soft matter, biophysics, fluid mechanics

Jonas Rønning

# Flow around topological defects in active nematic films

Jonas Rønning1 M. Cristina Marchetti 2 Mark J. Bowick3 and Luiza Angheluta1
1Njord Centre, Department of Physics, University of Oslo, P. O. Box 1048, 0316
Oslo, Norway
2Department of Physics, University of California Santa Barbara, Santa Barbara,
CA 93106, USA
3Kavli Institute for Theoretical Physics, University of California Santa
Barbara, Santa Barbara, CA 93106, USA<EMAIL_ADDRESS>
###### Abstract

We study the active flow around isolated defects and the self-propulsion
velocity of $+1/2$ defects in an active nematic film with both viscous
dissipation (with viscosity $\eta$) and frictional damping $\Gamma$ with a
substrate. The interplay between these two dissipation mechanisms is
controlled by the hydrodynamic dissipation length
$\ell_{d}=\sqrt{\eta/\Gamma}$ that screens the flows. For an isolated defect,
in the absence of screening from other defects, the size of the shear
vorticity around the defect is controlled by the system size $R$. In the
presence of friction that leads to a finite value of $\ell_{d}$, the vorticity
field decays to zero on the lengthscales larger than $\ell_{d}$. We show that
the self-propulsion velocity of $+1/2$ defects grows with $R$ in small systems
where $R<\ell_{d}$, while in the infinite system limit or when $R\gg\ell_{d}$,
it approaches a constant value determined by $\ell_{d}$.

###### keywords:

active nematics, topological defects, nematic liquid crystals, hydrodynamics

##  1 Introduction

Active matter consists of collections of individuals that dissipate energy
taken from the environment to generate motion and forces and self-organize
into a rich variety of ordered phases. Many active systems exhibit nematic
order interrupted by orientational defects and advected by spontaneous flows
driven by intrinsic activity of the self-propelled individuals. This behavior
is found in reconstituted systems, such as mixtures of cytoskeletal filaments
and motor proteins [1, 2, 3, 4], bacterial suspensions [5, 6] and cell sheets
[7, 8], as well as synthetic systems, like vertically vibrated layers of
granular rods [9, 6]. A central feature of active nematics is the feedback
between active stresses which distort orientational order and the spontaneous
flow generated by such distortions. In hydrodynamic descriptions [6], the
active stress $\sigma_{ij}^{a}$ exerted by elongated active entities on the
surrounding fluid is proportional to the nematic order parameter tensor
$Q_{ij}$, namely $\sigma^{a}_{ij}=\alpha_{0}Q_{ij}$ [10, 11]. The activity
coefficient $\alpha_{0}$ embodies the microscale biomolecular processes that
convert chemical energy into mechanical forces, and depends on the
concentration of active entities, which in general may vary in space and time
[12, 13]. The sign of $\alpha_{0}$ distinguishes between contractile
($\alpha_{0}>0$) stress generated by “puller" swimmers, such as the algae
Chlamydomonas, versus extensile ($\alpha_{0}<0$) stress generated by “pusher"
swimmers, e.g., most flagellated bacteria. Its magnitude controls the strength
of the active flow. Fluctuations in orientational order yield active stresses
and associated flows, which can in turn enhance the orientational distortions.
The resulting feedback loop destabilizes the nematic order, driving the system
to a state of self-sustained spatio-temporally chaotic flow, with
proliferation of topological defects, and termed active turbulence [14, 15].

The lowest-energy orientational defects in nematic films have half-integer
topological charge and opposite sign. The $+1/2$ defects have comet-like
shape, whilst the $-1/2$ defects have a tri-fold symmetry (see Figs. 3 and 4).
Defects strongly disrupt orientational order and induce long-range nematic
distortions. In active systems, such distortions generate flows with symmetry
and profiles controlled by the defect geometry. The nematic distortion created
by a $+1/2$ defect yields an active flow that is finite at the defect core. A
$+1/2$ defect then rides along with the flow it itself generates, behaving
like a motile particle with a non-vanishing self-propulsion velocity
$\mathbf{v}^{a}_{+}$, even in the absence of external drive [12, 16]. On the
other hand, the active backflow generated by a $-1/2$ defect vanishes at the
core due to the defect’s threefold symmetry (see Fig. 4). Thus $-1/2$ defects
behave like passive particles and have no spontaneous motility in the absence
of external driving. A simple estimate demonstrates that $\mathbf{v}^{a}_{+}$
is directed along the polar axis of the $+1/2$ defect and is proportional to
the activity $\alpha_{0}$. In an extensile medium $+1/2$ defects self-propel
in the direction of the head of the comet, while in a contractile system they
move towards the comet’s tail [12, 16, 17]. The direction of motion of $+1/2$
defects can then be used as a metric for determining the nature of active
stress in the system. Such measurements have for instance revealed the
surprising dominance of extensile stresses in confluent tissue composed of
tightly bound contractile individual cells [18, 19, 20, 21, 22].

The flow generated by defects and the resulting propulsive speed of the $+1/2$
also vary depending on the dissipative processes at play in the system and the
role of fluid incompressibility. Specifically, important differences exist
between “dry” systems, where dissipation is dominated by friction $\Gamma$
with a substrate or an external medium [23, 24] and “wet” systems where
dissipation is mainly controlled by viscosity $\eta$, resulting in long-range
hydrodynamic effects [17, 25, 14, 24]. In incompressible wet systems, activity
is also a source of pressure gradients, which in turn contribute alongside
with the nematic distortion to the self-motility of positive defects. In the
limit of viscous dominated flows with no friction with the substrate, the
self-propulsion speed scales as
$|\mathbf{v}^{a}_{+}|\sim\frac{|\alpha_{0}|}{\eta}\ell$, where $\ell$ is a
length scale given by the system size for an isolated defect [17] or by the
mean separation between defects, which is, in turn, controlled by the active
length scale $\ell_{a}=\sqrt{K/|\alpha_{0}|}$, with $K$ the nematic stiffness
[17]. In overdamped (dry) systems, where viscosity is negligible compared to
frictional damping with the substrate,
$|\mathbf{v}^{a}_{+}|\sim|\alpha_{0}|/(\xi\Gamma)$, where $\xi$ is the nematic
coherence length [23, 26, 24]. A complete calculation of the active flows
associated with defect configurations and of the propulsive speed of the
$+1/2$ defect that bridges between the two limits is, however, not available.
The need for such a calculation is further motivated by recent work that has
shown that tuning frictional damping relative to viscous dissipation leads to
different dynamical regimes and ordering behavior of interacting defects [27].

In this paper, we present a detailed calculation of the flow around isolated
$\pm 1/2$ defects and of the defect’s self-propulsion velocity in an
incompressible nematic film. We incorporate both viscous dissipation and
frictional damping and examine the interplay between the two, as well as the
long range hydrodynamic effects arising from incompressibility. We evaluate
the $+1/2$ self-propulsive speed $|\mathbf{v}_{+}^{a}|$ as a function of the
the hydrodynamic dissipation length $\ell_{d}=\sqrt{\eta/\Gamma}$ which
measures the competition between viscous dissipation and frictional damping.
The result is summarized in Fig. 1. When dissipation is controlled by friction
($\ell_{d}\ll\xi$), one recovers the simple dimensional estimate
$|\mathbf{v}^{a}_{+}|\sim|\alpha_{0}|/(\xi\Gamma)$. We show, however, that to
obtain this result it is not sufficient to consider the far flow field which
diverges near the defect, but one must resolve the full flow field near the
defect core. On the other hand, when viscous stresses dominate, the defect
propulsive speed depends on the order of limits. If $\Gamma=0$ from the
outset, then a simple estimate yields $v_{x}^{a}\sim r$ due to the long-range
nature of defect distortions. This limit, however, corresponds to a “floating”
layer and does not describe experimental situations where the active nematic
film is supported by a substrate [25] or in contact with other fluids. It has
been argued before that this unbounded growth should be cut off either by the
system size or by the defect separation [17]. Our work shows that a finite
friction cuts off the large scale divergence of the defect self-propulsion
speed at the scale $\ell_{d}$, with
$|\mathbf{v}_{+}^{a}|\sim\frac{|\alpha_{0}|}{\eta}\ell_{d}$ in the limit
$\zeta=\ell_{d}/\xi\gg 1$ where viscous dissipation exceeds frictional drag
and provides an analytical expression for the defect self-propulsion over all
values of friction and viscosity. We find that the structure of the flow field
around a defect is also affected by the competition between viscosity and
friction. At distances large compared to $\ell_{d}$, the flow velocity decays
in the far-field as $\sim 1/r$, due to friction with the substrate [25]. At
distances smaller than $\ell_{d}$, viscous dissipation dominates and smooths
out the velocity field near the defect core. Our work is relevant to defects
in thin film of microtubule nematics on a substrates, as well as to dense cell
layers.

Figure 1: The self-propulsion speed of a $+1/2$ defect as function of
$\ell_{d}/\xi$ in an unbounded system. The line is the exact analytical
solution, while the dotted line shows the asymptotic scaling with $\ell_{d}$,
i.e. $v_{x}^{a}\eta\sim\ell_{d}$ for $\ell_{d}>\xi$. The gray area corresponds
to the overdamped limit, where essentially $v_{x}^{a}$ depends solely on
friction.

In Sec. 2 we describe the hydrodynamic model. In Secs. 3 and 4, we provide
analytical derivations of closed expressions for the velocity and pressure
fields induced by $\pm 1/2$ defects in an infinite system. One implication of
the long-range interactions present in active nematics is that there are
strong finite-size effects on the single defect flow field. This is discussed
in Sec. 5, where we compare the analytical predictions with numerical
integration of the Stokes equations in a disk of finite radius. Finally, the
main results are discussed with concluding remarks in Sec. 6.

##  2 Hydrodynamic model

We consider a hydrodynamic model of an active nematic that couples flow
velocity $\mathbf{u}(\mathbf{r})$ to the nematic order parameter
$Q_{ij}=S(\hat{n}_{i}\hat{n}_{j}-\frac{1}{2}\delta_{ij})$, where $S$
quantifies the degree of order and
$\mathbf{\hat{n}}(\mathbf{r})=\left(\cos\theta(\mathbf{r}),\sin\theta(\mathbf{r})\right)$
is the orientational director field with head-tail symmetry. In the simplest
formulation, we consider that the Q-tensor is a minimizer of the de Gennes-
Landau free energy [6]

$\displaystyle\mathcal{F}=\int d\mathbf{r}\left[\frac{K}{2}|\nabla
Q|^{2}+\frac{g}{4}\left(1-\frac{1}{2}\textrm{Tr}\left(Q^{2}\right)\right)^{2}\right]\;,$
(1)

with isotropic elastic constant $K>0$ and $g$ the strength of the local
ordering potential. The uniform nematic ordered state corresponds to
$S_{0}^{2}=1$. The flow field satisfies a Stokes equation that balances forces
on a fluid element, given by [6]

$\displaystyle\left(\Gamma-\eta\nabla^{2}\right)\mathbf{u}=\alpha_{0}\nabla\cdot
Q(\mathbf{r})-\nabla p(\mathbf{r}),\qquad\nabla\cdot\mathbf{u}=0\;,$ (2)

where $\Gamma$ is a friction coefficient per unit area, $\eta$ is the dynamic
viscosity, and $\alpha_{0}$ is the activity parameter, with dimensions of
stress. For simplicity, we neglect the elastic stress as being of higher order
in the gradients of $\mathbf{Q}$ compared to the active stress and a more
important contribution for nematic textures with many defects. Here, we
consider the flow field generated by an isolated $\pm 1/2$ defect embedded in
an otherwise uniform nematic field.

In two dimensions, the traceless $\mathbf{Q}$-tensor has two independent
components and can be represented equivalently as a complex scalar order
parameter $\psi=Q_{xx}+iQ_{xy}$. The configuration of a defect located at the
origin can be written in terms of the $\psi$-field as
$\psi(\mathbf{r})=S(r)e^{2i\theta(\mathbf{r})}$, where $r\equiv|\mathbf{r}|$.
The detailed form of core function $S(r)$ depends on the specific terms
retained in the free energy, but it has the important generic asymptotic
behaviors that $S(r)\rightarrow 1$ for $r\gg\xi$ and $S(r)\approx ar/\xi$ when
$r\rightarrow 0$, where $\xi=\sqrt{K/g}$ is the coherence length which sets
the scale of the defect core and $a$ is a numerical constant $\mathcal{O}(1)$.
Below we set $a=1$, without loss of generality. The coherence length provides
an ultraviolet cutoff to separate inner core-solution from outer-core
solution. On long distances, the nematic orientation is a potential field that
has a branch cut starting at the origin where there is an isolated defect of
charge $q=\pm 1/2$ and can be written as [28, 24]

$\displaystyle\theta(\mathbf{r})=q\arctan\left(\frac{y}{x}\right)+\theta_{0},$
(3)

where $\theta_{0}$ is the uniform background orientation. Without loss of
generality, we set $\theta_{0}=0$.

We rescale the Stokes equation in units of the nematic relaxation time
$\tau=\gamma/g$ (where $\gamma$ is the inverse of the rotational diffusivity)
and the coherent length $\xi$, such that the dimensionless momentum equation
takes the form

$\left(1-\zeta^{2}\nabla^{2}\right)\mathbf{u}=\mathbf{F}^{\pm}-\nabla\tilde{p}(\mathbf{r}),\qquad\nabla\cdot\mathbf{u}=0,$
(4)

where $\mathbf{F}^{\pm}=\alpha\nabla\cdot Q$ is the active force generated by
a defect. The rescaled activity and pressure are given by
$\alpha=\alpha_{0}\gamma/(\Gamma K)$ and $\tilde{p}=p\gamma/(\Gamma K)$. The
dimensionless parameter $\zeta=\sqrt{\eta/(\Gamma\xi^{2})}=\ell_{d}/\xi$
measures the hydrodynamic dissipation length $\ell_{d}=\sqrt{\eta/\Gamma}$ in
units of the coherent length, $\xi$. In the following, we will omit the tilde
and all quantities are dimensionless unless otherwise stated.

The components of the $\mathbf{Q}$ tensor for an isolated $+1/2$ defect are
given by $Q_{xx}(\mathbf{r})=S(r)\frac{x}{r}$ and $Q_{xy}=S(r)\frac{y}{r}$.
The active force density then reduces to

$\displaystyle\mathbf{F}^{+}(\mathbf{r})=\begin{cases}2\alpha\mathbf{e}_{x},\quad
r\rightarrow 0,\\\ \frac{\alpha}{r}\mathbf{e}_{x},\quad r\gg 1.\end{cases}$
(5)

Similarly, for a negative defect $Q_{xx}=S(r)\frac{x}{r}$ and
$Q_{xy}=-S(r)\frac{y}{r}$, corresponding to an active force density given by

$\displaystyle\mathbf{F}^{-}(\mathbf{r})=\begin{cases}0,\quad r\rightarrow
0,\\\
-\alpha\frac{x^{2}-y^{2}}{r^{3/2}}\mathbf{e}_{x}+\alpha\frac{2xy}{r}\mathbf{e}_{y},\quad
r\gg 1\;.\end{cases}$ (6)

The solutions for the flow velocity and pressure can be written in term of the
corresponding Green functions as

$\displaystyle\mathbf{u}(\mathbf{r})=\frac{1}{2\pi\zeta^{2}}\int
d\mathbf{r}^{\prime}K_{0}(|\mathbf{r}-\mathbf{r}^{\prime}|/\zeta)\left[\mathbf{F}^{\pm}(\mathbf{r}^{\prime})-\nabla^{\prime}p(\mathbf{r}^{\prime})\right]\equiv\mathbf{u}^{a}+\mathbf{u}^{p},$
(7) $\displaystyle p(\mathbf{r})=\frac{1}{2\pi}\int
d\mathbf{r}^{\prime}\ln\left(|\mathbf{r}-\mathbf{r}^{\prime}|\right)\nabla^{\prime}\cdot\mathbf{F}^{\pm}(\mathbf{r}^{\prime}).$
(8)

where $\mathbf{u}^{a}$ and $\mathbf{u}^{p}$ are the contributions to the flow
velocity induced by the active stress and pressure gradients, respectively.
Note that the latter also depends (indirectly) on activity. In the limit of no
friction, Eqs. (7) and (8) reduce to Eqs. (3.7) and (3.8) of Ref. [17].

##  3 Positive nematic defect in an infinite system

###  3.1 Defect self-propulsion

The net active flow at the defect core acts as an advective velocity that
propels the defect with a velocity $\mathbf{v}^{a}$, which in turn is
controlled by both the active stress and pressure gradients. Thus we write
$\mathbf{v}^{a}=\mathbf{u}^{a}(0)+\mathbf{u}^{p}(0)$. The flow induced by the
active stress at the origin is given by Eq. (7) evaluated at $\mathbf{r}=0$.
The $y$-component vanishes due to symmetry considerations, and the
$x$-component is given by

$\displaystyle
u_{x}^{a}(0)=2\alpha\left[1-\frac{1}{\zeta}K_{1}(\zeta^{-1})\right]+\frac{\pi\alpha}{2\zeta}\left[1-\frac{1}{\zeta}\left(L_{-1}(\zeta^{-1})K_{0}(\zeta^{-1})+L_{0}(\zeta^{-1})K_{1}(\zeta^{-1})\right)\right]\;,$
(9)

where $\zeta=\ell_{d}/\xi$, $K_{n}(x)$ are modified Bessel functions and
$L_{n}(x)$ modified Struve function.

The integral determining the pressure field given by Eq. (8) can be performed
by a mapping to complex coordinates
$(x^{\prime},y^{\prime})\rightarrow(w,\bar{w})$, $(x,y)\rightarrow(z,\bar{z})$
and then using the substitution to polar coordinates $w=r^{\prime}\hat{w}$,
$\hat{w}=e^{i\theta^{\prime}}$. This yields

$\displaystyle p(\mathbf{r})=$
$\displaystyle-\frac{\alpha}{2i\pi}\int_{0}^{1}dr^{\prime}r^{\prime}\oint_{\gamma}d\hat{w}\left(\frac{1}{\hat{w}r^{\prime}(\hat{w}-zr^{{}^{\prime}-1})}-\frac{1}{\bar{z}(\hat{w}-r^{\prime}\bar{z}^{-1})}\right)$
$\displaystyle-\frac{\alpha}{4i\pi}\int_{1}^{\infty}dr^{\prime}\oint_{\gamma}d\hat{w}\left(\frac{1}{\hat{w}r^{\prime}(\hat{w}-zr^{{}^{\prime}-1})}-\frac{1}{\bar{z}(\hat{w}-r^{\prime}\bar{z}^{-1})}\right)$
(10)

with $\gamma$ a contour of unit radius centered at origin. The pole at
$\hat{w}=0$ is always inside the unit disk $|\hat{w}|<1$, whereas the poles at
$\hat{w}=zr^{{}^{\prime}-1}$ and $\hat{w}=r^{\prime}\bar{z}^{-1}$ are inside
the unit disk when $|z|<r^{\prime}$ or $|z|>r^{\prime}$, respectively. The
contour integrals are then evaluated using the residue theorem. Integrating
over $r^{\prime}$, we finally obtain

$p(\mathbf{r})=\begin{cases}\alpha x,&r<1,\\\ \frac{\alpha x}{r},&\text{if
}r>1.\end{cases}$ (11)

Consequently, the defect self-propulsion induced by pressure gradient has only
$x$-component which counteracts that induced by the active stress, and given
by

$u^{p}_{x}(0)=-\alpha\left(1-\frac{1}{\zeta}K_{1}\left(\zeta^{-1}\right)\right)-\frac{\pi\alpha}{4\zeta}\left[1-\frac{1}{\zeta}[L_{-1}(\zeta^{-1})K_{0}(\zeta^{-1})+L_{0}(\zeta^{-1})K_{1}(\zeta^{-1})]\right]=-\frac{u^{a}_{x}(0)}{2}\;.$
(12)

Combining these results, we find that the self-propulsion velocity of an
isolated $+1/2$ defect oriented along the $x$ axis is
$\mathbf{v}^{a}=v_{x}^{a}\hat{\mathbf{e}}$, where $v_{x}^{a}$ has the
following scaling form

$v_{x}^{a}=\alpha F(\zeta).$ (13)

where

$\displaystyle
F(\zeta)=\left(1-\frac{1}{\zeta}K_{1}\left(\zeta^{-1}\right)\right)+\frac{\alpha\pi}{4\zeta}\left[1-\frac{1}{\zeta}[L_{-1}(\zeta^{-1})K_{0}(\zeta^{-1})+L_{0}(\zeta^{-1})K_{1}(\zeta^{-1})]\right].$
(14)

When $\zeta\gg 1$, we can simplify the expression by expanding in powers of
$\zeta^{-1}$, and, to leading order, we obtain,

$F(\zeta)\underset{\zeta\gg
1}{\approx}\frac{\pi}{4\zeta}+\frac{1}{2\zeta^{2}}(\gamma-1-\ln(2\zeta))-\frac{1}{4\zeta^{2}}(2\gamma-1-2\ln(2\zeta))\;,$
(15)

where $\gamma\approx 0.577$ is the Euler-Mascheroni constant. Similarly, we
also take the other limit $\zeta\ll 1$, where the scaling function approaches
a constant value. The dependence of the scaling function $F$ on $\zeta$ is
plotted in Fig. 2 and its asymptotic scaling at $\zeta\gg 1$ as
$F\sim\zeta^{-1}$ is included as the dotted line. We can discuss the
implications of there results better, when we use dimensional quantities and
write the asymptotic behavior of the self-propulsion speed as

$v_{x}^{a}\approx\begin{cases}\frac{\pi}{4}\frac{\alpha_{0}}{\Gamma\ell_{d}}=\frac{\pi}{4}\frac{\alpha_{0}\ell_{d}}{\eta},&\zeta\gg
1\\\ \frac{\alpha_{0}}{\Gamma\xi},&\zeta\rightarrow 0\end{cases}$ (16)

As anticipated from dimensional analysis,
$v_{x}^{a}\sim\frac{\alpha_{0}}{\Gamma\xi}$, in the overdamped limit where
dissipation is controlled only by frictional drag [23, 26, 24]. In the
underdamped limit, where the effect of drag is much smaller than viscous
dissipation, hydrodynamic lenghscale becomes important in screening the
divergence of the self-propulsion speed with system size, such that
$v_{x}^{a}$ scales instead as $v_{x}^{a}\sim\alpha_{0}/\sqrt{\eta\Gamma}$. In
this case, the self-driven motion of $+1/2$ defect is reduced by both friction
and viscosity.

Figure 2: Scaling function $F(\zeta)$ as a function of $\zeta$. The gray
region corresponds to the overdamped limit where $v_{x}^{a}$ depends only on
friction. The doted black line is the asymptotic limit for $\zeta\gg 1$.

As discussed in the introduction, the presence of a finite drag always cuts
off the large-scale divergence of the speed of a single defect obtained in a
purely viscous $2D$ layer at the dissipation length $\ell_{d}$. When the flow
equations for a thin nematic film of thickness $h$ on a substrate are derived
via a lubrication approximation, the effective friction coefficient relates to
the film thickness and the viscosity of the substrate bulk fluid (oil), and
scales as $\Gamma\sim\bar{\eta}/h^{2}$ [29]. A more detailed calculation
relevant to active microtubule suspensions confined between water and oil
shows that the bulk viscosity plays an important role as an additional source
of dissipation in the nematic layer affecting the individual defect self-
propulsion [30], as well as the vortex statistics in the active turbulence
regime [31]. Notice that Ref. [30] shows that the $+1/2$ defect speed decays
algebraically with the bulk oil viscosity (that controls the drag) in the
regime where the flow dissipation comes from the viscous dissipation in the
nematic layer, consistent with our formulation. When the flow dissipation is
dominated by the oil bulk viscosity, there is, however, a logarithmic decay
with increasing oil viscosity and, indirectly, drag.

###  3.2 Flow field away from the defect

Outside the core, we treat the defect as a point source. From symmetry
considerations, the flow velocity due to $\sigma^{a}$ is again non-zero only
along the $x$-direction and it given by

$u_{x}^{a}(\mathbf{r})=\frac{\alpha}{2\pi\zeta^{2}}\int\frac{d\mathbf{r}^{\prime}}{r^{\prime}}K_{0}(|\mathbf{r}^{\prime}-\mathbf{r}|/\zeta)\;.$
(17)

The flow velocity associated with pressure gradients is finite also in the
$y$-direction and it is given by

$u^{p}_{i}(\mathbf{r})=-\frac{\alpha}{2\pi\zeta^{2}}\int
d\mathbf{r}^{\prime}K_{0}(|\mathbf{r}^{\prime}-\mathbf{r}|/\zeta)\left(\frac{\delta_{ix}}{r^{\prime}}-\frac{x^{\prime}r^{\prime}_{i}}{r^{\prime
3}}\right)\;.$ (18)

The term proportional to the $\delta$ function in Eq. 18 cancels
$\mathbf{u}^{a}_{x}$ from Eq. 17, such that the total active fluid flow is
entirely determined by pressure flow, with

$u_{i}(\mathbf{r})=\frac{\alpha}{2\pi\zeta^{2}}\int
d\mathbf{r}^{\prime}K_{0}(|\mathbf{r}^{\prime}-\mathbf{r}|/\zeta)\frac{x^{\prime}r^{\prime}_{i}}{r^{\prime
3}}\;.$ (19)

To evaluate this integral, we use a complex representation $u=u_{x}+iu_{y}$
and evaluate the resulting contour integrals as shown in appendix A where we
express them in terms of complete elliptic integrals of first and second kind.
We further use the power series representation of these elliptic integrals,
which allows us to write the active fluid velocity as a series expansion in
integrals over the zeroth order modified Bessel function, namely

$\displaystyle u^{+}(r,\phi)=$
$\displaystyle\frac{\alpha}{2\zeta^{2}}\sum_{n=0}^{\infty}\left(1-\frac{2n+1}{2n-1}e^{2i\phi}\right)\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\int_{0}^{r}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r^{\prime}}{r}\right)^{2n+1}$
$\displaystyle+$
$\displaystyle\frac{\alpha}{2\zeta^{2}}\sum_{n=0}^{\infty}\left(1-\frac{n}{n+1}e^{2i\phi}\right)\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\int_{r}^{\infty}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r}{r^{\prime}}\right)^{2n}\;.$
(20)

The $K_{0}(x)$ integrals are computed in appendix B. After some mathematical
manipulations the velocity reduces to

$u^{+}(r,\phi)=\frac{\alpha}{4\zeta}\left[\pi\left(I_{0}(r/\zeta)-I_{2}(r/\zeta)e^{2i\phi}\right)+\sum_{k,n=0}^{\infty}\left(\kappa^{+}_{1}(n,k)+\kappa^{+}_{2}(n,k)e^{2i\phi}\right)\frac{1}{(k!)^{2}}\left(\frac{r}{2\zeta}\right)^{2k+1}\right].$
(21)

with

$\displaystyle\kappa^{+}_{1}(n,k)$
$\displaystyle=\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\left(\frac{-(4n+1)(4k+3)}{(n+k+1)^{2}(2n-1-2k)^{2}}\right)\;,$
(22) $\displaystyle\kappa^{+}_{2}(n,k)$
$\displaystyle=\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\frac{[(2n-1)(4k(n+1)+1+n)-4k^{2}](4n+1)}{(n+1+k)^{2}(2n-1-2k)^{2}(n+1)(2n-1)}\;.$
(23)

The corresponding vorticity is given by

$\displaystyle\omega^{+}(r,\phi)=-\frac{\alpha}{8\zeta^{2}}\sin(\phi)\left(4\pi
I_{1}(r/\zeta)+\sum_{n,k}[(2k+1)\kappa^{+}_{1}(n,k)-(2k+3)\kappa^{+}_{2}(n,k)]\frac{1}{(k!)^{2}}\left(\frac{r}{2\zeta}\right)^{2k}\right).$
(24)

Both velocity and vorticity are shown in Fig. 3.

####  3.2.1 Asymptotic far-field flow:

The flow field greatly simplifies in the far-field $r/\zeta\gg 1$,
corresponding to distances much larger than the hydrodynamic dissipation
length. Then, the second term in Eq. (20) vanishes due to the exponential
decay of the Bessel function. In the first integral, we can replace the upper
limit $r$ with $\infty$ and perform it analytically with the result given as

$\displaystyle
u^{+}(r,\phi)=\frac{\alpha}{2r}\left(e^{2i\phi}+1+\left(\frac{\zeta}{r}\right)^{2}(1-3e^{2i\phi})\right)\;,$
(25)

where we have kept the two first terms in the expansion.

Figure 3: Flow streamlines (white arrow) around a $+1/2$ defect for $\alpha<0$
obtained from (a) full solution and (b) asymptotic one. The nematic director
field is shown in black lines and the background colormap denotes vorticity.
To show the structure of the near-field, the vorticity scale is saturated at
$\pm 0.2$ in a) and at $\pm 0.1$ in b). c) Cross section of vorticity obtained
from the exact solution (solid blue line) and the asymptotic limit (dotted
black line) at $x=0$ as a function of $y$. d) Cross section of the velocity
obtained from the exact solution (solid blue line) and the asymptotic limit
(dotted black line) at $y=0$ as a function of $x$.

The slow $1/r$-decay term in Eq. 25 is independent of viscosity $\eta$ and
identical to the one derived in Ref. [24] in the friction-dominated regime.
Corrections due to viscosity give rise to faster $1/r^{3}$ decay. The
corresponding far-field vorticity is

$\omega^{+}(r,\phi)=\frac{\alpha}{r^{2}}\sin{\phi}\left(1+3\left(\frac{\zeta}{r}\right)^{2}\right)\;.$
(26)

The far-field solutions are singular at the origin, which is not the case for
the full series solution that resolves the near core field. This is
demonstrated visually in Fig. 3 (c-d) where we plot cross-sections of the
velocity and vorticity profiles for both the full solution and the far-field
solution. The form of the expressions makes it natural to scale the position,
velocity and vorticity with $\zeta$, $\zeta/|\alpha|$ and $\zeta^{2}/|\alpha|$
respectively. The only free parameter is then the sign of $\alpha$. Panels (a)
and (b) show the flow streamlines and the vorticity field in the background
for the full and the far-field solutions, respectively, for an extensile
system ($\alpha<0$). The velocity magnitude is highest near the defect core
and decays as a power law following the far-field asymptote. The velocity
streamlines point towards the defect in the right half-plane, and away from
the defect in the left half-plane. For positive $\alpha$, the flow direction
is reversed. In an infinite system, the flow streamlines around an isolated
defect are not closed. On the other hand, as discussed later, in bounded
domains, the system size controls the size of the eddies formed around the
defect. For more realistic configurations with many defects, the system size
is typically replaced by the mean defect separation. It may be that other
intrinsic length scales controlled by elastic stresses are also important in
stabilizing finite-size vortices. These effects are left for future
investigation.

##  4 Negative nematic defect in an infinite system

By similar calculations as in Sec. 3, we find that the velocity induced by the
active stress at the position of the negative defect vanishes as expected from
symmetry consideration. After performing the integral in the complex plane and
subsequently integrating over the integrand with the Bessel function, we
determine the pressure field induced by the $-1/2$ defect vanishes inside the
defect core and non-zero outside given by

$\displaystyle p(\mathbf{r})=\begin{cases}0,\quad r<1,\\\
-\alpha\frac{x^{3}-3xy^{2}}{3r^{3}},\quad r>1.\end{cases}$ (27)

and its gradient vanishes at the origin, hence no advective pressure-flow of
the negative defect. Thus, an isolated $-1/2$ defect is stationary in a
uniform nematic field, regardless of activity.

###  4.1 Flow field away from the defect:

The flow field induced by the $-1/2$ defect can also be expressed analytically
as a series expansion of the elliptic integrals as detailed in appendix C,
with the resulting expression of the velocity field in the complex
representation $u^{-}=u^{-}_{x}+iu^{-}_{y}$ given as

$\displaystyle u^{-}(r,\phi)=$
$\displaystyle-\frac{\alpha}{2\zeta^{2}}\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\frac{2n+1}{2n-1}\left[e^{4i\phi}\frac{2n+3}{2n-3}-e^{-2i\phi}\right]\int_{0}^{r}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r^{\prime}}{r}\right)^{2n+1}$
$\displaystyle-\frac{\alpha}{2\zeta^{2}}\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\frac{n}{n+1}\left[e^{4i\phi}\frac{n-1}{n+2}-e^{-2i\phi}\right]\int_{r}^{\infty}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r}{r^{\prime}}\right)^{2n}.$
(28)

The integrals over the Bessel functions are evaluated in appendix B, and the
final expression is then given as

$\displaystyle
u^{-}(r,\phi)=\frac{\alpha}{8\zeta}\left(2\pi\left[I_{2}(r/\zeta)e^{-2i\phi}-I_{4}(r/\zeta)e^{4i\phi}\right]+\sum_{k,n}\left[\kappa^{-}_{1}(n,k)e^{-2i\phi}+\kappa^{-}_{2}(n,k)e^{4i\phi}\right]\frac{2}{(k!)^{2}}\left(\frac{r}{2\zeta}\right)^{2k+1}\right)$
(29)

with the coefficients

$\displaystyle\kappa^{-}_{1}(n,k)=\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\frac{(4n+1)[4k^{2}-(2n-1)(4k+1)(n+1)]}{(2n-1)(n+1)(n+1+k)^{2}(2n-1-2k)^{2}},$
(30)
$\displaystyle\kappa^{-}_{2}(n,k)=\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\Bigg{[}\frac{4n(n-1)}{(n+1)(n+2)(2n-1-2k)^{2}}-\frac{(2n+1)(2n+3)}{(2n-1)(2n-3)(n+k+1)^{2}}\Bigg{]}.$
(31)

The corresponding vorticity field as function of the polar coordinates follows
as,

$\omega^{-}(r,\phi)=-\frac{\alpha}{8\zeta^{2}}\sin(3\phi)\left(4\pi
I_{3}(r/\zeta)+\sum_{k,n}\left[(2k-1)\kappa^{-}_{1}(n,k)-(2k+5)\kappa^{-}_{2}(n,k)\right]\frac{1}{(k!)^{2}}\left(\frac{r}{2\zeta}\right)^{2k}\right)$
(32)

Figure 4: Flow streamlines (white arrow) around a $-1/2$ defect for $\alpha<0$
obtained from (a) full solution and (b) asymptotic one. The nematic director
field is shown in black lines and the background colormap denotes vorticity.
To show the structure of the near-field, the vorticity scale is saturated at
$\pm 0.2$ in a) and at $\pm 0.1$ in b). c) Cross section of vorticity obtained
from the exact solution (solid blue line) and the asymptotic limit (dotted
black line) at $x=0$ as a function of $y$. Notice that the vortices changes
sign rapidly at origin due to its multivalued phase (see Eq. 32). d) Cross
section of the velocity obtained from the exact solution (solid blue line) and
the asymptotic limit (dotted black line) at $y=0$ as a function of $x$.

###  4.2 Asymptotic far-field flow:

As with the $+1/2$ defect, the far-field asymptotic flow is dominated by the
leading order terms in the expansion, which can also be computed directly from
Eq. (28) in the limit of $r/\zeta\rightarrow\infty$. The result of this
calculation is that

$\displaystyle u^{-}(r,\phi)\underset{r/\zeta\gg
1}{\approx}=\frac{\alpha}{2r}\left[\left(\frac{\zeta}{r}\right)^{2}\left(15e^{4i\phi}+3e^{-2i\phi}\right)-\left(e^{4i\phi}+e^{-2i\phi}\right)\right].$
(33)

As for eq. (25) the $1/r$ term here was also obtained in Ref. [24]. The
vorticity related to this velocity is

$\omega^{-}(r,\phi)=\frac{3\alpha\sin(3\phi)}{r^{2}}\left(5\left(\frac{\zeta}{r}\right)^{2}-1\right).$
(34)

In this asymptotic approximation, the flow field is singular at the origin.
This singularity is however lifted by the higher order terms in the series
expansions, so that the exact flow is smooth everywhere. Fig. 4 shows the flow
streamlines with the vorticity field as the colormap for the asymptotic (in
panel a) and the exact solutions (in panel b), with the values scaled in the
same way as for fig. 3. Cross-sections of the vorticity and velocity at $y=0$
are plotted in panels (c)-(d) showing the singular behavior of the asymptotic
approximation at the origin, while it captures very well the far-field
behavior. The plots correspond to an extensile system with $\alpha<0$. The
$\sin(3\phi)$ factor in the vorticity divides the plane in six regions where
the sign of the vorticity is altered and making it multi-valued at the origin.
The size of the velocity is zero at origin as we discussed above. It increases
a bit outside before it starts to decay with increasing $r$ following the far-
field asymptotic behavior.

As for the $+1/2$ defect, the flow streamlines never closed in an infinite
system, thus there are no finite size vortices. In the next section, we
discuss how the picture changes once the defect is placed in bounded domain.

Figure 5: Flow streamlines (black lines) with vorticity as a colormap
background generated by a $+1/2$ (a,b) and a $-1/2$ (c,d) defect in disks of
different radii $R$ for an extensile system ($\alpha<0$). (a) and (c) are for
$R=1$ and (b) and (d) $R=50$.

##  5 Isolated defect in a bounded active nematic

The problem of finding the flow field around defects in a bounded domain is
challenging to solve analytically. Thus, we resort to numerical solutions of
the Stokes flow given by Eq. (4) in a disk of radius $R$ using finite element
methods and homogeneous boundary conditions (zero velocity). In addition, we
use the simplification that a single defect is imprinted in an uniform nematic
field, while the changes in the nematic orientation induced by confinement are
ignored [17].

The Stokes flow Eq. 4 is solved with FEniCS using Taylor-Hood elements, which
are quadratic for the velocity and linear for the pressure and vorticity [32,
33].

Fig. 5 shows the flow streamlines induced by a single $+1/2$ (a,b) and $-1/2$
(c,d) defect in a disk of radius $R$ for an extensile system for $\eta\neq 0$
and $\Gamma\neq 0$. The left and right columns correspond to $R=1$ and $R=50$,
respectively. In a bounded system, the vortical flows around each defect span
the system size, as also reported in Ref. [17] for $\Gamma=0$. However, due to
friction with the substrate, the flow decays on length scales larger than
$\ell_{d}$. This is evident by comparing the values in the far-field of
vorticity in left and right columns from Fig. 5, corresponding to $R=1$ (in
units of $\ell_{d}$) in (a,c) and $R=50$ (in units of $\ell_{d}$) in (b,d). We
notice that the center of a vortex is not fixed at the maximum of the
vorticity. This is due to the fact that the $\pm 1/2$ defects generate shear
flows which localise shear vorticity next to the defect cores. However, unlike
curvature vorticity in rotating flows which peaks at the vortex core, shear
vorticity is not necessarily an indication of the presence of vortices or
their location. In fact, with increasing $R$, the flow gradients near the
defect cores become sharper, the streamlines near the cores are "stretched" in
the radial direction, and the eyes of vortices move further from the origin.
In the limit $R\rightarrow\infty$, we expect vortices to get stretched out so
that flow streamlines close at infinity, and we recover the analytic flow
profiles.

In Fig. 6, we compare the cross-sectional profiles of velocity and vorticity
obtained from the numerical solution for a large system, $R=50$ ($\ell_{d}$)
to the analytical solutions. The analytical solutions is obtained by
truncating the summation in the full solution at $n=5000$ and $k=500$ up to
$r=15$ and then using the asymptotic solution for $r>15$. The plots of
velocity in panels (a, c) show that the numerical and analytical solutions
agree very well close to the defect cores, but deviate from each other near
the boundary. This is due to the imposed boundary conditions of vanishing
velocity field. The vorticity in Panels (b, d) agrees well in the entire
domain, with a small boundary effect due to vanishing velocity and vortices
spanning the system size. This effect is perhaps more visible for the negative
defect and decreases with increasing $R$.

Figure 6: Cross sections of the numerically obtained velocity (a,c) and
vorticity (b,d) profiles for $R=50$ (in units of $\ell_{d}$) (orange dotted
lines) are compared to the analytical solution for an infinite system (solid
blue lines) for a negative (a,b) and positive (c,d) defect. $x$ and $y$ are
also in units of $\ell_{d}$. The dashed black lines are the asymptotic
solutions.

The self-propulsion speed $v_{x}^{a}$ of the $+1/2$ defect is also affected by
the system size. If $\Gamma=0$ at the outset $v_{x}^{a}\sim R$, as noted in
Ref. [17]. Frictional damping screens out this divergence, yielding the finite
value given in Eq. (15) for $R\rightarrow\infty$. The numerical calculation
shows, however, that for smaller $R$ there are finite-size corrections to the
defect propulsive speed. These are displayed in Fig. 7, where we plot
$|v_{x}^{a}|$ as a function of $R$ obtained from the numerical solution of Eq.
4 for different values of $\eta$ and $\Gamma$. The horizontal dashed lines are
the analytical solution in the limit of infinite system, as given by
$v_{x}^{(a)}\approx\frac{\pi}{4}\frac{\alpha_{0}}{\sqrt{\eta\Gamma}}$, while
the dotted black line show the linear scaling with $R$ in the limit of zero
friction. We notice that viscosity $\eta$ determines the slope for $R$
dependence in small systems, while friction $\Gamma$ controls the cross-over
to the intrinsic constant speed. Note that the asymptotic constant values of
$v_{x}^{a}$ agree very well with the analytical prediction at $\zeta\gg 1$
because in the numerical computations the vortex core is actually set to zero
(hydrodynamic regime with $S=1$). For comparison, we also show in Fig. 7 (b)
the defect propulsion speed in the absence of friction $\Gamma=0$ from the
outset, where the speed increases linearly with the system size. The dotted
black line represents the analytical prediction as found in Ref.[17].

Figure 7: The self-propulsion speed of a $+1/2$ defect as a function of the
disk radius, $R$ for (a) different values of the parameters $\eta$ and
$\Gamma$ and (b) $\Gamma=0$. The black dotted line is a best fit line for the
system in (b), it is also plotted for the $\Gamma\neq 0$ systems ignoring the
constant term. Horizontal dashed lines are the analytical prediction for an
infinite system $v_{+}^{(a)}=\pi\alpha_{0}/(4\sqrt{\eta\tau})$ with the
rescaled activity $|\alpha_{0}|=1$.

##  6 Conclusion

In summary, we have evaluated the flow field induced by an isolated $\pm 1/2$
defect in an incompressible active nematic film on a substrate both for an
infinite system and finite-size disk. While the self-propulsion speed of a
$+1/2$ defect diverges with system size for an isolated film, we show
analytically that the presence of finite substrate friction $\Gamma$ cures
this divergence resulting in a finite speed
$v_{x}^{(a)}\approx\frac{\pi}{4}\frac{\alpha_{0}}{\sqrt{\eta\Gamma}}=\frac{\pi}{4}\frac{\alpha_{0}}{\eta}\ell_{d}$
that increases with the hydrodynamic dissipation length $\ell_{d}$. This is
also confirmed numerically in a finite disk with $R>\ell_{d}$. For small disks
with $R<\ell_{d}$, the active speed scales instead linearly with $R$.

Stable shear vortical flows are formed around the defects. In finite systems,
the size of the flow vortices is controlled by the dissipation length
$\ell_{d}$, hence spans the whole system if $\ell_{d}>R$. The eye of the
vortices shifts away from the defect core with increasing $R$. For infinite-
size systems, the flow streamlines close at infinity as predicted by the far-
field analytical solution. In the same limit, we showed that the absolute
value of the velocity decreases as $1/r$ for distances that are large compared
to the dissipation lengthscale, in agreement with previous studies. The $1/r$
far field decay of the flow created by defects may seem surprising as it
suggests that a defect acts like a point force. This behavior arises from the
long-range nature of the distortion of the texture created by defects. When
other defects are present (as required in the plane to guarantee zero net
topological charge), this decay is cut off by the defect separation. In finite
domains it is cutoff by the system size. The $1/r$ decay indicates, however,
that a multi-defect approach is needed to describe the defect gas, as
attempted in Refs. [34, 35].

In this work we have neglected the effect of the elastic stress. An
interesting extension would be to study the effects it would have on the flow
field, and also considering the effect of having multiple interacting defects.

No ethical dilemmas where encountered in the preparation of this paper.

This is primarily theoretical work and does not have any experimental data.
The computational data and codes for FEniCS are available on GitHub:
https://github.com/jonasron/Defect-Flows

J.R. derived the analytical results and performed finite-element simulations
for finite domains. L.A. verified all analytical calculations. All authors
contributed to a critical discussion of the analytical and numerical results
and participated in writing the manuscript. L.A. conceived and coordinated the
project.

We declare we have no competing interests.

J.R. and L.A. acknowledge support from the Research Council of Norway through
the Center of Excellence funding scheme, Project No. 262644 (PoreLab).

## Appendix:

##  Appendix A  Integrals for the $+1/2$ defect

Here, we provide the detailed steps that are taken to arrive at Eq. (20) from
Eq. (19). We start by changing to a complex representation $u=u_{x}+iu_{y}$
with complex coordinates $z=x+iy$ and $z^{\prime}=x^{\prime}+iy^{\prime}$. By
changing variables to $t=z^{\prime}-z$, and then to polar coordinates
$t=r^{\prime}e^{i\theta}=r^{\prime}\hat{z}$, we write Eq. (19) as

$u=\frac{\alpha}{4i\pi\zeta^{2}}\int
dr^{\prime}r^{\prime}K_{0}(r^{\prime}/\zeta)\oint_{\gamma}d\hat{z}\left(\frac{r^{\prime}\hat{z}^{2}+z\hat{z}}{(r^{\prime}+\bar{z}\hat{z})\sqrt{\hat{z}(r^{\prime}\hat{z}+z)(r^{\prime}+\bar{z}\hat{z})}}+\frac{1}{\sqrt{\hat{z}(r^{\prime}\hat{z}+z)(r^{\prime}+\bar{z}\hat{z})}}\right).$
(35)

Where $\gamma$ is the unit circle. We notice that the integral over $\hat{z}$
is over three branch points. $\hat{z}=0$ is always in the unit circle,
$\hat{z}=-z/r$ is inside when $|z|<r$ and $\hat{z}=-r/\bar{z}$ when $|z|>r$.
We consider the integral over $\hat{z}$ and start by looking at the last term.
Splitting up the square root, we write it as

$\frac{1}{\sqrt{\bar{z}r^{\prime}}}\oint_{\gamma}d\hat{z}\frac{1}{\sqrt{\hat{z}}\sqrt{(\hat{z}+z/r^{\prime})}\sqrt{(\hat{z}+r^{\prime}/\bar{z})}}.$
(36)

We see that for all values of $r^{\prime}$ we have two branch points inside of
the contour. Therefore, we write the integral as

$\frac{1}{\sqrt{\bar{z}r^{\prime}}}\oint_{\gamma}d\hat{z}\frac{1}{\sqrt{\hat{z}}\sqrt{(\hat{z}+a)}\sqrt{(\hat{z}+b)}}.$
(37)

From here $-a$ is the branch point inside of the contour, while $-b$ is the
point outside of the contour. We note that the complex numbers $a$ and $b$
have the same argument $\phi$ so we can write it as

$\frac{1}{\sqrt{\bar{z}r^{\prime}}}\oint_{\gamma}d\hat{z}\frac{1}{\sqrt{\hat{z}}\sqrt{(\hat{z}+ae^{i\phi})}\sqrt{(\hat{z}+be^{i\phi})}}.$
(38)

$a$ and $b$ are now either $r/r^{\prime}$ and $r^{\prime}/r$. If we change our
integral variable from $\hat{z}$ to $\hat{u}=\hat{z}e^{-i\phi}$, we get

$\frac{1}{\sqrt{\bar{z}r^{\prime}}}\oint_{\gamma}d\hat{u}e^{i\phi}\frac{1}{\sqrt{e^{i\phi}\hat{u}}\sqrt{(e^{i\phi}\hat{u}+ae^{i\phi})}\sqrt{(e^{i\phi}\hat{u}+be^{i\phi})}}=\frac{e^{-i\phi/2}}{\sqrt{\bar{z}r^{\prime}}}\oint_{\gamma}d\hat{u}\frac{1}{\sqrt{\hat{u}}\sqrt{(\hat{u}+a)}\sqrt{(\hat{u}+b)}}$
(39)

Note that all branch points $\hat{z}=0$,$-a$ and $-b$ are now located on the
real axis. We now have to consider what branch cuts we want to use to preform
this integral. We consider the integral over the domain as shown in Fig. 8.

Figure 8: Sketch of the keyhole contour in the complex plane associated with
the integral in Eq. 38.

Here we have cut out a area around the branch cut in order to avoid problems.
The key hole consists of a circle $C_{1}$ with radius $\epsilon$ around $-a$,
the circle $C_{2}$ around the origin and the lines connecting them which is
$\epsilon$ above or below the real line as shown in Fig. 8. Since there are no
poles in the domain between the two contours, the integral of them has to be
the same [36]. The contour integral becomes

$\displaystyle\int_{\gamma}\frac{e^{-i\phi/2}}{\sqrt{\bar{z}r^{\prime}}}d\hat{u}\frac{1}{\sqrt{\hat{u}}\sqrt{(\hat{u}+a)}\sqrt{(\hat{u}+b)}}$
$\displaystyle=\left(\int_{-a-i\epsilon\rightarrow-i\epsilon}+\int_{i\epsilon\rightarrow-a+i\epsilon}+\int_{C_{1}}+\int_{C_{2}}\right)\frac{e^{-i\phi/2}}{\sqrt{\bar{z}r^{\prime}}}\frac{d\hat{u}}{\sqrt{\hat{u}}\sqrt{(\hat{u}+a)}\sqrt{(\hat{u}+b)}}.$
(40)

When $\epsilon\rightarrow 0$, the integrals over $C_{1}$ and $C_{2}$
disappears. The integral above the real line is just above the branch cut and
therefore positive, while the one below is negative. We therefore get

$\oint_{\gamma}\frac{e^{-i\phi/2}}{\sqrt{\bar{z}r^{\prime}}}d\hat{u}\frac{1}{\sqrt{\hat{u}}\sqrt{(\hat{u}+a)}\sqrt{(\hat{u}+b)}}=-2\frac{e^{-i\phi/2}}{\sqrt{\bar{z}r^{\prime}}}\int_{-a}^{0}d\hat{u}\frac{1}{\sqrt{\hat{u}}\sqrt{(\hat{u}+a)}\sqrt{(\hat{u}+b)}}.$
(41)

Evaluating this integral, we obtain

$\oint_{\gamma}\frac{e^{-i\phi/2}}{\sqrt{\bar{z}r^{\prime}}}d\hat{u}\frac{1}{\sqrt{\hat{u}}\sqrt{(\hat{u}+a)}\sqrt{(\hat{u}+b)}}=4i\frac{e^{-i\phi/2}}{\sqrt{\bar{z}r^{\prime}}}\frac{1}{\sqrt{b}}K\left(\frac{a}{b}\right).$
(42)

Here $K$ is the complete elliptic integral of the first kind with the power
series

$K(x)=\frac{\pi}{2}\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}x^{n}.$
(43)

With the double factorial $(2n)!!=2\cdot 4\cdot 6...(2n-2)\cdot 2n$.

We now consider the other integral over $\hat{z}$ in Eq. (35):

$\oint_{\gamma}d\hat{z}\frac{r^{\prime}\hat{z}^{2}+z\hat{z}}{(r^{\prime}+\bar{z}\hat{z})\sqrt{\hat{z}(r^{\prime}\hat{z}+z)(r^{\prime}+\bar{z}\hat{z})}}$
(44)

This is not symmetric in the branch points as the other integral was. We can
write this as

$r^{\prime
1/2}\bar{z}^{-3/2}\oint_{\gamma}d\hat{z}\frac{\hat{z}^{1/2}(\hat{z}+e^{i\phi}r/r^{\prime})^{1/2}}{(\hat{z}+e^{i\phi}r^{\prime}/r)^{3/2}},$
(45)

and we can rotate the integral variable to $u=\hat{z}e^{-i\phi}$ and get

$r^{\prime
1/2}\bar{z}^{-3/2}\oint_{\gamma}d\hat{u}e^{i\phi}\frac{\hat{u}^{1/2}e^{i\phi/2}(\hat{u}e^{i\phi}+e^{i\phi}r/r^{\prime})^{1/2}}{(\hat{u}e^{i\phi}+e^{i\phi}r^{\prime}/r)^{3/2}}=r^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\oint_{\gamma}d\hat{u}\frac{\hat{u}^{1/2}(\hat{u}+r/r^{\prime})^{1/2}}{(\hat{u}+r^{\prime}/r)^{3/2}}.$
(46)

Which of the branch points that are inside of the integral depends on the
value of $r^{\prime}$. If $r^{\prime}>r$ then $r/r^{\prime}$ is inside of the
units circle. We can then use the same branch cuts and arguments as before and
find

$r^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\oint_{\gamma}d\hat{u}\frac{\hat{u}^{1/2}(\hat{u}+r/r^{\prime})^{1/2}}{(\hat{u}+r^{\prime}/r)^{3/2}}=-2r^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\int_{-r/r^{\prime}}^{0}d\hat{u}\frac{\hat{u}^{1/2}(\hat{u}+r/r^{\prime})^{1/2}}{(\hat{u}+r^{\prime}/r)^{3/2}}.$
(47)

We can preform the integral and find that it equals

$-2r^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\frac{2i}{\sqrt{r^{\prime}/r}}[(2r^{\prime}/r-r/r^{\prime})K(r^{2}/r^{\prime
2})-2r^{\prime}/rE(r^{2}/r^{\prime 2})]$ (48)

Here $E(x)$ is the complete elliptic integral of the second kind with the
power series

$E(x)=\frac{\pi}{2}\left(1-\sum_{n=1}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\frac{x^{n}}{2n-1}\right).$
(49)

The other possibility is that $r^{\prime}<r$. Now it is the $-3/2$ root that
is inside of the integral domain. In this case we cannot use the above
approach since the integrand does not go to zero when $\hat{z}$ goes to
$r^{\prime}/r$. However we can use the binomial expansion that is valid for
$|x|<1$. We have that

$r^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\oint_{\gamma}d\hat{u}\frac{\hat{u}^{1/2}(\hat{u}+a)^{1/2}}{(\hat{u}+b)^{3/2}}.$
(50)

Doing a change of variables to $h=\sqrt{u}$ with $dh=du/(2\sqrt{u})$. We then
get (remembering a factor half since because integrating once around $h$ we
have gone twice around $u$)

$r^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\oint_{\gamma}dh\frac{h^{2}(h^{2}+a)^{1/2}}{(h^{2}+b)^{3/2}}.$
(51)

We have that $|h|=1$, $a>1$ and $b<1$. We therefore write this

$r^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\oint_{\gamma}dh\frac{\sqrt{a}(1+h^{2}/a)^{1/2}}{h(1+b/h^{2})^{3/2}}.$
(52)

We now use the binomial expansion

$(1+x)^{r}=\sum_{n=0}^{\infty}{r\choose n}x^{n},$ (53)

and the integral becomes

$r^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\sqrt{a}\oint_{\gamma}\frac{dh}{h}\left(\sum_{n=0}^{\infty}{1/2\choose
n}\left(\frac{h^{2}}{a}\right)^{n}\right)\left(\sum_{k=0}^{\infty}{-3/2\choose
k}\left(\frac{b}{h^{2}}\right)^{k}\right).$ (54)

By the residue theorem only the terms where $k=n$ is contributing to the
integral. We therefore get

$\displaystyle r^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\sqrt{a}\oint_{\gamma}\frac{dh}{h}\left(\sum_{n=0}^{\infty}{1/2\choose
n}{-3/2\choose n}\left(\frac{b}{a}\right)^{n}\right)$ $\displaystyle=2\pi
ir^{\prime
1/2}\bar{z}^{-3/2}e^{i\phi/2}\sqrt{a}\left(\sum_{n=0}^{\infty}{1/2\choose
n}{-3/2\choose n}\left(\frac{b}{a}\right)^{n}\right)$ (55)

Inserting the expressions for $a$ and $b$ we find

$2\pi i\bar{z}^{-3/2}e^{i\phi/2}\sqrt{r}\sum_{n=0}^{\infty}{1/2\choose
n}{-3/2\choose n}\left(\frac{r^{\prime}}{r}\right)^{2n}.$ (56)

Now we have solved the integral over the angles $\hat{z}$. Inserting this into
the Eq. (35), we find

$\displaystyle
u=\frac{\alpha}{4i\pi\zeta^{2}}\int_{0}^{r}dr^{\prime}r^{\prime}K_{0}(r^{\prime}/\zeta)\Bigg{(}2\pi
i\bar{z}^{-3/2}e^{i\phi/2}\sqrt{r}\sum_{n=0}^{\infty}{1/2\choose
n}{-3/2\choose n}\left(\frac{r^{\prime}}{r}\right)^{2n}$
$\displaystyle+\frac{4ie^{-i\phi/2}}{\sqrt{\bar{z}r^{\prime}}}\sqrt{\frac{r^{\prime}}{r}}K\left(\frac{r^{\prime
2}}{r^{2}}\right)\Bigg{)}$
$\displaystyle+\frac{\alpha}{4i\pi\zeta^{2}}\int_{r}^{\infty}dr^{\prime}r^{\prime}K_{0}(r^{\prime}/\zeta)\Bigg{(}\frac{4ie^{i\phi/2}\sqrt{r}}{\bar{z}^{3/2}}\left[\left(\frac{r}{r^{\prime}}-2\frac{r^{\prime}}{r}\right)K\left(\frac{r^{2}}{r^{\prime
2}}\right)+2\frac{r^{\prime}}{r}E\left(\frac{r^{2}}{r^{\prime
2}}\right)\right]$
$\displaystyle+\frac{4ie^{-i\phi/2}}{\sqrt{\bar{z}r^{\prime}}}\sqrt{\frac{r}{r^{\prime}}}K\left(\frac{r^{2}}{r^{\prime
2}}\right)\Bigg{)}$ (57)

Using that $\bar{z}=re^{-i\phi}$ and inserting the expressions for the $K$ and
$E$ this becomes

$\displaystyle
u=\frac{\alpha}{2\zeta^{2}}\int_{0}^{r}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r^{\prime}}{r}\right)^{2n+1}\left(e^{2i\phi}\sum_{n=0}^{\infty}{1/2\choose
n}{-3/2\choose
n}+\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\right)$
$\displaystyle+\frac{\alpha}{2\zeta^{2}}\int_{r}^{\infty}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r}{r^{\prime}}\right)^{2n}\Bigg{(}\sum_{n=0}^{\infty}\left[\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}-2\left(\frac{(2n+1)!!}{(2n+2)!!}\right)^{2}\frac{2n+2}{2n+1}\right]e^{2i\phi}$
$\displaystyle+\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\Bigg{)}.$
(58)

We can simplify this by using that

$\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}-2\left(\frac{(2n+1)!!}{(2n+2)!!}\right)^{2}\frac{2n+2}{2n+1}=-\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\frac{n}{n+1},$
(59)

and for the binomial

${1/2\choose n}{-3/2\choose
n}=-\frac{2n+1}{2n-1}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}.$ (60)

Inserting this we finally arrive at

$\displaystyle
u=\frac{\alpha}{2\zeta^{2}}\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\left(1-\frac{2n+1}{2n-1}e^{2i\phi}\right)\int_{0}^{r}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r^{\prime}}{r}\right)^{2n+1}$
$\displaystyle+\frac{\alpha}{2\zeta^{2}}\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\left(1-\frac{n}{n+1}e^{2i\phi}\right)\int_{r}^{\infty}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r}{r^{\prime}}\right)^{2n}.$
(61)

##  Appendix B  Integrals over the Bessel function

We start a change of variables to $r^{\prime\prime}=\zeta^{-1}r^{\prime}$, and
then evaluate the integrals over the Bessel functions as

$\zeta^{2n+2}\int_{0}^{r/\zeta}dr^{\prime\prime}K_{0}(r^{\prime\prime})r^{\prime\prime
2n+1}=\frac{1}{4}r^{2n+2}n!\Bigg{[}n!{}_{2}\tilde{F}_{3}\left(1+n,1+n;1,2+n,2+n;\frac{r^{2}}{4\zeta^{2}}\right)\\\
-2{}_{1}\tilde{F}_{2}\left(1+n;1,2+n;\frac{r^{2}}{4\zeta^{2}}\right)\left(\gamma+\ln\left(\frac{r}{2\zeta}\right)\right)+2\frac{\partial}{\partial
a_{1}}{}_{2}\tilde{F}_{3}\left(a_{1},1+n;1,1,2+n;\frac{r^{2}}{4\zeta^{2}}\right)|_{a_{1}=1}\Bigg{]}\\\
=\frac{1}{4}r^{2n+2}F^{r^{\prime}<r}\left(n,r/\zeta\right),$ (62)

and

$\displaystyle\zeta^{1-2n}\int_{r/\zeta}^{\infty}dr^{\prime}K_{0}(r^{\prime})\left(\frac{1}{r^{\prime}}\right)^{2n}$
$\displaystyle=\frac{1}{4}r^{-2n+1}\Gamma\left(\frac{1}{2}-n\right)\Bigg{[}4^{-n}\Gamma\left(\frac{1}{2}-n\right)\left(2\left(\frac{r}{\zeta}\right)^{2n-1}-4^{n}{}_{2}\tilde{F}_{3}\left(\frac{1}{2}-n,\frac{1}{2}-n;1,\frac{3}{2}-n,\frac{3}{2}-n;\frac{r^{2}}{4\zeta^{2}}\right)\right)$
$\displaystyle+2{}_{1}\tilde{F}_{2}\left(\frac{1}{2}-n;1,\frac{3}{2}-n;\frac{r^{2}}{4\zeta^{2}}\right)\left(\gamma+\ln\left(\frac{r}{2\zeta}\right)\right)-2\frac{\partial}{\partial
a_{1}}{}_{2}\tilde{F}_{3}\left(a_{1},\frac{1}{2}-n;1,1,\frac{3}{2}-n;\frac{r^{2}}{4\zeta^{2}}\right)|_{a_{1}=1}\Bigg{]}$
$\displaystyle=\frac{1}{4}r^{-2n+1}F^{r^{\prime}>r}\left(n,r/\zeta\right).$
(63)

Here the regularized hypergeometric function is defined as

${}_{p}\tilde{F}_{q}(a_{1},..,a_{p};b_{1}...,b_{q};x)=\frac{1}{\Gamma(b_{1})...\Gamma(b_{q})}\sum_{k=0}^{\infty}\frac{(a_{1})_{k}...(a_{p})_{k}}{(b_{1})_{k}...(b_{q})_{k}}\frac{x^{k}}{k!}.$
(64)

Where $(a)_{k}=a\cdot(a+1)...(a+k-1)$ is the rising factorial. These
expressions can be simplified. Using that $\Gamma(x)=(x-1)\Gamma(x-1)$, it
follows that the Pochhammer symbol is given as

$(a)_{k}=\frac{\Gamma(a+k)}{\Gamma(a)},$ (65)

with the derivative

$\partial_{a}(a)_{k}|_{a=1}=(a)_{k}(\psi^{(0)}(a+k)-\psi^{(0)}(a))|_{a=1}=k!(\psi^{(0)}(1+k)-\psi^{(0)}(1)).$
(66)

Here $\psi^{(0)}(k)$ is the digamma function, that is the first derivative of
the logarithm of the gamma function. For integer arguments, it is given as

$\psi^{(0)}(n)=-\gamma+\sum_{l=1}^{n-1}\frac{1}{l}.$ (67)

Using the relations above, we find after some algebra that the moments are
given by the power series

$F^{r^{\prime}<r}\left(n,r/\zeta\right)=\sum_{k=0}^{\infty}\left[\frac{1}{(n+k+1)}-2\left(\gamma+\ln\left(\frac{r}{2\zeta}\right)\right)+2\sum_{l=1}^{k}\frac{1}{l}\right]\frac{1}{(n+k+1)(k!)^{2}}\left(\frac{r}{2\zeta}\right)^{2k}$
(68)

and

$\displaystyle F^{r^{\prime}>r}\left(n,r/\zeta\right)$
$\displaystyle=4\sum_{k=0}^{\infty}\left[\sum_{l=1}^{k}\frac{1}{l}-\left(\gamma+\ln\left(\frac{r}{2\zeta}\right)\right)-\frac{1}{2n-1-2k}\right]\frac{1}{(2n-1-2k)(k!)^{2}}\left(\frac{r}{2\zeta}\right)^{2k}$
$\displaystyle+\frac{2\pi}{((2n-1)!!)^{2}}\left(\frac{r}{\zeta}\right)^{2n-1}.$
(69)

##  Appendix C Integrals for the $-1/2$ defect

Here we provide details on the calculation leading to Eq. (28). Using the
complex representation with similar coordinate transformations as for the
$+1/2$-defect, the corresponding active flow velocity,
$u^{-}=u^{-}_{x}+iu^{-}_{y}$ induced by active stress and pressure gradient
from Eq. (7) reads as

$u^{-}=-\frac{\alpha}{4i\pi\zeta^{2}}\int
dr^{\prime}r^{\prime}K_{0}(r^{\prime}/\zeta)\oint_{\gamma}\frac{d\hat{z}}{\hat{z}}\left(\frac{1}{r^{\prime}\hat{z}+z}\sqrt{\frac{r^{\prime}\hat{z}^{-1}+\bar{z}}{r^{\prime}\hat{z}+z}}+\frac{r^{\prime}\hat{z}+z}{(r^{\prime}\hat{z}^{-1}+\bar{z})^{2}}\sqrt{\frac{r^{\prime}\hat{z}+z}{r^{\prime}\hat{z}^{-1}+\bar{z}}}\right).$
(70)

We start by looking at the integral over $\hat{z}$ and we will first consider
the second term:

$\oint_{\gamma}\frac{d\hat{z}}{\hat{z}}\frac{r^{\prime}\hat{z}+z}{(r^{\prime}\hat{z}^{-1}+\bar{z})^{2}}\sqrt{\frac{r^{\prime}\hat{z}+z}{r\hat{z}^{-1}+\bar{z}}}=\frac{r^{\prime
3/2}}{\bar{z}^{5/2}}\oint_{\gamma}d\hat{z}\frac{\hat{z}^{3/2}(\hat{z}+z/r^{\prime})^{3/2}}{(\hat{z}+r^{\prime}/\bar{z})^{5/2}}.$
(71)

Using that $z=re^{i\phi}$ and changing variable to $u=e^{i\phi}\hat{z}$ we get

$\frac{r^{\prime
3/2}}{\bar{z}^{5/2}}\oint_{\gamma}d\hat{z}\frac{\hat{z}^{3/2}(\hat{z}+e^{i\phi}r/r^{\prime})^{3/2}}{(\hat{z}+e^{i\phi}r^{\prime}/r)^{5/2}}=\frac{r^{\prime
3/2}}{r^{5/2}}e^{4i\phi}\oint_{\gamma}du\frac{u^{3/2}(u+\frac{r}{r^{\prime}})^{3/2}}{(u+\frac{r^{\prime}}{r})^{5/2}}.$
(72)

This integral has three branch points on the real axis. Two of these points
are inside the integration domain and gives troubles. If $r^{\prime}>r$ we
have the two points $u=0$ and $u=r/r^{\prime}$ inside of the unit circle. In
this case one use the same key hole contour technique as 41 and get

$\frac{r^{\prime
3/2}}{r^{5/2}}e^{4i\phi}\oint_{\gamma}du\frac{u^{3/2}(u+\frac{r}{r^{\prime}})^{3/2}}{(u+\frac{r^{\prime}}{r})^{5/2}}=-2\frac{r^{\prime
3/2}}{r^{5/2}}e^{4i\phi}\int_{-r/r^{\prime}}^{0}du\frac{u^{3/2}(u+\frac{r}{r^{\prime}})^{3/2}}{(u+\frac{r^{\prime}}{r})^{5/2}}.$
(73)

Preforming this integral, we arrive at

$\frac{4i}{3}e^{4i\phi}\frac{r^{\prime}}{r^{2}}\left(\left[3\left(\frac{r}{r^{\prime}}\right)^{2}-16+16\left(\frac{r^{\prime}}{r}\right)^{2}\right]K\left(\frac{r^{2}}{r^{\prime
2}}\right)+\left[8-16\left(\frac{r^{\prime}}{r}\right)^{2}\right]E\left(\frac{r^{2}}{r^{\prime
2}}\right)\right),$ (74)

where the functions $K$ and $E$ are defined in appendix A. Now let us look at
the integral when $r^{\prime}<r$. In this case, we can not use the contour
approach because the integrand diverges near the $r^{\prime}/r$ pole. We,
therefore, use the binomial expansion to evaluate this integral. We first
change the variable to $h=\sqrt{u}$. The integral is then

$\frac{r^{\prime
3/2}}{r^{5/2}}e^{4i\phi}\oint_{\gamma}du\frac{u^{3/2}(u+\frac{r}{r^{\prime}})^{3/2}}{(u+\frac{r^{\prime}}{r})^{5/2}}=\frac{r^{\prime
3/2}}{r^{5/2}}e^{4i\phi}\oint_{\gamma}dh\frac{h^{4}(h^{2}+a)^{3/2}}{(h^{2}+b)^{5/2}}.$
(75)

We have introduced $a=\frac{r}{r^{\prime}}$ and $b=\frac{r^{\prime}}{r}$ and
$|h|=1$, $|b|<1$ and $|a|>1$. We therefore write this as

$\frac{r^{\prime
3/2}}{r^{5/2}}a^{3/2}e^{4i\phi}\oint_{\gamma}dh\frac{h^{4}(1+h^{2}/a)^{3/2}}{h^{5}(1+b/h^{2})^{5/2}}=\frac{1}{r}e^{4i\phi}\oint_{\gamma}\frac{dh}{h}\left(\sum_{k=0}^{\infty}{3/2\choose
k}\left(\frac{h^{2}}{a}\right)^{k}\right)\left(\sum_{n=0}^{\infty}{-5/2\choose
n}\left(\frac{b}{h^{2}}\right)^{n}\right).$ (76)

The residual theorem makes it so that only the terms with $k=n$ is relevant.
The integral finally becomes

$\frac{2\pi i}{r}e^{4i\phi}\sum_{n=0}^{\infty}{3/2\choose n}{-5/2\choose
n}\left(\frac{r^{\prime}}{r}\right)^{2n}.$ (77)

We now turn to the first term in the integral over $\hat{z}$ in eq. (70). It
is

$\oint_{\gamma}\frac{d\hat{z}}{\hat{z}}\frac{1}{r^{\prime}\hat{z}+z}\sqrt{\frac{r^{\prime}\hat{z}^{-1}+\bar{z}}{r^{\prime}\hat{z}+z}}.$
(78)

We change variables to $t=1/\hat{z}$ with $dt=-d\hat{z}/\hat{z}^{2}$. In
addition there comes a negative sign because we must reverse the contour. This
integral is then

$\oint_{\gamma}\frac{dt}{t}\frac{1}{r^{\prime}t^{-1}+z}\sqrt{\frac{r^{\prime}t+\bar{z}}{r^{\prime}t^{-1}+z}},$
(79)

which is same integral as in Eq. (44) with $z$ and $\bar{z}$ interchanged. We
can therefore use the solution we found in appendix A with
$\phi\rightarrow-\phi$. The velocity field is then

$\displaystyle
u=-\frac{\alpha}{4i\pi\zeta^{2}}\int_{0}^{r}dr^{\prime}r^{\prime}K_{0}(r^{\prime}/\zeta)\Bigg{\\{}\frac{2\pi
i}{r}e^{4i\phi}\sum_{n=0}^{\infty}{3/2\choose n}{-5/2\choose
n}\left(\frac{r^{\prime}}{r}\right)^{2n}$ $\displaystyle+\frac{2\pi
i}{r}e^{-2i\phi}\sum_{n=0}^{\infty}{1/2\choose n}{-3/2\choose
n}\left(\frac{r^{\prime}}{r}\right)^{2n}\Bigg{\\}}$
$\displaystyle-\frac{\alpha}{4i\pi\zeta^{2}}\int_{r}^{\infty}dr^{\prime}r^{\prime}K_{0}(r^{\prime}/\zeta)\Bigg{\\{}\frac{4i}{3}e^{4i\phi}\frac{r^{\prime}}{r^{2}}\Bigg{(}\left[3\left(\frac{r}{r^{\prime}}\right)^{2}-16+16\left(\frac{r^{\prime}}{r}\right)^{2}\right]K\left(\frac{r^{2}}{r^{\prime
2}}\right)$
$\displaystyle+\left[8-16\left(\frac{r^{\prime}}{r}\right)^{2}\right]E\left(\frac{r^{2}}{r^{\prime
2}}\right)\Bigg{)}+4ie^{-2i\phi}\frac{1}{r}\left[\left(\frac{r}{r^{\prime}}-2\frac{r^{\prime}}{r}\right)K\left(\frac{r^{2}}{r^{\prime
2}}\right)+\frac{2r^{\prime}}{r}E\left(\frac{r^{2}}{r^{\prime
2}}\right)\right]\Bigg{\\}}$ (80)

Inserting the expressions for $K$ and $E$, and using Eq. (60) and

$\sum_{n=0}^{\infty}{3/2\choose n}{-5/2\choose
n}=\frac{(2n+1)(2n+3)}{(2n-1)(2n-3)}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}.$
(81)

We finally arrive at

$\displaystyle u^{-}(r,\phi)=$
$\displaystyle-\frac{\alpha}{2\zeta^{2}}\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\frac{2n+1}{2n-1}\left[e^{4i\phi}\frac{2n+3}{2n-3}-e^{-2i\phi}\right]\int_{0}^{r}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r^{\prime}}{r}\right)^{2n+1}$
$\displaystyle-\frac{\alpha}{2\zeta^{2}}\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^{2}\frac{n}{n+1}\left[e^{4i\phi}\frac{n-1}{n+2}-e^{-2i\phi}\right]\int_{r}^{\infty}dr^{\prime}K_{0}(r^{\prime}/\zeta)\left(\frac{r}{r^{\prime}}\right)^{2n}.$
(82)

## References

  * [1] Sanchez T, Chen DT, DeCamp SJ, Heymann M, Dogic Z. 2012 Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–434.
  * [2] Guillamat P, Ignés-Mullol J, Sagués F. 2017 Taming active turbulence with patterned soft interfaces. Nature communications 8, 1–8.
  * [3] Needleman D, Dogic Z. 2017 Active matter at the interface between materials science and cell biology. Nature reviews materials 2, 1–14.
  * [4] Kumar N, Zhang R, De Pablo JJ, Gardel ML. 2018 Tunable structure and dynamics of active liquid crystals. Science advances 4, eaat7779.
  * [5] Doostmohammadi A, Ignés-Mullol J, Yeomans JM, Sagués F. 2018 Active nematics. Nature Communications 9, 1–13.
  * [6] Marchetti MC, Joanny JF, Ramaswamy S, Liverpool TB, Prost J, Rao M, Simha RA. 2013 Hydrodynamics of soft active matter. Reviews of Modern Physics 85, 1143.
  * [7] Saw TB, Xi W, Ladoux B, Lim CT. 2018 Biological tissues as active nematic liquid crystals. Advanced materials 30, 1802579.
  * [8] Mueller R, Yeomans JM, Doostmohammadi A. 2019 Emergence of active nematic behavior in monolayers of isotropic cells. Physical review letters 122, 048004.
  * [9] Kudrolli A, Lumay G, Volfson D, Tsimring LS. 2008 Swarming and swirling in self-propelled polar granular rods. Physical review letters 100, 058001.
  * [10] Simha RA, Ramaswamy S. 2002 Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles. Physical Review Letters 89, 058101.
  * [11] Juelicher F, Kruse K, Prost J, Joanny JF. 2007 Active behavior of the cytoskeleton. Physics reports 449, 3–28.
  * [12] Giomi L, Bowick MJ, Ma X, Marchetti MC. 2013 Defect Annihilation and Proliferation in Active Nematics. Physical Review Letters 110, 228101\.
  * [13] Lemma LM, DeCamp SJ, You Z, Giomi L, Dogic Z. 2019 Statistical properties of autonomous flows in 2D active nematics. Soft matter 15, 3264–3272.
  * [14] Thampi SP, Golestanian R, Yeomans JM. 2014 Vorticity, defects and correlations in active turbulence. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, 20130366.
  * [15] Doostmohammadi A, Shendruk TN, Thijssen K, Yeomans JM. 2017 Onset of meso-scale turbulence in active nematics. Nature communications 8, 1–7.
  * [16] Pismen LM. 2013 Dynamics of defects in an active nematic layer. Physical Review E 88, 050502.
  * [17] Giomi L, Bowick MJ, Mishra P, Sknepnek R, Cristina Marchetti M. 2014 Defect dynamics in active nematics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, 20130365\.
  * [18] Kawaguchi K, Kageyama R, Sano M. 2017 Topological defects control collective dynamics in neural progenitor cell cultures. Nature 545, 327–331.
  * [19] Saw TB, Doostmohammadi A, Nier V, Kocgozlu L, Thampi S, Toyama Y, Marcq P, Lim CT, Yeomans JM, Ladoux B. 2017 Topological defects in epithelia govern cell death and extrusion. Nature 544, 212–216.
  * [20] Balasubramaniam L, Doostmohammadi A, Saw TB, Sankara G, Mueller R, Dang T, Thomas M, Gupta S, Sonam S, Toyama Y et al.. 2020 Nature of active forces in tissues: how contractile cells can form extensile monolayers. bioRxiv.
  * [21] Vafa F, Bowick MJ, Shraiman BI, Marchetti MC. 2021 Fluctuations can induce local nematic order and extensile stress in monolayers of motile cells. Soft Matter 17, 3068–3073.
  * [22] Killeen A, Bertrand T, Lee CF. 2021 Polar Fluctuations Lead to Extensile Nematic Behavior in Confluent Tissues. arXiv preprint arXiv:2107.03838.
  * [23] Shankar S, Ramaswamy S, Marchetti MC, Bowick MJ. 2018 Defect unbinding in active nematics. Physical Review Letters 121, 108002.
  * [24] Angheluta L, Chen Z, Marchetti MC, Bowick MJ. 2021 The role of fluid flow in the dynamics of active nematic defects. New Journal of Physics 23, 033009.
  * [25] Pismen LM, Sagués F. 2017 Viscous dissipation and dynamics of defects in an active nematic interface?. The European Physical Journal E 40, 92.
  * [26] Shankar S, Marchetti MC. 2019 Hydrodynamics of Active Defects: from order to chaos to defect ordering. Physical Review X 9, 041047.
  * [27] Thijssen K, Nejad MR, Yeomans JM. 2020 Role of Friction in Multidefect Ordering. Physical Review Letters 125, 218004.
  * [28] Pismen LM et al.. 1999 Vortices in nonlinear fields: From liquid crystals to superfluids, from non-equilibrium patterns to cosmic strings vol. 100. Oxford University Press.
  * [29] Maitra A, Srivastava P, Marchetti MC, Lintuvuori JS, Ramaswamy S, Lenz M. 2018 A nonequilibrium force can stabilize 2D active nematics. Proceedings of the National Academy of Sciences 115, 6934–6939.
  * [30] Guillamat P, Ignés-Mullol J, Shankar S, Marchetti MC, Sagués F. 2016 Probing the shear viscosity of an active nematic film. Physical review E 94, 060602.
  * [31] Martínez-Prat B, Alert R, Meng F, Ignés-Mullol J, Joanny JF, Casademunt J, Golestanian R, Sagués F. 2021 Scaling regimes of active turbulence with external dissipation. arXiv preprint arXiv:2101.11570.
  * [32] Alnæs M, Blechta J, Hake J, Johansson A, Kehlet B, Logg A, Richardson C, Ring J, Rognes ME, Wells GN. 2015 The FEniCS project version 1.5. Archive of Numerical Software 3, 9–23.
  * [33] Logg A, Mardal KA, Wells GN et al.. 2012 Automated Solution of Differential Equations by the Finite Element Method. Springer.
  * [34] Vafa F, Bowick MJ, Marchetti MC, Shraiman BI. 2020 Multi-defect Dynamics in Active Nematics. arXiv preprint arXiv:2007.02947.
  * [35] Zhang YH, Deserno M, Tu ZC et al.. 2020 Dynamics of active nematic defects on the surface of a sphere. Physical Review E 102, 012607.
  * [36] Kreyszig E. 2011 Advanced Engineering Mathematics (International Student Version). Hoboken: John Wiley & Sons, inc 10 edition.