Weakly nonlinear analysis of pattern formation in active suspensions

Ohm, Laurel; Shelley, Michael

doi:10.1017/jfm.2022.392

Cited by 9 publications

(10 citation statements)

References 57 publications

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“…Our results on stability may be viewed as complementary to the wealth of computational literature on dynamics in the unstable pusher region, where perturbations to the uniform isotropic equilibrium can be seen to give rise to the emergence of collective swimmer motion and large-scale flows [27,31,40,43,45,44,46,47,48,51]. In particular, our results highlight the complex role of swimming in these collective dynamics.…”

Section: Introductionsupporting

confidence: 67%

“…In particular, our results highlight the complex role of swimming in these collective dynamics. Without swimming, the isotropic state in pusher suspensions is always unstable for 0 ≤ ν, κ ≪ 1 [40], and, as we see here, swimming has a clear stabilizing effect. However, as noted in [45], swimming is also a necessary ingredient for the particle density fluctuations observed in simulations.…”

Section: Introductionsupporting

confidence: 63%

“…The implicit dispersion relation (2.24) can be solved numerically for λ as a function of the parameter ψ (see [27,45,44,47]; note that a different nondimensionalization from (A.4) is commonly used) from which we observe that (2.24) has a solution with Re(λ) ≥ 0 for ψ ≥ ψ * for some ψ * . In this situation the linearized equations (1.7)-(1.8) have growing modes which give rise to pattern formation and "bacterial turbulence" observed in numerical simulations [40,45,46,47,48].…”

Section: 20)mentioning

confidence: 99%

“…Remark 2.3. Note that the integral equation (2.24) is precisely the dispersion relation arising in the eigenvalue problem for the linearized operator (1.7), which has been studied in detail by various authors [27,40,44,45,47,51,52]. In particular, any λ satisfying (2.24) is an eigenvalue of the linearized operator (1.7).…”

Section: 20)mentioning

confidence: 99%

“…In the pusher case (ι = −), the stability threshold ψ * , defined in Lemma 2.2, corresponds precisely to the eigenvalue crossing studied in the computational literature [27,40,44,45,47,51]. It arises in the Penrose condition (terminology borrowed from kinetic theory) on the Volterra equation for ∇u in the proof of Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

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On the stabilizing effect of swimming in an active suspension

Albritton¹,

Ohm²

2022

Preprint

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We consider a kinetic model of an active suspension of rod-like microswimmers. In certain regimes, swimming has a stabilizing effect on the suspension. We quantify this effect near homogeneous isotropic equilibria ψ = const. Notably, in the absence of particle (translational and orientational) diffusion, swimming is the only stabilizing mechanism. On the torus, in the non-diffusive regime, we demonstrate linear Landau damping up to the stability threshold predicted in the applied literature. With small diffusion, we demonstrate nonlinear stability of arbitrary equilibrium values for pullers (rear-actuated swimmers) and enhanced dissipation for both pullers and pushers (front-actuated swimmers) at small concentrations. On the whole space, we prove nonlinear stability of the vacuum equilibrium due to generalized Taylor dispersion. Contents 1. Introduction 1 2. Landau damping 9 3. Taylor dispersion 15 4. Enhanced dissipation 23 Appendix A. Nondimensionalization 32 Appendix B. Strong solution theory 33 Appendix C. Proof of Lemma 4.3 36 References 37

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Section: Introductionsupporting

confidence: 67%

Section: Introductionsupporting

confidence: 63%

Section: 20)mentioning

confidence: 99%

Section: 20)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the stabilizing effect of swimming in an active suspension

Albritton¹,

Ohm²

2022

Preprint

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Orientation Mixing in Active Suspensions

Coti Zelati,

Dietert,

Gérard-Varet

2023

Ann. PDE

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We study a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles where the particles are described by a distribution in space and orientation. The uniform distribution of particles is the stationary state of incoherence which is known to exhibit a phase transition. We perform an extensive study of the linearised evolution around the incoherent state. We show (i) in the non-diffusive regime corresponding to spectral (neutral) stability that the suspensions experience a mixing phenomenon similar to Landau damping and we provide optimal pointwise in time decay rates in weak topology. Further, we show (ii) in the case of small rotational diffusion $$\nu $$ ν that the mixing estimates persist up to time scale $$\nu ^{-1/2}$$ ν - 1 / 2 until the exponential decay at enhanced dissipation rate $$\nu ^{1/2}$$ ν 1 / 2 takes over. The interesting feature is that the usual velocity variable of kinetic models is replaced by an orientation variable on the sphere. The associated orientation mixing leads to limited algebraic decay for macroscopic quantities. For the proof, we start with a general pointwise decay result for Volterra equations that may be of independent interest. While, in the non-diffusive case, explicit formulas on the sphere allow to conclude the desired decay, much more work is required in the diffusive case: here we prove mixing estimates for the advection-diffusion equation on the sphere by combining an optimized hypocoercive approach with the vector field method. One main point in this context is to identify good commuting vector fields for the advection-diffusion operator on the sphere. Our results in this direction may be useful to other models in collective dynamics, where an orientation variable is involved.

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Variational bounds and nonlinear stability of an active nematic suspension

Weady

2024

J. Fluid Mech.

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We use the entropy method to analyse the nonlinear dynamics and stability of a continuum kinetic model of an active nematic suspension. From the time evolution of the relative entropy, an energy-like quantity in the kinetic model, we derive a variational bound on relative entropy fluctuations that can be expressed in terms of orientational order parameters. From this bound we show isotropic suspensions are nonlinearly stable for sufficiently low activity, and derive upper bounds on spatiotemporal averages in the unstable regime that are consistent with fully nonlinear simulations. This work highlights the self-organising role of activity in particle suspensions, and places limits on how organised such systems can be.

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Weakly nonlinear analysis of pattern formation in active suspensions

Cited by 9 publications

References 57 publications

On the stabilizing effect of swimming in an active suspension

On the stabilizing effect of swimming in an active suspension

Orientation Mixing in Active Suspensions

Variational bounds and nonlinear stability of an active nematic suspension

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