Cascade or not cascade? Energy transfer and elastic effects in active nematics
CCascade or not cascade? Energy Transfer and Elastic Effects in Active Nematics
Livio Nicola Carenza, Luca Biferale, and Giuseppe Gonnella Dipartimento di Fisica, Università degli Studi di Bari and INFN, via Amendola 173, Bari, I-70126, Italy Dipartimento di Fisica and INFN, Università di Roma “Tor Vergata”, Via Ricerca Scientifica 1, 00133 Roma, Italy
We numerically study the multi-scale properties of a d active gel to address the energy transfermechanism. We find that activity is able to excite long-ranged distortions of the nematic patterngiving rise to spontaneous laminar flows and to a chaotic regime by further increasing the rateof active energy injection. By means of a scale-to-scale spectral analysis we find that the gel isbasically driven by the local balancing between active injection and viscous dissipation, withoutany signal of non-linear hydrodynamical transfer and turbulent cascades. Furthermore, elasticitymay qualitatively play an important role by transferring energy from small to larger scales throughnemato-hydrodynamic interactions. INTRODUCTION
Active fluids are driven to an out-of-equilibrium stateby the injection of energy at microscopic length-scales [1,2]. Systems of biological origin, such as cytoskeletal orbacterial suspensions, can be made active by dispers-ing ATP or tuning oxygen concentration [3, 4], whilesynthetic realizations (Janus particles [5] or polyacrylicacid hydrogels [6]) are able to convert chemical energyinto motion. When activated, the individual constituentstend to arrange in a liquid-crystalline fashion, developingorientational (polar or nematic) order [7], in accordanceto their intrinsic (vector or head-tail) symmetry. Thevisco-elastic response of active gels to internal energy in-put has been widely analyzed in the past decade [2, 8],as we briefly summarize in the following. Varying thestrength of small-scale active injection leads to differentdynamical behaviors ranging from a quiescent state (seeFig. 1a) at low energy injection rate, dominated by elas-tic relaxation, to a spontaneous flow regime (Fig. 1b),where long-ranged elastic instabilities, induced by activ-ity, are able to produce and autonomously sustain lam-inar flows [7, 9–12] and give rise to many unexpectedbehaviors [13–18].Experiments on active gels have shown that activitymay further lead to a chaotic state by strengtheningelastic deformations, until the threshold of productionof topological defect pairs is reached [19]. This regime,characterized by the development of vortical flows, iscommonly –and misleadingly (as we will discuss in thisLetter)– addressed as active turbulence [3, 20–26], due tothe qualitative resemblance to hydrodynamic turbulenceat high Reynolds numbers Re [27].The mechanism at the base of such chaotic behavioris yet not fully understood. In particular, the ques-tion concerning how (and if ) energy may be transferredamong different length-scales by means of some non-linear interactions –as it happens in classic hydrody-namic turbulence– still remains unanswered. The tran-sition from the laminar towards the chaotic state hasbeen widely analyzed in terms of the topological prop- erties [33–35] and classified as belonging to the directpercolation universality class [21], for a system confinedin a channel with rigid walls. Furthermore, the spec-tral properties of active gels have been investigated indifferent experimental realizations [3, 20, 36, 37] andstudied by means of different numerical and analyticalmodels [31, 34, 38–41] which often do not agree witheach other and do not fit into the scenario of turbu-lence at high Re . In particular, in a d isotropic tur-bulent fluid –where non-linear advection overcomes vis-cous dissipation– a counter directional dual cascade takesplace with inverse energy cascade towards larger length-scales and direct enstrophy cascade to smaller ones [27].To speak about turbulence you need to be in presence ofat least two ingredients: a highly chaotic spatio-temporalbehaviour and an energy transfer (cascade) over a con-trollable scale separation between injection and dissipa-tive mechanisms. Conversely, active gels flow at negligi-ble Re ( (cid:46) − ), a regime where hydrodynamic advec-tion is not likely playing a significant role, precluding hy-drodynamic non-linearities to be responsible for the on-set of the instability in dense active suspensions [20, 42].Still, one may expect energy to be transferred by means ofnon-linear elastic interactions, analogously to what hap-pens in polymer solutions flowing at low Re [43–45].A systematic approach to address the energy transfermechanism in active gels was recently used by Urzay etal. [31] which showed the absence of hydrodynamic turbu-lence making use of the full nemato-hydrodynamic theoryfor active nematics. Later on, Alert et al. [41] analyticallyand numerically studied the energy balance in Fourierspace in a minimal model for uniaxial defect-free activenematics, finding an universal power-law scaling k − ofthe kinetic energy spectrum at small wave-numbers dueto the long-ranged visco-elastic interactions, without anyenergy cascade. The absence of hydrodynamic advec-tion was also reported by the authors of this paper forthe case of an active polar emulsion, even if no scale-invariance was found [46]. Interestingly, Linkmann etal. [47] have shown that the nemato-hydrodynamic equa-tions used in the aforementioned papers can be mappedinto the Eulerian Słomka-Dunkel model [3, 36, 48] where a r X i v : . [ c ond - m a t . s o f t ] J a n Figure 1.
Dynamical regimes of active nematics.
Numerical simulations of an active nematics with elastic constant K = 0 . at varying the intensity of the activity ζ . The picture is a summary of the three dynamical regimes occurringin active gels (whose spectral properties we discuss in this paper) as previously reported in [28–31]. Left panel shows theconfiguration of the nematic field (white rods) during the relaxing dynamics in proximity of two oppositely charged semi-integer defects in the quiescent regime ( i.e. at low activity ζ = 10 − ). Central panel shows a configuration at ζ = 5 × − inthe spontaneous flow regime. The nematic field undergoes an activity-induced banding instability that breaks the rotationalsymmetry, producing bands of laminar flow (colored arrows) which is stronger where elastic deformations are more pronounced.Right panel shows a typical chaotic configuration at strong activity ζ = 5 × − , where rotational symmetry is statisticallyrestored. This state is characterized by the formation of walls , narrow regions of strong banding deformations, which eventuallygive rise to the enucleation of topological defects [32]. Simulations are performed on a square grid of size L = 512 and differentportions of the systems are shown in the three panels. the range of scales and the rate of energy injection can beselected by tuning some model parameters. Controver-sially, this single-fluid model predicts energy to be trans-ferred between length-scales by means of advective in-teractions [38, 49, 50], thus leaving open the question ifthe chaotic regime found in the models of active gels canbe characterized as a turbulent cascade. The question isnot semantic, as in the presence of turbulence we mustbe able to control the intensity and the scale-extension ofthe fluctuating fields, and we expect universal behaviourindependent of the details of the forcing mechanisms.The goal of this paper is to study the energy transfermechanism in a well established model for active ne-matics [31, 34, 51], in order to disentangle all possiblecontributions behind the complex multi-scale behaviourincluding the –so far– elusive role of elasticity. Westart from the full nemato-hydrodynamic theory foractive nematics and we perform a systematic spectralanalysis to elucidate the role of reactive, elastic, kine-matic and dissipative contributions to the multi-scaleenergy dynamics. We confirm that kinematic advectivecontributions are factually negligible and dynamics tobe mostly driven by the mutual balancing of activeinjection and viscous dissipation without any importantturbulent energy transfer across scales, except for a smallcontribution given by elasticity which moves energyfrom small scales towards larger ones, giving rise to an effective small non-linear inverse energy transfer. Model
We model the dynamics of a bidimensional ac-tive gel by means of the Landau-De Gennes theory forLC [52]. We consider ρ and v , respectively the densityand the velocity of the fluid. The ordering properties ofthe LC are encoded in the nematic (trace-less and sym-metric) tensor Q αβ , whose principle eigenvector n –thedirector– defines the local preferential direction of align-ment of the LC. The dynamics of the active gel is ruledby the following set of equations: ( ∂ t + v · ∇ ) Q − S ( ∇ v , Q ) = Γ − H . (1) ρ ( ∂ t + v · ∇ ) v = −∇ p + ∇ · (cid:2) σ pass + σ act (cid:3) , (2)The first is the Beris-Edwards equation which defines therelaxation of the LC. Here Γ = 1 is the rotational viscos-ity, while S ( ∇ v , Q ) = ( ξ D + Ω )( Q + I / Q + I / ξ D − Ω ) − ξ ( Q + I / T r ( Q ∇ v ) , is the strain rotational derivatives where D αβ = ( ∂ α v β + ∂ β v α ) / is the strain rate tensor, Ω αβ = ( ∂ α v β − ∂ β v α ) / is the vorticity tensor and ξ is the dimensionless align-ment parameter, related to the shape of the suspended Figure 2.
Spectral properties of the turbulent state.
Panel (a) shows the normalized energy spectra ˜ E k = E k U O / ( l a (cid:82) dk S actk ) for some values of the activity parame-ter ζ and K = 0 . , where U is the typical flow velocity (seetext). Panel (c) shows the corresponding spectra of activeenergy input S actk . Panels (b) and (d) show the continuousbehavior respectively of the Weissenberg number and the en-tropy production at varying ζ for different values of the elasticconstant K . Simulations are performed on a square grid ofsize L = 512 . particles. Unless otherwise stated, we choose ξ = 0 . ,corresponding to flow aligning rods [53]. The tracelessmolecular field H ij = − (cid:16) δ F δQ ij (cid:17) tl is derived from theLandau-De Gennes free energy F = (cid:82) df , with the freeenergy density f given by the sum of a bulk contribution f bulk = A (cid:2) (cid:0) − χ (cid:1) Q − χ Q + χ Q (cid:3) and an elasticterm f el = K ( ∇ Q ) , where A = 0 . and K are respec-tively the bulk and elastic constant and χ is a parame-ter controlling the isotropic-nematic transition, occurringwhen χ > . [52]. Eq. (2) is the Navier-Stokes equation,where p is the ideal fluid pressure and the stress ten-sor has been divided in a passive ( σ pass ) and an active ( σ act ) part. The former accounts for dissipative and reac-tive effects and can be expressed as the sum of a viscouscontribution σ visc = 2 η D , with η the fluid viscosity, andan elastic one σ elαβ = − ξH αγ (cid:18) Q γβ + 13 δ γβ (cid:19) − ξ (cid:18) Q αγ + 13 δ αγ (cid:19) H γβ + 2 ξ (cid:18) Q αβ − δ αβ (cid:19) Q γµ H γµ + Q αγ H γβ − H αγ Q γβ . (3)The active stress is given by σ act = − ζ Q [54, 55]. Theconstant ζ , the activity, tunes the intensity of the ac- Figure 3.
Energy balance in Fourier space.
Time av-eraged power spectra at K = 0 . , for the spontaneous flowregime at ζ = 5 × − are shown in panel (a) and for thechaotic regime at ζ = 5 × − in panel (b). Notice that thestrength of active injection (and viscous dissipation) increasesby a factor ∼ . tive doping and describes extensile particles, if ζ > , orcontractile ones otherwise.We numerically integrate Eqs. (1) and (2) on a 2Dsimulation box of size L = 512 by means of a hybridlattice Boltzmann method [51], so that the fluid may bein principle slightly compressible. However, the Machnumber M a = ¯ v/c s (cid:28) in our simulations (where ¯ v denotes the average fluid velocity and c s the speed ofsound) and density fluctuations δρ ∼ M a , are to allpractical effect negligible. Therefore, in the following weassume density to be homogeneous ( ρ = 1 ) so that theflow field satisfies the condition of incompressibility ∇ · v = 0 . Simulations units can be mapped onto physicalones by fixing the grid spacing ∆ x = 5 µm , the time-step ∆ t = 20 ms and the force-scale f ∗ = 2 µN . In oursimulations the viscosity is set to η = 5 / correspondingto . kPas . OUR RESULTS
We vary the intensity of extensile activity and the elas-tic constant of the LC, moving from the quiescent regimeof Fig. 1(a) to the chaotic case shown in panel (c). It isknown that the dynamical equations (1) and (2) exhibitan elastic instability to bending deformations at increas-ing the activity parameter ζ [8, 34, 56]. This first occurswhen the active length-scale l a ∼ (cid:112) K/ζ drops under halfthe size of the system, being able to excite a long wave-length modulation of the nematic pattern, as shown inFig. 1(b) for the case at ζ = 5 × − . The deformationof the LC injects energy in the system by means of theactive stress σ act in the direction normal to the bend-ing. Our approach to quantitatively study the responseof elastic and dissipative contributions to active injectionis to consider a balance equation for the kinetic energy Figure 4.
Elastic energy transfer.
Elastic spectra S elk forsome values of ζ and K = 0 . . The color scale is the sameas in Fig. 2. The red (blue) line in the inset shows the ratiobetween the typical flow scale l v and the active injection l a (elastic l el = L/k el ) length-scale. in Fourier space: ∂ t E k + T k = S visck + S elk + S actk . (4)Here E k = (cid:104)| v k | (cid:105) / is the energy spectrum, (cid:104)·(cid:105) denot-ing the sum on shells of equal momentum ( | k | = k ), and T k = (cid:104) v ∗ k · J k (cid:105) is the rate at which energy is transferredby non-linear advective interactions, with J k the Fourierrepresentation of the hydrodynamic flux −∇ p + v · ∇ v .The terms on the right-hand side of Eq. (4) are respec-tively given by S visck = 2 πi (cid:104) v ∗ k ⊗ k : σ visc k (cid:105) /L = − π ηL k E k S elk = 2 πi (cid:104) v ∗ k ⊗ k : σ el k (cid:105) /L S actk = 2 πi (cid:104) v ∗ k ⊗ k : σ act k (cid:105) /L and represent the rate at which energy is absorbed, in-jected or –eventually– transferred among length-scales,by viscous, elastic and active contributions.By comparing the energy spectrum E k and the spec-trum of active injection S actk , respectively shown inFig. 2(a) and (c) for the case with smallest activity(spontaneous flow regime) at ζ = 5 × − , we observethat the flow develops fluctuations at the same length-scale where energy is injected, giving rise to the lam-inar flow observed in panel (b) of Fig. 1. As activityincreases, the active stress injects energy at larger andlarger wave-numbers (see Fig. 2(c)), thus strengtheningthe bending of the active nematics, leading to the pres-ence of narrow regions, walls [29, 34], characterized by Figure 5.
Elastic relaxation.
Dynamical response of theelastic spectrum S elk as activity is switched off starting from achaotic steady-state configuration at ζ = 5 × − . The timeiterations are reported in the legend in lattice units. strong deformations which eventually produce a prolif-eration of topological defects (point-like disclinations).These are singular regions where the nematic order islost and it is not possible to define the mean orientationof the LC molecules. Walls and disclinations play a rel-evant role on the onset of the chaotic regime since thestrong distortions in their neighborhood generate flowswhich are in turn responsible for the deformation of thewalls and the unbinding of more defects pairs [28, 57],thus driving the system towards the apparently turbu-lent state. The increasing amount of energy injected inthe system has the important effect of strengthening theflows, which develop on a wider range of scales, as sug-gested by the behavior of the energy spectra at largeactivity in Fig. 2(a). In order to characterize the hy-drodynamic response of the active nematic to elastic de-formations, we introduce an active Weissenberg number W i = U τ el /l a [44], where U = (cid:113)(cid:82) dkE k is the typicalflow velocity, τ el = U / ( (cid:82) dk |S elk | ) is the relaxation timeof the LC and l a = L/k a is the typical length-scale ofactive injection, with k a = ( (cid:82) dk S actk k ) / ( (cid:82) dk S actk ) . Thebehavior of W i at varying ζ (see Fig. 2(b)) reflects thecontinuous nature of the transition from the spontaneousflow towards the chaotic regime, without any singularityneither in the hydrodynamic nor in the energetic prop-erties of the system, regardless of the strength of the LCelasticity. This is also confirmed by the power-law scal-ing ∼ ζ of the entropy production s = η ( ∇ v tl ) + H ,shown in Fig. 2(d).As shown by the previous discussion, energy spectra donot provide enough information to disentangle the intri-cate transfer mechanism in a complex fluid. In particular,it is not possible on the basis of the spectrum only to un-derstand what are the physical mechanisms behind theformation of given structures at a given scale. In orderto achieve it, we analyzed the energy balance in Fourierspace by considering the scale-to-scale contributions ofthe terms in Eq. (4), shown in Fig. 3. We observe thatin the spontaneous flow regime (left panel, correspond-ing to Fig. 1(b)) only small wave-numbers contribute todynamics. This is because activity is only able to ex-cite smooth long-ranged deformations of the LC patternwhich set up and autonomously maintain laminar flowsspanning the whole system. The energy injected by ac-tivity (ochre line) is either dissipated by viscosity (gray)or used to sustain deformations in the LC pattern (red),leading to a localized balance, scale-by-scale, for the en-ergy dynamics.As ζ is increased and the system enters the chaoticregime, the rate of active injection considerably increasesas activity excites modes at larger and larger wave-numbers, while the energy scale of elastic effects remainsroughly unaltered, as suggested by a direct comparisonof the two cases shown in Fig. 3. In agreement with pre-vious works [31, 41], the dynamics is basically driven bythe local balancing between active injection and viscousdissipation, while hydrodynamic convective interactionsare practically negligible. Interestingly enough, we ob-serve here that the elastic term change sign developinga (small) non linear energy transfer from large towardssmall wave-numbers (see right panel of Fig. 3), due tocross-triadic interactions in S elk between the nematic ten-sor in Eq. (3) and the velocity field, thus opening the roadto have an equilibrium parameter in the system to con-trol the flow response. Thus, it is of interest to furtheranalyze the behavior of the elastic contribution at vary-ing the elastic constant K . For K (cid:46) − , the elasticterm S elk counters active injection, regardless of the in-tensity of the active pumping, by absorbing energy at anylength-scale, analogously to what happens in the sponta-neous flow regime. This picture drastically changes whenstiffer LC ( K (cid:38) − ) are considered. In this case, thebehavior of the elastic contribution develops a positivebranch at small wave-numbers and a negative one at large k (Fig. 3(b) and Fig. 4), giving rise to an effective energytransfer from small towards larger length-scales. Fig. 4shows the elastic spectrum S elk for K = 0 . for some val-ues of ζ . We observe that the non-linear energy transfersets up as ζ (cid:38) × − , corresponding to the threshold ofthe chaotic regime for the specific value of the elastic con-stant here considered. The amount of energy transferredamong scales grows with ζ and the elastic term behavesas an energy source, since the positive branch at smallwave-numbers is always larger than its negative counter-part at larger k . Interestingly the wave-number k el defin-ing the crossover between the source and the absorbentbranch of the elastic spectrum grows towards larger k asactivity is increased together with the injection length-scale l a (see Fig. 4). Moreover, we observe that the ratiobetween the typical flow scale l v = L ( (cid:82) dkE k / ( (cid:82) dkE k k ) and the injection scale slowly increases with ζ , contraryto what happens in fully developed d hydrodynamic tur-bulence, where the separation between the two length-scales can grow indefinitely [27]. Hence, the elastic non-linear term moves an amount of energy which is small ,if compared with the other contribution, over a limitedrange of scales so that energy is mostly dissipated at thesame scale where it is injected. Moreover, if the elastictransfer would play a relevant role, one would expect thestatistical properties of the flow to develop scale-invariantfeatures – i.e. a power low decay of the energy spectrumas in hydrodynamic turbulence– in contrast with our ob-servations. These results suggest that the elastic termdoes not establish an energy cascade even contributingto the overall dynamics with a small non-linear transfer.This is the result –more than the cause– of the tendencyof the LC to relax deformations induced by activity atsmall scales ( ∼ l a < l el = L/k el ), giving rise to the typi-cal configuration of Fig. 1(c) where narrow walls of strongbend deformations in the nematic pattern are spaced bywider aligned regions.The role of the elasticity is confirmed by a dynamicalexperiment consisting in observing the relaxing dynamicsof the LC. We start from the fully chaotic configurationof Fig. 1(c) at ζ = 5 × − and we switch the activity offto track the evolution of the elastic spectrum S elk duringrelaxation, as shown in Fig. 5. As activity does not sup-port any energy input anymore, the negative branch atlarge wave-numbers immediately fades away. The onlyenergy source is the elastic deformations of the nematicpattern, which progressively relax into an aligned equilib-rium configuration. In this case, we observe an inversionof the behavior of the elastic spectrum with a negativebranch at small k which removes energy from large-scalesto transfer it towards smaller scales, where it is dissipatedby viscosity. CONCLUSION
In this Letter we have numerically analyzed the con-tinuous transition from spontaneous flow towards thechaotic regime of an active nematics. We found thatthe transition is driven by activity by exciting distor-tions at larger and larger wave-numbers. By means ofa scale-to-scale analysis, we showed that non-linear hy-drodynamic interactions do not influence the dynamicsof the active gel which is basically ruled by the local bal-ancing between active energy injection and viscous dissi-pation, meaning that energy is mostly dissipated at thoselength-scales at which it has been injected, in line withthe small Reynolds number Re , measured from simula-tion which never exceeds . . This is in strict contrastwith the results obtained by means of a Słomka-Dunkelapproach [36, 38, 47], where the turbulent regime hashydrodynamic origin as it arises from an inverse turbu-lent cascade triggered by an ad hoc linear instability inthe NSE which leads to small scale injection through theaction of a hyper-viscous term. However, in this casethe Reynolds number Re is consistently greater than –a regime where inertial effects actually overcome viscousdissipation. The energy spectra in our simulations do notexhibit power-law scaling, contrary to what was reportedby Alert et al. in [41] for the simplified case of an uniaxialdefect-free active nematics. This may be due to the factthat biaxial fluctuations in the ordering properties of theLC may consistently alter the long-ranged interactions inthe chaotic state, leading to the loss of scale invariance–an aspect which deserves further investigation. Further-more, we found that the non-linear elastic terms mayqualitatively play an important role by moving energybetween different scales, developing a positive branch atsmall wave-numbers and a negative one at large k , thusleading to non-linear energy transfer. The rate at whichenergy is transferred among scales, though, is small com-pared with the injection/dissipation rate and the elasticcontributions cannot sustain any energy cascade. Hence,our results show that the full nemato-hydrodynamic the-ory for active gels does not exhibit typical turbulent dy-namics. Accumulation of energy at large scales occursas the effect of non-trivial couplings between the bandedpatterns of the LC –which lead to active energy input–and the velocity field. Nonetheless, the question whetherit exists a range of parameters where it is actually possi-ble to observe a fully developped inverse cascade still re-mains open. In such a case we expect the elastic energytransfer to drive the dynamics of the system and even-tually develop power-law scaling in an extended range ofscales comprised between the small scales of active injec-tion and the larger scales where the velocity field developsvortical structures. [1] S. Ramaswamy. The mechanics and statistics of activematter. Annu. Rev. Condens. Matter Phys. , 1:323, 2010.[2] M.C. Marchetti, J.F. Joanny, S. Ramaswamy, T.B. Liver-pool, J. Prost, M. Rao, and R.A. Simha. Hydrodynamicsof soft active matter.
Rev. Mod. Phys , 85:1143, 2013.[3] H.H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher,H. Lowen R.E. Goldstein, and J.M. Yeomans. Meso-scaleturbulence in living fluids.
Proc. Natl. Acad. Sci. , 109,2012.[4] P. Guillamat, Z. Kos, J. Hardoüin, M. Ravnik, andR. Sagués. Active nematic emulsions.
Science Advances ,4:4, 2018.[5] S. Ebbens, D.A. Gregory, G. Dunderdale, J.R. Howse,Y. Ibrahim, T.B. Liverpool, and R. Golestanian. Elec-trokinetic effects in catalytic platinum-insulator janusswimmers.
Europhys. Lett. , 106:5, 2014.[6] P.A. Korevaar, C.N. Kaplan, A. Grinthal, R.M. Rust,and J. Aizenberg. Non-equilibrium signal integration inhydrogels.
Nat. Comm. , 11:386, 2010. [7] T. Sanchez, D.T.N. Chen, S.J. Decamp, M. Heymann,and Z. Dogic. Spontaneous motion in hierarchically as-sembled active matter.
Nature , 491:431–434, 2012.[8] K. Kruse, J.F. Joanny, F. Jülicher, J. Prost, and K. Seki-moto. Asters, vortices, and rotating spirals in active gelsof polar filaments.
Phys. Rev. Lett. , 92:078101, 2004.[9] K.-T. Wu, J.B. Hishamunda, D.T.N. Chen, S.J. DeCamp, Y.-W. Chang, A. Fernández-Nieves, S. Fraden,and Z. Dogic. Transition from turbulent to coherentflows in confined three-dimensional active fluids.
Science ,355(6331), 2017.[10] R. Voituriez, J.F. Joanny, and J. Prost. Spontaneous flowtransition in active polar gels.
E.P.L. , 70:404, 2005.[11] L.N. Carenza, G. Gonnella, D. Marenduzzo, and G. Ne-gro. Rotation and propulsion in 3d active chiral droplets.
Proc. Natl. Acad. Sci. , 116(44):22065–22070, 2019.[12] G. Negro, A. Lamura, G. Gonnella, and D. Marenduzzo.Hydrodynamics of contraction-based motility in a com-pressible active fluid.
EPL , 127(5):58001, 2019.[13] A. Loisy, J. Eggers, and T.B. Liverpool. Active suspen-sions have nonmonotonic flow curves and multiple me-chanical equilibria.
Phys. Rev. Lett. , 121, 2018.[14] Shuo Guo, Devranjan Samanta, Yi Peng, Xinliang Xu,and Xiang Cheng. Symmetric shear banding and swarm-ing vortices in bacterial superfluids.
Proc. Natl. Acad.Sci. , 2018.[15] L. Giomi, T. B. Liverpool, and M. C. Marchetti. Shearedactive fluids: Thickening, thinning, and vanishing viscos-ity.
Phys. Rev. E , 81:051908, 2010.[16] G. Negro, L.N. Carenza, A. Lamura, A. Tiribocchi, andG. Gonnella. Rheology of active polar emulsions: fromlinear to unidirectional and unviscid flow, and intermit-tent viscosity.
Soft Matter , 15:8251–8265, 2019.[17] L.F. Cugliandolo, P. Digregorio, G. Gonnella, andA. Suma. Phase coexistence in two-dimensional pas-sive and active dumbbell systems.
Phys. Rev. Lett. ,119:268002, 2017.[18] L.N. Carenza, G. Gonnella, A. Lamura, D. Marenduzzo,G. Negro, and T. Tiribocchi. Soft channel formation andsymmetry breaking in exotic active emulsions.
Sci. Rep. ,(on press), 2020.[19] S.J. DeCamp, G.S. Redner, A. Baskaran, M.F. Hagan,and Z. Dogic. Orientational order of motile defects inactive nematics.
Nat. Mater. , 14:11110, 2015.[20] C. Dombrowski, L. Cisneros, S. Chatkaew, R.E. Gold-stein, and J.O. Kessler. Self-concentration and large-scale coherence in bacterial dynamics.
Phys. Rev. Lett. ,93:098103, 2004.[21] A. Doostmohammadi, T.N. Shendruk, K. Thijssen, andJ.M. Yeomans. Onset of meso-scale turbulence in activenematics.
Nat. Comm. , 8, 2017.[22] J. Dunkel, S. Heidenreich, K. Drescher, H.H. Wensink,M. Bär, and R.E. Goldstein. Fluid dynamics of bacterialturbulence.
Phys. Rev. Lett. , 110:228102, 2013.[23] G. Duclos, R. Adkins, D. Banerjee, M.S.E. Peterson,M. Varghese, I. Kolvin, A. Baskaran, R.A. Pelcovits, T.R.Powers, A. Baskaran, F. Toschi, M.F. Hagan, S.J. Stre-ichan, V. Vitelli, D.A. Beller, and Z. Dogic. Topologicalstructure and dynamics of three-dimensional active ne-matics.
Science , 367(6482):1120–1124, 2020.[24] P. Guillamat, J. Ignés-Mullol, and F. Sagués. Tamingactive turbulence with patterned soft interfaces.
Nat.Comm. , 8, 12 2017. [25] L.N. Carenza, G. Gonnella, D. Marenduzzo, and G. Ne-gro. Chaotic and periodical dynamics of active chiraldroplets.
Physica A , 559:125025, 2020.[26] G. Negro, L.N. Carenza, P. Digregorio, G. Gonnella, andA. Lamura. Morphology and flow patterns in highlyasymmetric active emulsions.
Physica A: Statistical Me-chanics and its Applications , 503:464 – 475, 2018.[27] A. Alexakis and L. Biferale. Cascades and transitionsin turbulent flows.
Phys. Rep. , 767-769:1 – 101, 2018.Cascades and transitions in turbulent flows.[28] L. Giomi, M.J. Bowick, P. Mishra, R. Sknepnek, andM.C. Marchetti. Defect dynamics in active nematics.
Philos. Trans. Royal Soc. A , 372, 2014.[29] S.P. Thampi, R. Golestanian, and J.M. Yeomans. Insta-bilities and topological defects in active nematics.
EPL ,105(1):18001, 2014.[30] L. Giomi, L. Mahadevan, B. Chakraborty, and M.F. Ha-gan. Excitable patterns in active nematics.
Phys. Rev.Lett. , 106:218101, May 2011.[31] J. Urzay, A. Doostmohammadi, and J.M. Yeomans.Multi-scale statistics of turbulence motorized by activematter.
Journal of Fluid Mechanics , 822, 07 2017.[32] F. Bonelli, L.N. Carenza, G. Gonnella, D. Marenduzzo,E. Orlandini, and A. Tiribocchi. Lamellar ordering,droplet formation and phase inversion in exotic activeemulsions.
Sci. Rep. , 9:2801, 2019.[33] S.P. Thampi, R. Golestanian, and J.M. Yeomans. Vortic-ity, defects and correlations in active turbulence.
Philo-sophical Transactions of the Royal Society of LondonA: Mathematical, Physical and Engineering Sciences ,372(2029), 2014.[34] L. Giomi. Geometry and topology of turbulence in activenematics.
Physical Review X , 5, 2015.[35] A.J. Tan, E. Roberts, S.A. Smith, U.A. Olvera,J. Arteaga, S. Fortini, K.A. Mitchell, and L.S. Hirst.Topological chaos in active nematics.
Nat. Phys. , 8 2019.[36] V. Bratanov, F. Jenko, and E. Frey. New class ofturbulence in active fluids.
Proc. Natl. Acad. Sci. ,112(49):15048–15053, 2015.[37] A. Creppy, O. Praud, X. Druart, P.L. Kohnke, andF. Plouraboué. Turbulence of swarming sperm.
Phys.Rev. E , 92:032722, 2015.[38] M. Linkmann, G. Boffetta, M.C. Marchetti, and B. Eck-hardt. Phase Transition to Large Scale Coherent Struc-tures in Two-Dimensional Active Matter Turbulence.
Phys. Rev. Lett. , 122:214503, 2019.[39] C.W. Wolgemuth. Collective swimming and the dynam-ics of bacterial turbulence.
Biophys. J. , 95(4):1564 –1574, 2008.[40] S.P. Thampi, R. Golestanian, and J.M. Yeomans. Veloc-ity correlations in an active nematic.
Phys. Rev. Lett. ,111:118101, Sep 2013. [41] R. Alert, J.-F. Joanny, and J. Casademunt. Universalscaling of active nematic turbulence.
Nat. Phys. , 3 2020.[42] T. Ishikawa, N. Yoshida, H. Ueno, M. Wiedeman,Y. Imai, and T. Yamaguchi. Energy transport in aconcentrated suspension of bacteria.
Phys. Rev. Lett. ,107:028102, Jul 2011.[43] A. Groisman and V. Steinberg. Elastic turbulence in apolymer solution flow.
Nature , 405:53–55, 2000.[44] V. Steinberg. Scaling relations in elastic turbulence.
Phys. Rev. Lett. , 123:234501, 2019.[45] A.N. Morozov and W. van Saarloos. An introductory es-say on subcritical instabilities and the transition to tur-bulence in visco-elastic parallel shear flows.
Phys. Rep. ,447(3):112 – 143, 2007.[46] L.N. Carenza, L. Biferale, and G. Gonnella. Multiscalecontrol of active emulsion dynamics.
Phys. Rev. Fluids ,5:011302, Jan 2020.[47] M. Linkmann, M.C. Marchetti, G. Boffetta, and B. Eck-hardt. Condensate formation and multiscale dynamicsin two-dimensional active suspensions.
Phys. Rev. E ,101:022609, Feb 2020.[48] J. Słomka and J. Dunkel. Generalized navier-stokes equa-tions for active suspensions.
Eur. Phys. J. , 224, 7 2015.[49] J. Słomka, P. Suwara, and J. Dunkel. The nature oftriad interactions in active turbulence.
J. Fluid Mech. ,841:702–731, 2018.[50] J. Słomka and J. Dunkel. Spontaneous mirror-symmetrybreaking induces inverse energy cascade in 3d active flu-ids.
Proc. Natl. Acad. Sci. , 114(9):2119–2124, 2017.[51] L.N. Carenza, G. Gonnella, A. Lamura, G. Negro, andA. Tiribocchi. Lattice boltzmann methods and activefluids.
Eur. Phys. J. E , 42(6):81, 2019.[52] P.G. de Gennes and J. Prost.
The physics of liquid crys-tals . The International series of monographs on physics83 Oxford science publications. Oxford University Press,2nd ed edition, 1993.[53] D. Marenduzzo, E. Orlandini, M.E. Cates, and J.M. Yeo-mans. Steady-state hydrodynamic instabilities of activeliquid crystals: Hybrid lattice boltzmann simulations.
Phys. Rev. E , 76:031921, 2007.[54] T.J. Pedley and J.O. Kessler. Hydrodynamic Phenomenain Suspensions of Swimming Microorganisms.
Annu. Rev.Fluid Mech. , 24(1):313, 1992.[55] Y. Hatwalne, S. Ramaswamy, M. Rao, and R.A. Simha.Rheology of active-particle suspensions.
Phys. Rev. Lett. ,92:118101, 2004.[56] R.A. Simha and S. Ramaswamy. Hydrodynamic fluc-tuations and instabilities in ordered suspensions of self-propelled particles.
Phys. Rev. Lett. , 89:058101, 2002.[57] T.N. Shendruk, A. Doostmohammadi, K. Thijssen, andJ.M. Yeomans. Dancing disclinations in confined activenematics.