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Figure 1. Stationary patterns upon the spontaneous flow instability. (A) Growth rate Eq. (8) of perturbations transverse to the director (¢ = 7/2) for a contractile system (S = —1) with R = 1. The crit- ical wavelength ~ @, decreases with the activity number A. Points indicate numerical results (Methods), with discrete wave numbers for a system of size L. (B) Stripe pattern and the corresponding spon- taneous shear flow for A = 200. (C) Stable vortex pattern at higher activity number A = 500. Small bars indicate the director (top). Lines with arrows indicate streamlines (bottom). Vortex patterns).

Figure 1 Stationary patterns upon the spontaneous flow instability. (A) Growth rate Eq. (8) of perturbations transverse to the director (¢ = 7/2) for a contractile system (S = —1) with R = 1. The crit- ical wavelength ~ @, decreases with the activity number A. Points indicate numerical results (Methods), with discrete wave numbers for a system of size L. (B) Stripe pattern and the corresponding spon- taneous shear flow for A = 200. (C) Stable vortex pattern at higher activity number A = 500. Small bars indicate the director (top). Lines with arrows indicate streamlines (bottom). Vortex patterns).

Related Figures (6)

Figure 2. Disordered pattern of director domains with a characteristic wavelength. (A) Snapshot of the angle field for a system size of 2048 x 2048 grid points at activity number Amax = 3.2 - 10°. (B) The spectrum of the Frank elastic energy Eq. (4) (Eq. (S29), units of K/Lmax) is peaked at an intrinsic wavelength \; ~ £- independent of system size. Here, £. is held fixed (Cc = Lmax/WAmax © Dmax /566), such that the activity number A = L* /£2 increases with system size L = n&. The wave number is rescaled by the largest system size Lomax, Such that the axis shows the mode number in the largest system (n = 2048).
Figure 2. Disordered pattern of director domains with a characteristic wavelength. (A) Snapshot of the angle field for a system size of 2048 x 2048 grid points at activity number Amax = 3.2 - 10°. (B) The spectrum of the Frank elastic energy Eq. (4) (Eq. (S29), units of K/Lmax) is peaked at an intrinsic wavelength \; ~ £- independent of system size. Here, £. is held fixed (Cc = Lmax/WAmax © Dmax /566), such that the activity number A = L* /£2 increases with system size L = n&. The wave number is rescaled by the largest system size Lomax, Such that the axis shows the mode number in the largest system (n = 2048).
Figure 3. Spectra of the four contributions to the energy balance Eq. (9) (Eq. (S32)). These results are for the largest system size with 2048 x 2048 grid points at activity number A = 3.2 - 10°. Fig. S2 shows results for smaller systems. The active energy injection is en- tirely balanced by dissipation at every scale; no energy is transferred between scales. The energy injection rate is maximal at the selected wavelength \,;.
Figure 3. Spectra of the four contributions to the energy balance Eq. (9) (Eq. (S32)). These results are for the largest system size with 2048 x 2048 grid points at activity number A = 3.2 - 10°. Fig. S2 shows results for smaller systems. The active energy injection is en- tirely balanced by dissipation at every scale; no energy is transferred between scales. The energy injection rate is maximal at the selected wavelength \,;.
Figure 4. Universal scaling of the flow spectra at large scales. (A) Snapshot of the stream function field for a system size of 2048 x 2048 grid points at activity number Amax = 3.2 - 10°. The small background ripples correspond to coherent vortices of typical size given by the scale of maximal energy injection 4;. However, large-scale correlated flow patches also form due to the non-local character of viscous flow. Lines with arrows indicate a few streamlines that highlight these large-scale circulations. (B-C) The spectra of kinetic energy and enstrophy Eq. (10) (Eqs. (S35) and (S38)) exhibit a peak at the maximal injection scale A;. At smaller and, importantly, also at larger scales, the spectra feature distinctive universal scaling regimes. As in Fig. 2B, @. is held fixed (€¢ = Dmax/VAmax © Dmax/566), such that the activity number A = L*/€2 increases with system size L = nA. The wave number is rescaled by the largest system size Lmax, such that the axis shows the mode number in the largest system (n = 2048).
Figure 4. Universal scaling of the flow spectra at large scales. (A) Snapshot of the stream function field for a system size of 2048 x 2048 grid points at activity number Amax = 3.2 - 10°. The small background ripples correspond to coherent vortices of typical size given by the scale of maximal energy injection 4;. However, large-scale correlated flow patches also form due to the non-local character of viscous flow. Lines with arrows indicate a few streamlines that highlight these large-scale circulations. (B-C) The spectra of kinetic energy and enstrophy Eq. (10) (Eqs. (S35) and (S38)) exhibit a peak at the maximal injection scale A;. At smaller and, importantly, also at larger scales, the spectra feature distinctive universal scaling regimes. As in Fig. 2B, @. is held fixed (€¢ = Dmax/VAmax © Dmax/566), such that the activity number A = L*/€2 increases with system size L = nA. The wave number is rescaled by the largest system size Lmax, such that the axis shows the mode number in the largest system (n = 2048).
Figure S1. Probability distribution of the active power J for different system sizes L = nA. Here, ¢. is held fixed (C2 = Dmax/VAmax © Imax /566), such that the activity number A = L? / 2 increases with system size L. Negative and positive values of I respectively cor- respond to active energy absorption and injection into the system. Therefore, on average, active stresses inject energy to drive the flows.
Figure S1. Probability distribution of the active power J for different system sizes L = nA. Here, ¢. is held fixed (C2 = Dmax/VAmax © Imax /566), such that the activity number A = L? / 2 increases with system size L. Negative and positive values of I respectively cor- respond to active energy absorption and injection into the system. Therefore, on average, active stresses inject energy to drive the flows.
Figure $2. Spectra of the four contributions to the energy balance Eq. (9) (Eq. (S32)) for different system sizes L = nA. Here, @ is held fixed (€. = Lmax/WVAmax © Dmax/566), such that the activity number A = L?/€? increases with system size L. The wave number is rescaled by the largest system size max, such that the axis shows the mode number in the largest system (n = 2048).
Figure $2. Spectra of the four contributions to the energy balance Eq. (9) (Eq. (S32)) for different system sizes L = nA. Here, @ is held fixed (€. = Lmax/WVAmax © Dmax/566), such that the activity number A = L?/€? increases with system size L. The wave number is rescaled by the largest system size max, such that the axis shows the mode number in the largest system (n = 2048).
Figure S3. Power spectrum of the vorticity source Eq. (S43) for different system sizes L = nA. Here, @. is held fixed (@. = Lmax/V Amax © LDmax/566), such that the activity number A = L? /€2 increases with system size L. The wave number is rescaled by the largest system size Lmax, such that the axis shows the mode number in the largest system (n = 2048).
Figure S3. Power spectrum of the vorticity source Eq. (S43) for different system sizes L = nA. Here, @. is held fixed (@. = Lmax/V Amax © LDmax/566), such that the activity number A = L? /€2 increases with system size L. The wave number is rescaled by the largest system size Lmax, such that the axis shows the mode number in the largest system (n = 2048).
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