic stir bar mixer driven by an external rotating
magnetic field. On the other hand, the solution inside a gap bounded by two
coaxially positioned disks may prove to be useful, for instance, in the
modeling of ionic transport in small-scale capacitors.

The present results may be extended to further explore the rotational motion
of a spherical particle of finite size, with a radius comparable to the radii
of the confining disks. For that purpose, the solution of the flow problem
could, in principle, be formulated in bipolar coordinates. Another possible
extension of the present work could be to address the general problem of
rotational motion near one or two no-slip disks for arbitrary positioning of
the singularity and arbitrary orientation of the torque. These steps could be
the subject of possible future investigations.

##  Appendix A Solution of the integral equation (20)

In this Appendix, we show that the solution of the resulting integral equation
Eq. (20) can be cast in the form of solution of a classical Abel integral
equation. The general form of an Abel integral equation can be presented as
Hochstadt (2011)

$\int_{0}^{x}\frac{\phi(s)}{\left(x-s\right)^{\frac{1}{2}}}\,\mathrm{d}s=g(x)\,,$
(66)

the solution of which is given by

$\phi(x)=\frac{1}{\pi}\frac{\mathrm{d}}{\mathrm{d}x}\int_{0}^{x}\frac{g(u)}{\left(x-u\right)^{\frac{1}{2}}}\,\mathrm{d}u\,.$
(67)

Using the changes of variables $s=t^{2}$ and $x=r^{2}$, Eq. (20) can be
written as

$\int_{0}^{x}\frac{s^{-\frac{1}{4}}\,\hat{\omega}(s^{\frac{1}{2}})}{\left(x-s\right)^{\frac{1}{2}}}\,\mathrm{d}s=\left(2\pi\right)^{\frac{1}{2}}x^{\frac{1}{2}}f(x^{\frac{1}{2}})\,.$
(68)

By identification with Eq. (66), we get
$\phi(s)=s^{-\frac{1}{4}}\,\hat{\omega}(s^{\frac{1}{2}})$ and
$g(x)=\left(2\pi\right)^{\frac{1}{2}}x^{\frac{1}{2}}f(x^{\frac{1}{2}})$. Using
the solution form given by Eq. (67), we then obtain

$x^{-\frac{1}{4}}\hat{\omega}(x^{\frac{1}{2}})=\left(\frac{2}{\pi}\right)^{\frac{1}{2}}\frac{\mathrm{d}}{\mathrm{d}x}\int_{0}^{x}\frac{u^{\frac{1}{2}}f(u^{\frac{1}{2}})}{\left(x-u\right)^{\frac{1}{2}}}\,\mathrm{d}u\,.$
(69)

Next, applying the change of variable $r=x^{\frac{1}{2}}$, the latter equation
can readily be expressed as

$\hat{\omega}(r)=\frac{1}{2}\left(\frac{2}{\pi
r}\right)^{\frac{1}{2}}\frac{\mathrm{d}}{\mathrm{d}r}\int_{0}^{r^{2}}\frac{u^{\frac{1}{2}}f(u^{\frac{1}{2}})}{\left(r^{2}-u\right)^{\frac{1}{2}}}\,\mathrm{d}u\,.$
(70)

Finally, the change of variable $v=u^{\frac{1}{2}}$ yields

$\hat{\omega}(r)=\left(\frac{2}{\pi
r}\right)^{\frac{1}{2}}\frac{\mathrm{d}}{\mathrm{d}r}\int_{0}^{r}\frac{v^{2}f(v)}{\left(r^{2}-v^{2}\right)^{\frac{1}{2}}}\,\mathrm{d}v\,,$
(71)

which exactly corresponds to Eq. (21) rewriting $r=t$.

##  Appendix B Comparison with numerical calculations using the finite-element
method

To confirm our analytical solutions, we perform numerical simulations using
the finite-element method. Formulated in cylindrical coordinates,
$\bm{v}=(v_{r},v_{\theta},v_{z})$, we can take advantage of the fact that the
solution is constant in angular direction, $\partial_{\theta}\bm{v}=\bm{0}$.
This reduces the problem to a two-dimensional equation formulated in the
$r/z$-plane for the angular component $v_{\theta}$ only. Since there is also
no coupling to the pressure field, the problem reduces to a scalar one. In the
following, we illustrate our approach for the two-disk geometry and denote by

$\begin{split}\Omega&=\Big{(}(0,R_{max})\times(-Z_{max},Z_{max})\Big{)}\\\
&\qquad\setminus\Big{(}(0,R)\times\\{H/2\\}\cup(0,R)\times\\{-H/2\\}\Big{)}\end{split}$
(72)

the numerical domain, artificially restricted to $0<r<R_{max}$ and
$-Z_{max}<z<Z_{max}$. To limit the impact of the artificial outer boundaries,
we set $R_{max}=Z_{max}=6.625$. We could not identify a significant effect by
further extending these limits.

The variational formulation of the problem is then given by von Wahl  _et al._
(2021)

$\int_{\Omega}\eta r\Big{(}\frac{\partial v_{\theta}}{\partial
r}\frac{\partial\phi}{\partial r}+\frac{\partial v_{\theta}}{\partial
z}\frac{\partial\phi}{\partial
z}+\frac{1}{r}v_{\theta}\phi\Big{)}\,\text{d}r\,\text{d}z=F(\phi)\,,$ (73)

$\forall\phi\in H^{1}_{0}(\Omega;D)$, where we denote by $H^{1}_{0}(\Omega;D)$
the space of square integrable functions with weak derivatives that are zero
on the two discs $D=(0,R)\times\\{-H/2\\}\cup(0,R)\times\\{H/2\\}$. On the
right-hand side of Eq. (73), the problem is driven by a Dirac form as

$F(\phi)=\phi(\bm{r}_{h})\,,$ (74)

centered in a point close to the $z$-axis $\bm{r}_{h}=(r_{0},z_{h})$,
$z_{h}=1/128$.

The equation is discretized with quadratic finite-elements using piecewise
quadratic elements on a quadrilateral mesh Richter (2017) featuring about
$1\,750\,000$ unknowns. All computations are performed in the finite-element
software library Gascoigne 3d Becker  _et al._ (2020).

##  Appendix C Mobility between two disks in the limit $R\ll 1$

In the range of small values of $R\ll 1$, we attempt to find an approximate
expression of the scaled correction factor. For $t\ll 1$, it follows from Eqs.
(19) and (51) that $\mathcal{K}_{-}(t)\sim t^{\frac{1}{2}}$ and
$\mathcal{S}(t)\sim t^{-\frac{1}{2}}$, respectively. Therefore, to ensure the
convergence of the system of integral equations (50) at the lower limit $t=0$,
we require that the unknown functions $\hat{f}_{1}(t)$ and $\hat{f}_{2}(t)$
scale (at least) as $t^{\frac{1}{2}}$ as $t\to 0$.

We now use the ansatz $\hat{f}_{1}(t)=\alpha_{1}t^{\frac{1}{2}}$ and
$\hat{f}_{2}(t)=\alpha_{2}t^{\frac{1}{2}}$, where $\alpha_{1}$ and
$\alpha_{2}$ are two real numbers to be subsequently determined. Inserting
these expressions into Eqs. (50) evaluated at $r=\beta R$, with
$\beta\in(0,1)$, performing analytically the integration, expanding the
resulting expressions into Taylor series of $R$, and solving for $\alpha_{1}$
and $\alpha_{2}$ yields

$\alpha_{1}=-\left(\frac{\pi}{2}\right)^{\frac{1}{2}}\frac{\beta
R}{\left(\frac{1}{2}+h\right)^{3}}\,,\quad\alpha_{2}=\left(\frac{\pi}{2}\right)^{\frac{1}{2}}\frac{\beta
R}{\left(\frac{1}{2}-h\right)^{3}}\,.$ (75)

Next, by substituting the above expressions of $\hat{f}_{1}(t)$ and
$\hat{f}_{2}(t)$ into Eq. (58), evaluating the integral analytically and
performing a series expansion about $R=0$, we obtain

$k_{2}\simeq\frac{R^{2}}{2}\left(\frac{2}{\pi}\right)^{\frac{1}{2}}\left(\frac{\alpha_{2}}{\left(\frac{1}{2}-h\right)^{3}}-\frac{\alpha_{1}}{\left(\frac{1}{2}+h\right)^{3}}\right)\,.$
(76)

Finally, by inserting the expressions of $\alpha_{1}$ and $\alpha_{2}$ stated
by Eqs. (75) into Eq. (76), the correction factor in the range $R\ll 1$ can be
obtained as

$k_{2}\simeq\frac{\beta
R^{3}}{2}\left(\left(\frac{1}{2}-h\right)^{-6}+\left(\frac{1}{2}+h\right)^{-6}\right)\,.$
(77)

By setting $\beta=5/6$, the latter approximate expression is found to be in
good agreement with the numerical results.

## Acknowledgments

A.D.M.I. and H.L. gratefully acknowledges support from the DFG (Deutsche
Forschungsgemeinschaft) through the projects DA 2107/1-1 and LO 418/23-1.
A.M.M. thanks the DFG for support through the Heisenberg Grant ME 3571/4-1.

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