3.1 Dilute suspension

We first examine the case of a confined dilute suspension by neglecting both the steric interaction stress ( $\unicode[STIX]{x1D701}=0$ ) and the constraining stress ( $\unicode[STIX]{x1D6FD}=0$ ), and start simulations from an initially near-isotropic state. As shown in figure 1(a–f), where $R_{1}=1.0$ and $R_{2}=2.0$ are fixed, characteristic nematic structures and flow patterns are highlighted at different regimes of $|\unicode[STIX]{x1D6FC}|$ . As shown in (a), a unidirectional circulating flow first appears when $|\unicode[STIX]{x1D6FC}|$ goes beyond a certain critical value $\unicode[STIX]{x1D6FC}_{cr}$ (in this case, $\unicode[STIX]{x1D6FC}_{cr}=-1.23$ ). While the nematic order is still low, as shown in (d), it clearly shows that particles become aligned in the azimuthal direction due to fluid shear (Saintillan & Shelley Reference Saintillan and Shelley2008; Thampi et al. Reference Thampi, Doostmohammadi, Golestanian and Yeomans2015), with a stronger alignment near rigid walls. As $|\unicode[STIX]{x1D6FC}|$ further increases up to 10.0 in (b), a circulating flow pattern still dominates but the streamlines exhibit periodic bending deformations in the radial direction to form travelling waves, leading to counterclockwise (CCW) and clockwise (CW) vortices near the inner and outer boundaries respectively. Such oscillatory patterns have also been seen in other active liquid crystal systems due to bending (or splay) deformations (Voituriez, Joanny & Prost Reference Voituriez, Joanny and Prost2005; Giomi et al. Reference Giomi, Mahadevan, Chakraborty and Hagan2011, Reference Giomi, Mahadevan, Chakraborty and Hagan2012). In (e), the flow-induced alignment is further enhanced to form low-order structures (blue colour). At even higher values of $|\unicode[STIX]{x1D6FC}|$ , the flow pattern in (c) becomes seemingly chaotic. The resultant nematic field exhibits a typical active liquid-crystalline phase (Sanchez et al. Reference Sanchez, Chen, Decamp, Heymann and Dogic2012; Giomi et al. Reference Giomi, Bowick, Ma and Marchetti2013; Thampi, Golestanian & Yeomans Reference Thampi, Golestanian and Yeomans2014; Gao et al. Reference Gao, Blackwell, Glaser, Betterton and Shelley2015b ; Shelley Reference Shelley2016), in which $\pm 1/2$ disclination defects stream around, and are constantly generated/annihilated during interactions (see f).

Figure 1. (a–c) Characteristic flow patterns (streamline overlaid on the colourmap of vorticity $\unicode[STIX]{x1D6FA}$ ) and (d–f) nematic structures (nematic director $\boldsymbol{m}$ field overlaid on the colourmap of scalar order parameter $s$ ) when choosing $R_{1}=1.0$ and $R_{2}=2.0$ . The insets in (a) and (d) are comparisons between the analytical (solid lines) and numerical (open circles) results for normalized $u_{\unicode[STIX]{x1D703}}$ and $D_{r\unicode[STIX]{x1D703}}$ components taken along the white lines when $|\unicode[STIX]{x1D6FC}|$ slightly goes beyond the critical value $|\unicode[STIX]{x1D6FC}|_{cr}=1.23$ for the isotropy–circulation instability. (g) Phase diagram of ( $|\unicode[STIX]{x1D6FC}|,R_{2}-R_{1}$ ) when fixing $R_{1}=1.0$ . The solid and dashed lines characterize the isotropy–circulation instability in an annulus and a straight channel respectively. The dash–dotted line characterizes the circulation–travelling-wave instability in a straight channel for sharply aligned rods by neglecting the rotational diffusion.

It should be noted that while our model is apolar, the observed phenomena are very similar to those reported for polar active suspensions where effects of self-swimming motions of microparticles (e.g. bacteria) are taken into account (Lushi, Wioland & Goldstein Reference Lushi, Wioland and Goldstein2014; Wioland et al. Reference Wioland, Lushi and Goldstein2016; Theillard et al. Reference Theillard, Alonso-Matilla and Saintillan2017). It is essential to recognize that the nonlinear dynamics is governed by a concatenation of hydrodynamic instabilities as $\unicode[STIX]{x1D6FC}$ increases, and can be characterized by a series of instabilities including isotropy to circulation, circulation to travelling wave and travelling wave to chaos. An overview of the instabilities is presented in a phase diagram plotted in the ( $|\unicode[STIX]{x1D6FC}|,R_{2}-R_{1}$ ) plane in (g), where $R_{1}$ is fixed to be 1.0. Furthermore, the marginal stability curve that delineates the isotropy–circulation instability can be calculated analytically in the polar coordinates. To do this, we perturb the near-isotropic solutions as $\unicode[STIX]{x1D63F}=\unicode[STIX]{x1D644}/2+\unicode[STIX]{x1D716}\unicode[STIX]{x1D63F}^{\prime }(r)$ and $\boldsymbol{u}=\unicode[STIX]{x1D716}\boldsymbol{u}^{\prime }(r)$ ( $\unicode[STIX]{x1D716}\ll 1$ ) by assuming azimuthal symmetry. Following the procedure of Woodhouse & Goldstein (Reference Woodhouse and Goldstein2012), we examine an initially exponential growth $\unicode[STIX]{x2202}\unicode[STIX]{x1D63F}^{\prime }/\unicode[STIX]{x2202}t=\unicode[STIX]{x1D705}\unicode[STIX]{x1D63F}^{\prime }$ with the growth factor $\unicode[STIX]{x1D705}\geqslant 0$ . After some algebra, we find $D_{rr}^{\prime }=D_{\unicode[STIX]{x1D703}\unicode[STIX]{x1D703}}^{\prime }=0$ , and the hydrodynamic instability arises from the following coupled linearized equations for $u_{\unicode[STIX]{x1D703}}^{\prime }$ and $D_{r\unicode[STIX]{x1D703}}^{\prime }$ :

where $\unicode[STIX]{x1D6FE}=-((16d_{R}+4\unicode[STIX]{x1D705}+\unicode[STIX]{x1D6FC})/4d_{T})$ . We are able to seek the solutions $D_{r\unicode[STIX]{x1D703}}^{\prime }=-(A_{0}/4\unicode[STIX]{x1D6FE}d_{T}r^{2})+A_{1}(\unicode[STIX]{x2202}\text{J}_{2}(\sqrt{\unicode[STIX]{x1D6FE}}r)/\unicode[STIX]{x2202}r)+A_{2}(\unicode[STIX]{x2202}\text{N}_{2}(\sqrt{\unicode[STIX]{x1D6FE}}r)/\unicode[STIX]{x2202}r)$ and $u_{\unicode[STIX]{x1D703}}^{\prime }=-(A_{0}/2r)(1+\unicode[STIX]{x1D6FC}/4\unicode[STIX]{x1D6FE}d_{T})+(2A_{1}/\sqrt{\unicode[STIX]{x1D6FE}})\text{J}_{1}(\sqrt{\unicode[STIX]{x1D6FE}}r)+(2A_{2}/\sqrt{\unicode[STIX]{x1D6FE}})\text{N}_{1}(\sqrt{\unicode[STIX]{x1D6FE}}r)+A_{3}r$ , where $\text{J}_{j}$ and $\text{N}_{j}$ are respectively Bessel functions of the first and second kinds, with coefficients $A_{j}$ to be determined. By applying boundary conditions $\unicode[STIX]{x2202}D_{r\unicode[STIX]{x1D703}}^{\prime }/\unicode[STIX]{x2202}r=0$ and $u_{\unicode[STIX]{x1D703}}^{\prime }=0$ on the two walls, we then solve the eigenvalues of $\unicode[STIX]{x1D6FE}$ numerically at $\unicode[STIX]{x1D705}=0$ (solid line in g), as well as the velocity and $\unicode[STIX]{x1D63F}$ field (insets in a,d), which, again, confirm our simulation results.

Next, we reveal the nature of the hydrodynamic instabilities and gain physical insight into the formation of the observed coherent structures in figure 1. As shown below, while analytical approaches are feasible for the isotropy–circulation and circulation–travelling-wave transitions, the travelling-wave–chaos instability is highly nonlinear and hence can only be explored numerically. Here, we avoid tedious mathematical manipulations in the polar coordinates to simplify our analysis by considering confinement in a straight periodic channel (i.e. in the limit $R_{1,2}\rightarrow \infty$ ), where very similar collective dynamics and hydrodynamic instabilities are observed (Theillard et al. Reference Theillard, Alonso-Matilla and Saintillan2017). For the two cases considered, we employ a homogeneous base-state solution of $\unicode[STIX]{x1D63F}_{0}$ without background flow (i.e. $\boldsymbol{u}_{0}=\mathbf{0}$ , $p_{0}=0$ ), and introduce perturbations ( $\unicode[STIX]{x1D63F}^{\prime },\boldsymbol{u}^{\prime },p^{\prime }$ ) to yield the following linearized equations:

with boundary conditions $\unicode[STIX]{x2202}\unicode[STIX]{x1D63F}^{\prime }/\unicode[STIX]{x2202}y=0$ and $\boldsymbol{u}^{\prime }=0$ on the upper ( $y=h$ ) and bottom ( $y=-h$ ) walls. (It should be noted here that $D_{11}^{\prime }+D_{22}^{\prime }=0$ .) We then take a Fourier transform in the $x$ direction of all perturbed quantities $\boldsymbol{f}^{\prime }$ such that $\boldsymbol{f}^{\prime }(x,y)=\tilde{\boldsymbol{f}}(k,y)\exp (\text{i}kx+\unicode[STIX]{x1D70E}t)$ , where $\tilde{\boldsymbol{f}}$ is the amplitude function, $k$ is the wavenumber and $\unicode[STIX]{x1D70E}$ is the complex-valued growth rate.

Case 1: isotropy–circulation instability. In this case, the instability grows from an initially isotropic state, i.e. $\unicode[STIX]{x1D63F}_{0}=\unicode[STIX]{x1D644}/2$ . The linearized equations in (3.3) and (3.5) become

where $\unicode[STIX]{x1D6FF}^{(n)}=\text{d}^{n}/\text{d}y^{n}$ represents the $n$ th derivative of $y$ , and $\unicode[STIX]{x1D714}=(\unicode[STIX]{x1D70E}+4d_{R})/d_{T}+k^{2}$ . We find that the general solution of the above equations has the vector form

where $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D714}+\unicode[STIX]{x1D6FC}/4d_{T}$ , and $\boldsymbol{F}_{j}$ can be written as

By implementing the eight boundary conditions $\unicode[STIX]{x1D6FF}\tilde{D}_{11}=\unicode[STIX]{x1D6FF}\tilde{D}_{12}=\unicode[STIX]{x1D6FF}\tilde{v}=\tilde{v}=0$ at $y=\pm h$ , we obtain a linear system for $B_{j}$ . The existence of non-trivial solutions gives $\unicode[STIX]{x1D714}$ and hence $\unicode[STIX]{x1D70E}$ . On the other hand, we find that the long-wave solution has to be solved separately due to a singularity at $k=0$ . By making use of the fact that the solution is unidirectional, we are able to obtain the equilibrium solution, $(\tilde{D}_{11},\tilde{D}_{12},\tilde{v})=(0,\sin (\sqrt{(\unicode[STIX]{x1D6FC}+4\unicode[STIX]{x1D70E}+16d_{R})/4d_{T}}y),\cos (\sqrt{(\unicode[STIX]{x1D6FC}+4\unicode[STIX]{x1D70E}+16d_{R})/4d_{T}}y))$ , which yields the growth rate

As shown by the comparisons in figure 2(a), we find that the real parts of the growth rates of the long-wave modes (diamond symbols) are always larger than those of the corresponding finite-wavelength ones (solid lines), and hence dominate the instability. Moreover, (3.14) suggests a marginal stability condition, $|\unicode[STIX]{x1D6FC}|>4(\unicode[STIX]{x03C0}/2h)^{2}d_{T}+16d_{R}$ , which recovers the theoretical predictions for unbounded suspensions in the limit $h\rightarrow \infty$ (Gao et al. Reference Gao, Blackwell, Glaser, Betterton and Shelley2015b ). When making connections to the annulus geometry by replacing the gap width $2h$ by $R_{2}-R_{1}$ , we find that the marginal instability curve in figure 1(g) can be accurately fitted as a function of the form $K(R_{2}-R_{1})^{-2}d_{T}+16d_{R}$ but with a different constant $K$ due to curved geometries (in this case, $K\approx 28.7$ ). It is apparent that the prediction in the straight periodic channel (dashed line in figure 1 g) agrees well with that of the curved annulus, suggesting that (3.14) in fact provides a reasonable estimate of the time scale of the isotropy–circulation instability under a general parallel confinement.

Figure 2. (a) The growth rate $\text{Re}(\unicode[STIX]{x1D70E})$ for the long-wave (diamonds) and finite-wavelength (lines) instabilities as a function of the wavenumber $k$ for isotropy–circulation instabilities in a periodic straight channel when choosing $h=0.5$ , $d_{T}=d_{R}=0.025$ . (b) Streamlines overlaid on the colourmap of the $u$ velocity component at $\unicode[STIX]{x1D6FC}=-4.0$ . (c) Velocity profiles at $\unicode[STIX]{x1D6FC}=-2.0$ ( $u_{max}=0.07$ ) and $\unicode[STIX]{x1D6FC}=-4.0$ ( $u_{max}=0.13$ ) respectively.

Figure 3. (a) The growth rate $\text{Re}(\unicode[STIX]{x1D70E})$ as a function of the wavenumber $k$ for bending instabilities of the sharply aligned extensors when confined in a periodic straight channel when choosing $h=0.5$ , $d_{R}=0$ and $d_{T}=0.025$ . The maximum growth rates at the critical wavenumbers $k_{cr}$ are marked by solid circles. (b,c) The vorticity field $\unicode[STIX]{x1D6FA}$ as the bending instability starts to grow at $\unicode[STIX]{x1D6FC}=-4.0$ and $\unicode[STIX]{x1D6FC}=-8.0$ respectively. The solid black scales represent the length scales $\unicode[STIX]{x03C0}/k_{cr}$ that match the separation distance between the neighbouring vortices.

Case 2: circulation–travelling-wave instability. It needs to kept in mind that the circulation–travelling-wave instability is in fact a secondary instability developed from a shear-induced aligned state whose analytical form, however, is lacking. Nevertheless, we consider a reduced model where all of the extensor particles are initially aligned in the $x$ direction, and neglect the rotational diffusion by choosing $d_{R}=0$ (Gao et al. Reference Gao, Betterton, Jhang and Shelley2017). For such a ‘sharply aligned’ configuration, the homogeneous base-state solution is simply $\unicode[STIX]{x1D63F}_{0}=\operatorname{diag}\{1,0\}$ . After some algebra, we are able to eliminate several unknown variables, and show that the hydrodynamic instability is determined by the following sixth-order ordinary differential equation for the off-diagonal component $\tilde{D}_{12}$ :

which admits the general solution $\tilde{D}_{12}=\sum _{j=1}^{6}C_{j}\exp (\unicode[STIX]{x1D706}_{j}y)$ . The eigenvalues $\unicode[STIX]{x1D706}_{j}$ and hence the growth rate $\unicode[STIX]{x1D70E}$ are then solved numerically by applying the converted boundary conditions $(\unicode[STIX]{x1D70E}+d_{T}k^{2})\tilde{D}_{12}-d_{T}\unicode[STIX]{x1D6FF}^{(2)}\tilde{D}_{12}=0$ and $\unicode[STIX]{x1D6FF}\tilde{D}_{12}=\unicode[STIX]{x1D6FF}^{(3)}\tilde{D}_{12}=0$ at $y=\pm h$ , and seeking non-trivial solutions. The real parts of the growth rates as a function of $k$ are plotted in figure 3(a) at different values of $\unicode[STIX]{x1D6FC}$ , showing that the maximum growth rates occur at finite critical wavenumbers $k_{cr}>0$ . In (b,c), we set up the corresponding numerical simulations with the same initial conditions, and highlight the snapshots of the vorticity field $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y-\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x$ . The simulation results show the generation of fluid jets that are perpendicular to the particle alignment direction due to the nematic bending instability, which leads to uniformly spaced vortices of alternating signs. The separation distance between neighbouring vortex pairs can be accurately measured by the characteristic length scale $\unicode[STIX]{x03C0}/k_{cr}$ .

While the above analysis is for a reduced model by assuming particle alignment in the $x$ direction without background flow, it reveals the finite-wavelength nature of the bending instability, and provides quantitative measurements for the intrinsic time and length scales selected by the parallel confinement. Similarly to case 1, the predicted circulation–travelling-wave borderline (dash–dotted line in (g)) in the straight channel indeed agrees with the simulation results for the annulus geometry, especially in the regime of high $|\unicode[STIX]{x1D6FC}|$ and small $R_{2}-R_{1}$ , where particles are strongly aligned and hence close to the sharp-alignment approximation. Beyond the travelling-wave regime, the transition towards chaotic flows is highly nonlinear, and can only be determined by numerical simulations.

3.2 Concentrated suspension

In this section, we examine the nonlinear dynamics of concentrated suspensions where both $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D701}$ are non-zero. A mean-field torque is introduced to govern the rotational dynamics of extensor particles in the kinetic model through a Maier–Saupe steric potential (Maier & Saupe Reference Maier and Saupe1958). As shown by Gao et al. (Reference Gao, Blackwell, Glaser, Betterton and Shelley2015a ,Reference Gao, Blackwell, Glaser, Betterton and Shelley b , Reference Gao, Betterton, Jhang and Shelley2017), for rod-like particles, the resultant enhanced steric interactions spontaneously drive the system away from an initially isotropic state to form a nematically aligned state when $\unicode[STIX]{x1D701}>4d_{R}$ .

Compared with the dilute cases where isotropy is the only admissible equilibrium state, we have obtained the steady-state solutions of $\unicode[STIX]{x1D63F}$ without flows at relatively small values of $|\unicode[STIX]{x1D6FC}|$ , which exhibit much more complicated nematic configurations, as shown in figure 4. For passive particles, when choosing $\unicode[STIX]{x1D6FC}=0$ , figure 4(a) reveals that the system quickly relaxes to a homogeneous nematic state where the equilibrium solution satisfies a 2D distribution of Bingham form (Gao et al. Reference Gao, Blackwell, Glaser, Betterton and Shelley2015b , Reference Gao, Betterton, Jhang and Shelley2017; Gao & Li Reference Gao and Li2017),

where $\unicode[STIX]{x1D719}$ is the rod orientation angle and $\unicode[STIX]{x1D6FF}$ is a function of the coefficient $\unicode[STIX]{x1D709}=2\unicode[STIX]{x1D701}/d_{R}$ . Therefore, the degree of alignment is in fact determined by the ratio between $\unicode[STIX]{x1D701}$ and $d_{R}$ . As $\unicode[STIX]{x1D6FC}$ increases, it is seen that the nematic field lines in figure 4(b) become distorted to spiral outwards to the annulus boundaries, reminiscent of the defect structures of active nematics when confined in a circular disk (Woodhouse & Goldstein Reference Woodhouse and Goldstein2012; Gao et al. Reference Gao, Betterton, Jhang and Shelley2017). As $|\unicode[STIX]{x1D6FC}|$ approximately goes beyond 0.5, the nematic structures in figure 4(c,d) switch to become azimuthal-symmetric and independent of the values of $\unicode[STIX]{x1D6FC}$ , which can be solved analytically by setting the right-hand side of (2.5) to be zero and then seeking the steady-state solution of $\unicode[STIX]{x1D63F}$ as a function of $r$ only. The observed variations in the equilibrium nematic states can be illustrated by examining the system transient dynamics that evolves from near-isotropy. One interesting example is shown in figure 5 when choosing $\unicode[STIX]{x1D6FC}=-2.0$ . An isotropy–circulation instability first occurs in figure 5(a) to reorient particles, due to the combined effect of the circulating flow and the Maier–Saupe steric interactions. However, in this case, the confinement effect in the thin gap is so strong that it suppresses the bending deformation occurring in the radial direction. As a result, the induced circulating flow gradually diminishes and the system eventually relaxes to an axisymmetric equilibrium state where all particles are approximately aligned in the azimuthal direction; see the insets in figure 5(b–d).

Figure 4. Equilibrium solutions of nematically aligned states at different values of $\unicode[STIX]{x1D6FC}$ when choosing $R_{1}=1.0$ and $R_{2}=2.0$ ; the nematic director field is overlaid on the colourmap of $s$ . Inset in (d): comparisons between the analytical (solid line) and numerical (open symbols) results at $\unicode[STIX]{x1D6FC}=-0.5,-2.0$ for the $D_{rr}$ component. The numerical data are taken along the white line marked in (c).

Figure 5. Evolution of the flow patterns for a concentrated extensor suspension when choosing $\unicode[STIX]{x1D6FC}=-2.0$ , $R_{1}=1.0$ and $R_{2}=2.0$ . Insets: the nematic director field overlaid on the colourmap of $s$ .

Figure 6. (a–d) Flow patterns (streamline overlaid on the colourmap of $\unicode[STIX]{x1D6FA}$ ) and (e–h) nematic structures (nematic director field overlaid on the colourmap of $s$ ) that characterize the long-time dynamics of a confined extensor suspension when choosing $\unicode[STIX]{x1D6FC}=-2.0$ , $R_{1}=0.75$ and $R_{2}=2.0$ .

Figure 7. (a–d) Short-time evolution of the flow (velocity vector overlaid on the colourmap of $\unicode[STIX]{x1D6FA}$ ) and (e–h) nematic fields (nematic director overlaid on the colourmap of $s$ ) for the case shown in figure 6. In (e–h), the black arrows show the direction of the crack formation and $+1/2$ defect movement.

Increase of the gap width relaxes the confinement effect to facilitate the active flow generation through the bending instability, which can be similarly explained by the analysis in the dilute cases above. As shown in figure 6, the system long-time dynamics clearly exhibits an isotropy–circulation (b,f) and then a circulation–travelling-wave (c,g) instability. Compared with the stable structures observed in the dilute cases, the structure formation and evolution in concentrated suspensions are much more complicated. In (c), it is shown that the active flows bend the streamlines to form travelling waves, leading to six evenly spaced ‘incipient cracks’ growing from the inner wall into the bulk regime shown in (g). Disclination defects of charge $\pm 1/2$ are seen to be born along these cracks, move around and undergo nucleation/annihilation when interacting with each other as well as with rigid boundaries. Interestingly, the system gradually approaches a quasi-steady-state where four $+1/2$ defects oscillate around their equilibrium positions; see (h). The flow field remains travelling-wave-like but also periodically switches directions; see (d). The entire history of the complex dynamics is shown in the supplementary movie S1 (available at https://doi.org/10.1017/jfm.2017.759).

As shown in figure 7, a close look at the short-time dynamics of oscillating $+1/2$ defects and the accompanying swirling flow reveals the subtle interplay among the nematic structure variation, the active flow generation and their interactions with the curved walls. The swirling flow drives the upward movement of the $+1/2$ disclination defect and, in the mean time, induces an opposite bending deformation nearby to form an incipient crack. As shown in (f,g), the crack ‘head’ keeps moving downward to become a $+1/2$ defect (see (h)), while its ‘tail’ gradually merges with the up-moving  $+1/2$ defect before it hits the wall. Corresponding to the nematic structure change, the resultant CCW (CW) vorticity is seen to be strengthened (weakened) when the crack and defect interact, and then weakened (strengthened) during their annihilation. We find that such an oscillating state is very robust when choosing $R_{1}$ less than 1.5, and the gap width $R_{2}-R_{1}$ between 1.0 and 1.5 (see also movie S2 for the similar dynamics observed in a smaller annulus). As shown in movie S3, more complex defect dynamics and seemingly chaotic flows dominate when $R_{2}-R_{1}>2.0$ , resembling the active nematic flows observed in an unbounded domain (Gao et al. Reference Gao, Blackwell, Glaser, Betterton and Shelley2015b , Reference Gao, Betterton, Jhang and Shelley2017).