Spontaneous flow created by active topological defects

Abstract

Topological defects are at the root of the large-scale organization of liquid crystals. In two-dimensional active nematics, two classes of topological defects of charges $\pm 1/2$ are known to play a major role due to active stresses. Despite this importance, few analytical results have been obtained on the flow-field and active-stress patterns around active topological defects. Using the generic hydrodynamic theory of active systems, we investigate the flow and stress patterns around these topological defects in unbounded, two-dimensional active nematics. Under generic assumptions, we derive analytically the spontaneous velocity and stall force of self-advected defects in the presence of both shear and rotational viscosities. Applying our formalism to the dynamics of monolayers of elongated cells at confluence, we show that the non-conservation of cell number generically increases the self-advection velocity and could provide an explanation for their observed role in cellular extrusion and multilayering. We finally investigate numerically the influence of the Ericksen stress. Our work paves the way to a generic study of the role of topological defects in active nematics, and in particular in monolayers of elongated cells.

Graphical abstract

Appendices

Appendix A: Modified Bessel functions

1.1 Appendix A.1: Definition

The modified Bessel functions $I_\alpha ,K_\alpha$ are the two general solutions of the following equation:

$$\begin{aligned} r^2 f''(r) + r f'(r) - (\alpha ^2+r^2) f(r) = 0\ . \end{aligned}$$

(A.1)

A related equation to the modified Bessel Eq. (A.1) is given by Bowman [48]:

$$\begin{aligned} r^2 f''(r) + (1-2a) r f'(r) - (b^2 c^2 r^{2c} - a^2 + d^2 c^2) f(r) = 0\ . \nonumber \\ \end{aligned}$$

(A.2)

Two independent solutions of Eq. (A.2) are $r^a I_d(b r^c)$ and $r^a K_d(b r^c)$.

1.2 Appendix A.2: Derivatives

The first-order derivatives of the modified Bessel functions are given by:

$$\begin{aligned} I_\alpha '(r)&= I_{\alpha -1}(r)-\frac{\alpha }{r}I_\alpha \end{aligned}$$

(A.3)

$$\begin{aligned} K_\alpha '(r)&= -\left( K_{\alpha -1}(r)+\frac{\alpha }{r}K_\alpha \right) \end{aligned}$$

(A.4)

$$\begin{aligned} I_0'(r)&= I_1(r) \end{aligned}$$

(A.5)

$$\begin{aligned} K_0'(r)&= -K_1(r) \end{aligned}$$

(A.6)

1.3 Appendix A.3: Asymptotic expansions

We also use the following asymptotic expansions:

$$\begin{aligned} I_\alpha (r)&\underset{r \rightarrow 0}{\simeq }\frac{1}{\Gamma (\alpha +1)}\left( \frac{r}{2} \right) ^\alpha \end{aligned}$$

(A.7)

$$\begin{aligned} K_\alpha (r)&\underset{r \rightarrow 0}{\simeq } {\left\{ \begin{array}{ll} \frac{\Gamma (\alpha )}{2} \left( \frac{2}{r} \right) ^\alpha \quad \text {if } \alpha >0 \\ -\ln (\frac{r}{2}) - \gamma \quad \text {if } \alpha =0 \end{array}\right. } \end{aligned}$$

(A.8)

where $\gamma$ denotes here the Euler’s constant. At infinity, we use

$$\begin{aligned} I_\alpha (r)&\underset{r \rightarrow +\infty }{\simeq } \frac{e^r}{\sqrt{2\pi r}} \end{aligned}$$

(A.9)

$$\begin{aligned} K_\alpha (r)&\underset{r \rightarrow +\infty }{\simeq } \sqrt{\frac{\pi }{2r}} e^{-r} \end{aligned}$$

(A.10)

Appendix B: Velocity field

1.1 Appendix B.1: Limit of a vanishing rotational viscosity

This appendix is dedicated to the full computation of the velocity starting from Eq. (9). As justified in the main text, the stream function reads $\tilde{\psi }({\tilde{r}},\theta ) = \tilde{\psi }({\tilde{r}}) \sin \theta$. Integrating one Laplace operator gives:

$$\begin{aligned} \Delta \tilde{\psi }({\tilde{r}},\theta ) - \tilde{\psi }({\tilde{r}},\theta ) = \left[ A {\tilde{r}} + \frac{B}{{\tilde{r}}} + s \right] \sin \theta \ , \end{aligned}$$

(B.11)

where A and B are integration constants. The radial dependence of the stream function then satisfies:

$$\begin{aligned} \frac{\!\mathrm {d}^2 \tilde{\psi }({\tilde{r}})}{\!\mathrm {d}{\tilde{r}}^2} + \frac{1}{\tilde{r}}\frac{\!\mathrm {d}\tilde{\psi }({\tilde{r}})}{\!\mathrm {d}{\tilde{r}}} - \left( 1+\frac{1}{{\tilde{r}}^2}\right) \tilde{\psi }({\tilde{r}}) = A {\tilde{r}} + \frac{B}{{\tilde{r}}} + s\ . \nonumber \\ \end{aligned}$$

(B.12)

The homogeneous solution $\tilde{\psi }^0$ to this equation reads:

$$\begin{aligned} \tilde{\psi }^0({\tilde{r}}) = A_0 I_1({\tilde{r}}) + B_0 K_1({\tilde{r}})\ , \end{aligned}$$

(B.13)

where $A_0$ and $B_0$ are integration constants. The Wronskian associated with Eq. (B.12) reads

$$\begin{aligned} I_1({\tilde{r}}) K'_1({\tilde{r}}) - I'_1({\tilde{r}}) K_1({\tilde{r}}) = -\frac{1}{{\tilde{r}}}\ , \end{aligned}$$

(B.14)

which leads to:

$$\begin{aligned} {\tilde{\psi }}({\tilde{r}})&=\left\{ I_1({\tilde{r}})\left( A_0 - \int _{{\tilde{r}}}^{+\infty }{K_1(u)(Au^2 + su + B)\, \!\mathrm {d}u} \right) \right. \nonumber \\&+\left. K_1({\tilde{r}}) \left( B_0 - \int _0^{{\tilde{r}}}{I_1(u)(Au^2 + su + B)\, \!\mathrm {d}u} \right) \right\} \ . \end{aligned}$$

(B.15)

The velocity field $\mathbf {{\tilde{v}}} = {\tilde{v}}_r({\tilde{r}}) \cos \theta \, \mathbf {e_r} + {\tilde{v}}_\theta ({\tilde{r}}) \sin \theta \, \mathbf {e_\theta }$ is obtained from the derivatives of the stream function: ${\tilde{v}}_r({\tilde{r}}) = {\tilde{\psi }}({\tilde{r}})/ {\tilde{r}}$ and ${\tilde{v}}_\theta ({\tilde{r}}) = -\!\mathrm {d}{\tilde{\psi }}({\tilde{r}})/\!\mathrm {d}{\tilde{r}}$. We obtain

$$\begin{aligned}&\tilde{v}_r({\tilde{r}}) = {\tilde{r}}^{-1} \left[ I_1({\tilde{r}}) \left( A_0 - \int _{{\tilde{r}}}^{+\infty }{K_1(u)(Au^2 + su + B)\, \!\mathrm {d}u} \right) \right. \nonumber \\&\quad +\left. K_1({\tilde{r}}) \left( B_0 - \int _0^{{\tilde{r}}}{I_1(u)(Au^2 + su + B)\, \!\mathrm {d}u} \right) \right] \end{aligned}$$

(B.16)

$$\begin{aligned}&\tilde{v}_\theta ({\tilde{r}}) =\! -\left[ \left( I_0(\tilde{r})\!-\frac{I_1({\tilde{r}})}{{\tilde{r}}} \right) \!\! \left( A_0 - \int _{\tilde{r}}^{+\infty }\!{K_1(u)(Au^2 + su + B)\, \!\mathrm {d}u}\! \right) \right. \nonumber \\&\quad -\left. \left( K_0({\tilde{r}}) + \frac{K_1(\tilde{r})}{{\tilde{r}}} \right) \left( B_0 - \int _0^{{\tilde{r}}}{I_1(u)(Au^2 + su + B)\, \!\mathrm {d}u} \right) \right] \ . \end{aligned}$$

(B.17)

Imposing a finite velocity at the origin leads to $B_0 = 0$ and $B=0$, as a finite velocity at infinity leads to $A_0=0$. We then obtain $\tilde{v}_r({\tilde{r}}=0)=-(A + s\pi /4)$ and $\tilde{v}_\theta ({\tilde{r}}=0)= A + s\pi /4$ at the origin, and $\tilde{v}_r({\tilde{r}}=\infty )= -A$ and $\tilde{v}_\theta (\tilde{r}=\infty )= A$ at infinity. Imposing ${\tilde{v}}_r(0) = \tilde{v}_\theta (0) = 0$ yields $A=-s\pi /4$, and imposing ${\tilde{v}}_r(\tilde{r}=\infty ) = -{\tilde{v}}_0$ and ${\tilde{v}}_\theta ({\tilde{r}}=\infty ) = {\tilde{v}}_0$ yields $A = \tilde{v}_0$. Coming back to physical units, these lead to the self-advection velocity given by Eq. (11).

1.2 Appendix B.2: Velocity field with a finite rotational viscosity

We start the computation from Eq. (26) with the dependence ${\bar{\psi }}({\bar{r}},\theta ) = {\bar{\psi }}({\bar{r}}) \sin \theta$. Integrating one Laplace operator gives:

$$\begin{aligned} \frac{\!\mathrm {d}^2 {\bar{\psi }}(r)}{\!\mathrm {d}{\bar{r}}^2} + \frac{(1-\lambda )}{{\bar{r}}}\frac{\!\mathrm {d}{\bar{\psi }}(r)}{\!\mathrm {d}{\bar{r}}} - (1+\frac{1}{{\bar{r}}^2}) {\bar{\psi }} =A\,{\bar{r}} + \frac{B}{{\bar{r}}} + s \ . \nonumber \\ \end{aligned}$$

(B.18)

The left-hand side of this equation is of the form (A.2) with $a = \lambda /2$, $b=1$, $c=1$, and $d=\sqrt{1+\lambda ^2/4}$. Following Appendix A.1, two independent homogeneous solutions to Eq. (B.18) are ${\bar{r}}^{\lambda /2}I_{\alpha }({\bar{r}}),{\bar{r}}^{\lambda /2}K_{\alpha }({\bar{r}})$, with $\alpha = \sqrt{1+\lambda ^2/4}$. Following a similar procedure as in Appendix B.1, the dimensionless velocity field has the form:

$$\begin{aligned} {\bar{v}}_r = {\bar{r}}^{\lambda /2-1}&\left\{ I_\alpha ({\bar{r}}) \left( A_0 - \int _{\bar{r}}^{+\infty }{u^{-\lambda /2}K_\alpha (u)(Au^2 + su + B)\, \!\mathrm {d}u}\right) \right. \nonumber \\&+\left. K_\alpha ({\bar{r}}) \left( B_0 - \int _0^{\bar{r}}{u^{-\lambda /2}I_\alpha (u)(Au^2 + su + B)\, \!\mathrm {d}u}\right) \right\} \nonumber \\&\times \cos \theta \end{aligned}$$

(B.19)

$$\begin{aligned} {\bar{v}}_\theta = {\bar{r}}^{\lambda /2}&\left\{ \left( \frac{\alpha - \lambda /2}{{\bar{r}}} I_\alpha ({\bar{r}}) - I_{\alpha -1}({\bar{r}}) \right) \times \right. \nonumber \\&\left. \quad \left( A_0 - \int _{\bar{r}}^{+\infty }{u^{-\lambda /2}K_\alpha (u)(Au^2 + su + B)\, \!\mathrm {d}u}\right) \right. \nonumber \\&+ \left. \left( \frac{\alpha - \lambda /2}{{\bar{r}}} K_\alpha ({\bar{r}}) + K_{\alpha -1}({\bar{r}})\right) \right. \nonumber \\&\left. \quad \times \left( B_0 - \int _0^{\bar{r}}{u^{-\lambda /2}I_\alpha (u)(Au^2 + s u + B)\, \!\mathrm {d}u}\right) \right\} \sin \theta \ . \end{aligned}$$

(B.20)

The boundary conditions still set $A_0=0$, $B_0=0$, $B=0$, and $A={\bar{v}}_0$, similar as in Appendix B.1. The asymptotic expansion of the velocity field close to the core reads:

$$\begin{aligned} {\bar{v}}_r({\bar{r}})&\underset{{\bar{r}}\ll 1}{\simeq } -\frac{2^{-\alpha }}{\Gamma (\alpha +1)}(sC_1^\lambda + {\bar{v}}_0 C_2^\lambda ) {\bar{r}}^{\lambda /2+\alpha -1} \end{aligned}$$

(B.21)

$$\begin{aligned} {\bar{v}}_\theta ({\bar{r}})&\underset{{\bar{r}}\ll 1}{\simeq } \frac{2^{-\alpha }(\lambda /2 + \alpha )}{\Gamma (\alpha +1)} (sC_1^\lambda + {\bar{v}}_0 C_2^\lambda ) {\bar{r}}^{\lambda /2+\alpha -1} \ , \end{aligned}$$

(B.22)

with

$$\begin{aligned} C_1^\lambda&= \int _0^\infty {\!\mathrm {d}u \, u^{1-\lambda /2} K_\alpha (u)}\nonumber \\&=2^{- \lambda /2}\Gamma \left[ 1-\frac{1}{2}\left( \alpha +\frac{\lambda }{2} \right) \right] \, \Gamma \left[ 1+\frac{1}{2}\left( \alpha - \frac{\lambda }{2} \right) \right] \end{aligned}$$

(B.23)

$$\begin{aligned} C_2^\lambda&= \int _0^\infty {\!\mathrm {d}u \, u^{2-\lambda /2} K_\alpha (u)}\nonumber \\&= 2^{1- \lambda /2}\Gamma \left[ \frac{1}{2}\left( 3 - \alpha - \frac{\lambda }{2} \right) \right] \, \Gamma \left[ \frac{1}{2}\left( 3 + \alpha - \frac{\lambda }{2} \right) \right] \ . \end{aligned}$$

(B.24)

Contrary to the case of a vanishing rotational viscosity, the velocity field in the defect reference frame vanishes close to the core for any finite value of ${\bar{v}}_0$, since the exponent $\lambda /2+\alpha -1$ is positive. To set the value of ${\bar{v}}_0$, we must consider the tangential stress $\sigma _{r \theta }$. Using the angular dependence of the stream function and force balance, its dimensionless version reads

$$\begin{aligned} {\bar{\sigma }}_{r \theta }({\bar{r}},\theta )= \left( 2\frac{\lambda - 1}{{\bar{r}}}(\bar{v}_\theta ({\bar{r}})+{\bar{v}}_r({\bar{r}})) - {\bar{r}} {\bar{v}}_r({\bar{r}}) - {\bar{v}}_0\,{\bar{r}} \right) \sin \theta \ . \end{aligned}$$

(B.25)

Since $\lambda /2+\alpha -2$ is negative, ${\bar{v}}_0$ must equal $-sC_1^\lambda /C_2^\lambda$ for this tangential stress to remain finite at the origin. This leads to Eq. (27) in physical units.

Appendix C: Stall force

1.1 Appendix C.1: Limit of vanishing rotational viscosity

In the limit of vanishing rotational viscosity, the stall force is determined by the velocity and pressure solutions to Eqs. (13) and (14), together with the incompressibility condition $\nabla \cdot \mathbf {v}= 0$. Quantities defined inside the core $r<a$ bear the superscript ‘c.’ Using the dimensionless units of Sect. 2, the stream function satisfies:

$$\begin{aligned} \Delta \left[ \Delta \tilde{\psi } - \tilde{\psi }\right]&= -s\frac{\sin \theta }{\tilde{r}^2} \qquad {\tilde{r}}>{\tilde{a}} \end{aligned}$$

(C.26)

$$\begin{aligned} \Delta \left[ \Delta \tilde{\psi }^\mathrm {c} - \tilde{\psi }^\mathrm {c}\right]&= -\tilde{f}\frac{\sin \theta }{\tilde{r}^2} \qquad {\tilde{r}}<{\tilde{a}}\ , \end{aligned}$$

(C.27)

with ${\tilde{a}}=a/L$ and $\tilde{f} = f/(\pi a \zeta \Delta \mu )$ the normalized core radius and overall force applied to the core. The solution for the velocity field outside the core is given by Eqs. (B.16) and (B.17) with $A_0=0$ and $A=\tilde{v}_0$ as in Appendix B.1. The velocity field inside the core, however, reads:

$$\begin{aligned} \tilde{v}_r^\mathrm {c}({\tilde{r}})&= {\tilde{r}}^{-1} \left[ I_1(\tilde{r}) \left( A_0^\mathrm {c} + \int _{0}^{\tilde{r}}{K_1(u)(A^\mathrm {c}u^2 + \tilde{f} u + B^\mathrm {c})\, \!\mathrm {d}u} \right) \right. \nonumber \\&\qquad +\left. K_1({\tilde{r}}) \left( B^\mathrm {c}_0 - \int _0^{\tilde{r}}{I_1(u)(A^\mathrm {c}u^2 + \tilde{f}u + B^\mathrm {c})\, \!\mathrm {d}u} \right) \right] \end{aligned}$$

(C.28)

$$\begin{aligned} \tilde{v}^\mathrm {c}_\theta ({\tilde{r}})&= -\left[ \left( I_0({\tilde{r}})-\frac{I_1({\tilde{r}})}{{\tilde{r}}} \right) \times \right. \nonumber \\&\qquad \quad \left. \left( A^\mathrm {c}_0 + \int _{0}^{\tilde{r}}{K_1(u)(A^\mathrm {c}u^2 + \tilde{f} u + B^\mathrm {c})\, \!\mathrm {d}u} \right) \right. \nonumber \\&\quad - \left( K_0({\tilde{r}}) + \frac{K_1({\tilde{r}})}{{\tilde{r}}} \right) \times \nonumber \\&\qquad \quad \left. \left( B^\mathrm {c}_0 - \int _0^{\tilde{r}}{I_1(u)(A^\mathrm {c}u^2 + \tilde{f}u + B^\mathrm {c})\, \!\mathrm {d}u} \right) \right] \ . \end{aligned}$$

(C.29)

The boundary condition $\mathbf {v}^\mathrm {c}({\tilde{r}}=0)=\mathbf {0}$ imposes that $B^\mathrm {c}$, $B^\mathrm {c}_0$, and $A_0^\mathrm {c}$ vanish. The other integration constants are set by imposing the continuity of the velocity and stress fields at the boundary of the core $r=a$. Introducing $\tilde{P}$, the pressure normalized by $|\zeta | \Delta \mu /2$ and taking the divergence of the force balance Eqs. (13) and (14), we obtain:

$$\begin{aligned} \Delta \tilde{P}&= s\frac{\cos \theta }{{\tilde{r}}^2}\qquad \tilde{r}>{\tilde{a}} \end{aligned}$$

(C.30)

$$\begin{aligned} \Delta \tilde{P}^\mathrm {c}&= \tilde{f}\frac{\cos \theta }{\tilde{r}^2}\qquad {\tilde{r}}<{\tilde{a}}\ . \end{aligned}$$

(C.31)

These equations are solved by:

$$\begin{aligned} \tilde{P} ({\tilde{r}},\theta )&= \left[ A_* {\tilde{r}} + \frac{B_*}{{\tilde{r}}} - s \right] \cos \theta \end{aligned}$$

(C.32)

$$\begin{aligned} \tilde{P}^\mathrm {c} (r,\theta )&= \left[ A_*^\mathrm {c} {\tilde{r}} + \frac{B_*^\mathrm {c}}{{\tilde{r}}} - \tilde{f}\right] \cos \theta \ , \end{aligned}$$

(C.33)

where $A_*$, $B_*$, $A_*^\mathrm {c}$, and $B_*^\mathrm {c}$ are integration constants. Pressure and velocity are linked by force balance, imposing

$$\begin{aligned} B_*&= B \end{aligned}$$

(C.34)

$$\begin{aligned} B_*^\mathrm {c}&= B^\mathrm {c} \end{aligned}$$

(C.35)

$$\begin{aligned} A_*&= 0 \end{aligned}$$

(C.36)

$$\begin{aligned} A_*^\mathrm {c}&= A^\mathrm {c} - \tilde{v}_0\ . \end{aligned}$$

(C.37)

The normal and tangential components of the stress read $\sigma _{rr}=2\eta \partial _r v_r - P$ and $\sigma _{r\theta }=\eta [(\partial _\theta v_r -v_\theta )/r + \partial _r v_\theta ]$, respectively. We impose the continuity of the velocity and stress fields at the boundary of the core in $r=a$ in the limit of a small core size ${\tilde{a}}\ll 1$, in which case we can make use of the following asymptotic expressions:

$$\begin{aligned} \tilde{v}_r^\mathrm {c}({\tilde{r}})&\underset{{\tilde{r}}\ll 1}{\simeq } \frac{A^\mathrm {c}}{8} {\tilde{r}}^2 + \frac{{\tilde{f}}}{3} {\tilde{r}} \end{aligned}$$

(C.38)

$$\begin{aligned} \tilde{v}_r({\tilde{r}})&\underset{{\tilde{r}}\ll 1}{\simeq } -s\frac{C^0_1}{2}-\tilde{v}_0\frac{C^0_2}{2}+\frac{B_0}{\tilde{r}^2}+\frac{B}{2}\log {\tilde{r}} \end{aligned}$$

(C.39)

$$\begin{aligned} \tilde{v}_\theta ^\mathrm {c}({\tilde{r}})&\underset{{\tilde{r}}\ll 1}{\simeq } -\frac{3A^\mathrm {c}}{8} {\tilde{r}}^2 - \frac{2{\tilde{f}}}{3} {\tilde{r}} \end{aligned}$$

(C.40)

$$\begin{aligned} \tilde{v}_\theta ({\tilde{r}})&\underset{{\tilde{r}}\ll 1}{\simeq } s\frac{C^0_1}{2} + \tilde{v}_0\frac{C^0_2}{2}+\frac{B_0}{\tilde{r}^2}-\frac{B}{2}\log {\tilde{r}} \ , \end{aligned}$$

(C.41)

with $C_1^0$ and $C_2^0$ given, respectively, by eqs. (B.23) and (B.24) with $\lambda =0$. Finally, we obtain the stall force by imposing $\tilde{v}_0=0$. The result is given by Eq. (15), to leading order in $\tilde{a}=a/L=a\sqrt{\xi /\eta }$.

1.2 Appendix C.2: Finite rotational viscosity

The procedure to determine the stall force with a finite rotational viscosity resembles that presented in Appendix C.1. Since we assume no nematic order within the core region, the fields inside the core are unchanged as compared with Appendix C.1. The pressure field outside the core does not depend on the rotational viscosity and is unchanged. The quantity that changes is the velocity field outside the core region, now given by Eqs. (B.21–B.24). Close to the core, for $\bar{a} < \bar{r} \ll 1$, the velocity components have the following asymptotic expressions:

$$\begin{aligned} \bar{v}_r(\bar{r}) \underset{\bar{r}\ll 1}{\simeq }&-\frac{2^{-\alpha }}{\Gamma (\alpha +1)}(s\,C^\lambda _1+{\bar{v}}_0 \, C^\lambda _2) \bar{r}^{\lambda /2+\alpha -1} \nonumber \\&+2^{\alpha -1}B_0 \Gamma (\alpha ) \bar{r}^{\lambda /2-\alpha -1} - \frac{B}{\lambda } \end{aligned}$$

(C.42)

$$\begin{aligned} \bar{v}_\theta (\bar{r}) \underset{\bar{r}\ll 1}{\simeq }&\frac{2^{-\alpha }(\lambda /2+\alpha )}{\Gamma (\alpha +1)}(s\,C^\lambda _1+{\bar{v}}_0 \, C^\lambda _2) \bar{r}^{\lambda /2+\alpha -1}\nonumber \\&+2^{\alpha -1}(\alpha -\lambda /2)B_0 \Gamma (\alpha ) \bar{r}^{\lambda /2-\alpha -1} + \frac{B}{\lambda }\ , \end{aligned}$$

(C.43)

with the notations of Appendix B.2. Using the expressions Eqs. (C.32), (C.33), (C.38), (C.40), (C.42), and (C.43), continuity at the core boundary leads to the stalling force given by Eq. (29) for ${\bar{v}}_0=0$¹.

Appendix D: Cell division/extrusion

In Sect. 2.3, we introduce the Helmholtz decomposition $\mathbf {v} = \mathbf {\nabla } \times (\psi \ \mathbf {e}_z)+\mathbf {\nabla } \phi$. In dimensionless units, ${\tilde{\phi }}$ satisfies Eq. (20), which is solved by:

$$\begin{aligned} {\tilde{\phi }}({\tilde{r}})&=\left\{ I_1\left( \frac{\tilde{r}}{\delta }\right) \left( A^0_\phi - \int _{\tilde{r}/\delta }^{+\infty }{K_1(u)(A_\phi u^2 + su + B_\phi )\, \!\mathrm {d}u} \right) \right. \nonumber \\&+\left. K_1\left( \frac{{\tilde{r}}}{\delta }\right) \left( B^0_\phi - \int _0^{{\tilde{r}}/\delta }{I_1(u)(A_\phi u^2 + su + B_\phi )\, \!\mathrm {d}u} \right) \right\} \,, \end{aligned}$$

(D.44)

where $\delta =\sqrt{(\eta +\kappa )/\eta }$. The velocity then reads:

$$\begin{aligned} \tilde{v}_r({\tilde{r}})&= \left( \frac{\tilde{\psi }(\tilde{r})}{\tilde{r}} + \frac{\!\mathrm {d}\phi (\tilde{r})}{\!\mathrm {d}\tilde{r}}\right) \cos \theta \end{aligned}$$

(D.45)

$$\begin{aligned} \tilde{v}_\theta ({\tilde{r}})&= -\left( \frac{\!\mathrm {d}{\tilde{\psi }}(\tilde{r})}{\!\mathrm {d}\tilde{r}} + \frac{\tilde{\phi }(\tilde{r})}{{\tilde{r}}}\right) \sin \theta \ , \end{aligned}$$

(D.46)

where ${\tilde{\psi }}$ is given by Eq. (B.15) as in Appendix B.1. The force balance condition (8) then imposes:

$$\begin{aligned} B&= \frac{B_\phi }{\delta } \end{aligned}$$

(D.47)

$$\begin{aligned} A&+ \frac{A_\phi }{\delta } = \tilde{v}_0\ . \end{aligned}$$

(D.48)

Imposing that the divergence of the velocity field—or equivalently the proliferation rate k—does not diverge at the core nor at infinity, we find that $A^0_\phi$, $A_0$, $B^0_\phi$, $B_0$, B, and $B_\phi$ must all vanish. At the center of the defect ${\tilde{r}} =0$, the components of velocity then read $\tilde{v}_r({\tilde{r}} = 0) = -(\tilde{v}_0 + s (\pi /4)(1+1/\delta ))$ and $\tilde{v}_\theta ({\tilde{r}} = 0) = \tilde{v}_0 + s (\pi /4)(1+1/\delta )$. Imposing $\tilde{\mathbf {v}}({\tilde{r}} = 0) = \mathbf {0}$ in the reference frame of the default yields the self-advection velocity given by Eq. (21).

The divergence of the velocity field is then given by:

$$\begin{aligned} \nabla \cdot \tilde{\mathbf {v}} = -\frac{s}{\delta ^2}&\left[ I_1\left( \frac{{\tilde{r}}}{\delta }\right) \int _{{\tilde{r}}/\delta }^{+\infty }{K_1(u)\, u \, \!\mathrm {d}u} \right. \nonumber \\&\left. + K_1\left( \frac{{\tilde{r}}}{\delta }\right) \int _0^{\tilde{r}/\delta }{I_1(u)\,u\, \!\mathrm {d}u} -1\right] \cos \theta \ . \end{aligned}$$

(D.49)

It is represented in Fig. 3.

Appendix E: -1/2 defects

1.1 Appendix E.1: Velocity field

This appendix is dedicated to the computation of the velocity field around a $-1/2$ defect, plotted in Fig. 2. We add the superscript ‘-’ to denote the quantities associated with a $-1/2$ defect. Force balance reads:

$$\begin{aligned} \eta \Delta \mathbf {v}^-&- \mathbf {\nabla } P^- - \xi \mathbf {v}^- \nonumber \\&-\frac{\zeta \Delta \mu }{2r} \left( -\cos (3\theta )\, \mathbf {e}_r + \sin (3\theta )\, \mathbf {e}_\theta \right) = \mathbf {0} \ . \nonumber \\ \end{aligned}$$

(E.50)

Here the defect is immotile, such that the velocity field satisfies:

$$\begin{aligned} \mathbf {v}^-(+\infty ,\theta )&= \mathbf {0} \end{aligned}$$

(E.51)

$$\begin{aligned} \mathbf {v}^-(0,\theta )&= \mathbf {0} \ . \end{aligned}$$

(E.52)

The curl of Eq. (E.50) gives:

$$\begin{aligned} \Delta \left[ \eta \Delta \psi ^- - \xi \psi ^- \right] = \frac{3}{2}\zeta \Delta \mu \frac{\sin 3\theta }{r^2} \ . \end{aligned}$$

(E.53)

Using a normalization by a characteristic length $L=\sqrt{\eta /\xi }$ and a characteristic time $\tau ^- = 2\eta /(3|\zeta | \Delta \mu )$, we introduce the dimensionless stream function ${\tilde{\psi }}^- = (\tau ^-/L^2) \psi ^-$ and spatial variable ${\tilde{r}} = r/L$. We then have:

$$\begin{aligned} \Delta \left[ \Delta {\tilde{\psi }}^- - {\tilde{\psi }}^- \right] = s\frac{\sin 3\theta }{r^2}\ , \end{aligned}$$

(E.54)

where $s=\text {sign}(\zeta )$ and $\psi ^-$ is of the form $\tilde{\psi }^-({\tilde{r}},\theta ) = {\tilde{\psi }}^-({\tilde{r}}) \sin 3 \theta$. Integrating one Laplace operator in this equation leads to:

$$\begin{aligned} \frac{\!\mathrm {d}^2 {\tilde{\psi }}^-({\tilde{r}})}{\!\mathrm {d}{\tilde{r}}^2} + \frac{1}{{\tilde{r}}}\frac{\!\mathrm {d}{\tilde{\psi }}^-({\tilde{r}})}{\!\mathrm {d}\tilde{r}} - \left( 1+\frac{3}{{\tilde{r}}^2}\right) {\tilde{\psi }}^-({\tilde{r}}) = A^- {\tilde{r}} + \frac{B^-}{{\tilde{r}}} - s \ . \nonumber \\ \end{aligned}$$

(E.55)

Solving this equation leads to the following dimensionless velocity field:

$$\begin{aligned} {\tilde{v}}_r^- = s&\left\{ I_3({\tilde{r}}) \int _{{\tilde{r}}}^{+\infty }{ K_3(u) u \, \mathrm {d}u}\right. \nonumber \\&+\left. K_3({\tilde{r}}) \int _0^{{\tilde{r}}}{ I_3(u)u\, \mathrm {d}u} \right\} \cos 3\theta \end{aligned}$$

(E.56)

$$\begin{aligned} {\tilde{v}}_\theta ^- = s&\left\{ \left( \frac{3}{{\tilde{r}}}I_3(\tilde{r})-I_2({\tilde{r}}) \right) \int _{{\tilde{r}}}^{+\infty }{ K_3(u) u \, \mathrm {d}u}\right. \nonumber \\&+\left. \left( \frac{3}{{\tilde{r}}}K_3({\tilde{r}})+K_2({\tilde{r}}) \right) \int _0^{{\tilde{r}}}{ I_3(u)u\, \mathrm {d}u} \right\} \sin 3\theta \ , \end{aligned}$$

(E.57)

accounting for the boundary conditions Eqs. (E.51) and (E.52). This velocity field is plotted in Fig. 2b, d.

The pressure field $P^-$ is obtained by taking the divergence of Eq. (E.50):

$$\begin{aligned} \Delta P^- = \frac{3 \zeta \Delta \mu }{2r^2}\cos 3\theta \ , \end{aligned}$$

(E.58)

which yields, with the boundary conditions (E.51)–(E.52):

$$\begin{aligned} P^-(r,\theta ) = -\frac{3 \zeta \Delta \mu }{2}\cos 3\theta \ . \end{aligned}$$

(E.59)

1.2 Appendix E.2: Cell division/extrusion

We derive in this section the divergence of the velocity field represented in Fig. 3. The derivation resembles that for +1/2 defects as presented in Sect. 2.3 with a pressure-dependent proliferation rate given by Eq. (18). The velocity field is decomposed into $\mathbf {v}^- = \mathbf {\nabla } \times (\psi ^-\ \mathbf {e}_z)+\mathbf {\nabla } \phi ^-$. The divergence-free and curl-free parts of the velocity, respectively, satisfy:

$$\begin{aligned} \Delta \left[ \Delta \tilde{\psi }^- - \tilde{\psi }^-\right]&= s\frac{\sin 3\theta }{\tilde{r}^2} \end{aligned}$$

(E.60)

$$\begin{aligned} \Delta \left[ \Delta \tilde{\phi }^- - \frac{1}{\delta ^2}\tilde{\phi }^- \right]&= s\frac{\cos {3\theta }}{\delta ^2 \tilde{r}^2} \ . \end{aligned}$$

(E.61)

Solving Eq. (E.61), we get the divergence profile of the velocity field represented in Fig. 3b,d as:

$$\begin{aligned} \nabla \cdot \tilde{\mathbf {v}}^- = \frac{s}{\delta ^2}&\left[ I_3\left( \frac{{\tilde{r}}}{\delta }\right) \int _{{\tilde{r}}/\delta }^{+\infty }{K_3(u)\, u \, \!\mathrm {d}u} \right. \nonumber \\&\left. + K_3\left( \frac{{\tilde{r}}}{\delta }\right) \int _0^{\tilde{r}/\delta }{I_3(u)\,u\, \!\mathrm {d}u}-1\right] \cos 3\theta \ . \end{aligned}$$

(E.62)