Sparse nonlinear models of chaotic electroconvection
aa r X i v : . [ phy s i c s . c o m p - ph ] S e p Sparse nonlinear models of chaotic electroconvection
Yifei Guan ∗ , Steven L. Brunton , and Igor Novosselov Department of Mechanical Engineering, Rice University, Houston, TX, 77005, United States Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, United States
Abstract
Convection is a fundamental fluid transport phenomenon, where the large-scale motion ofa fluid is driven, for example, by a thermal gradient or an electric potential. Modeling con-vection has given rise to the development of chaos theory and the reduced-order modeling ofmultiphysics systems; however, these models have been limited to relatively simple thermalconvection phenomena. In this work, we develop a reduced-order model for chaotic electrocon-vection at high electric Rayleigh number. The chaos in this system is related to the standardLorenz model obtained from Rayleigh-Benard convection, although our system is driven by amore complex three-way coupling between the fluid, the charge density, and the electric field.Coherent structures are extracted from temporally and spatially resolved charge density fieldsvia proper orthogonal decomposition (POD). A nonlinear model is then developed for the chaotictime evolution of these coherent structures using the sparse identification of nonlinear dynamics(SINDy) algorithm, constrained to preserve the symmetries observed in the original system.The resulting model exhibits the dominant chaotic dynamics of the original high-dimensionalsystem, capturing the essential nonlinear interactions with a simple reduced-order model.
In convection, a body force acting on the fluid can lead to the formation of coherent flow structures,with examples including thermal Rayleigh-B´enard convection (RBC) [1–4], surface tension drivenMarangoni effects [5–7], solar magneto-convection [8, 9], magnetohydrodynamic convection [10,11], and electroconvection (EC). Quantitative analysis of thermal convection phenomena was firstoffered by Lord Rayleigh in the early twentieth century [2]. The initially structured convectivepatterns were subjected to symmetry breaking perturbations and developed into chaotic motionwhen a critical parameter was reached. Lorenz, Fetter, and Hamilton [12] later discovered a simplereduced-order model for RBC, explaining that the transition to chaotic flow is driven by a thermalgradient, i.e., a two-way coupling between the body force and the fluid motion. This landmarkresult changed how researchers view the role of numerical simulations and reduced-order modelingin science, and it established the modern field of chaos theory. However, this type of simplifieddynamical system model has not yet been developed for convective systems with more complexcoupling mechanisms. Electroconvection is a natural system to extend these analyses, as it involvesa nontrivial three-way, multiphysics coupling between the fluid, the electric field, and the chargedensity. In this work, we demonstrate a modern data-driven approach to identify sparse nonlinearreduced-order models for electroconvection. Our approach, in many ways, automates the methodsemployed by Lorenz, leveraging recent techniques in sparse regression and optimization. Margaret Hamilton and Ellen Fetter were graduate students at MIT and assisted Lorenz in programming theLGP-30 computer that was used for simulations. Although Lorenz included them both in the acknowledgements ofhis papers, by modern standards, Hamilton and Fetter would likely be co-authors on the paper. modes . High-dimensional simulations often obscure this simplicity, making optimization and control tasks pro-hibitively expensive [44]. Thus, there is a need for accurate and efficient reduced-order modelsthat describe EC phenomena, similar to the Lorenz model for RBC [12]. The Lorenz model wasobtained by Galerkin projection of the governing equations onto a dramatically reduced set of threemodes, given by the first three Fourier modes proposed by Saltzman [45]. Since Lorenz, this ap-proach has been applied extensively to RBC [46–54]. Instead of extracting these modes manually,modes are now typically extracted automatically, for example via proper orthogonal decomposition(POD) [41, 55–58] . Galerkin projection of the governing partial differential equation onto PODmodes is a more modern approach to obtain a reduced set of ordinary differential equations [41, 64–66]. Recently, it was shown that closely related nonlinear reduced-order models could be obtainedfor fluids purely from measurement data, and without recourse to the governing equations, byapplying the sparse identification of nonlinear dynamics (SINDy) algorithm [67–70] to time-seriesdata of POD mode amplitudes. Loiseau et al. [68, 69] showed that it is also possible to incorporatepartially known physics, such as energy conservation, as a constraint in the SINDy regression.In this work, we demonstrate a data-driven framework to obtain reduced-order models of EC,shown in Fig. 1. In particular, we examine a non-equilibrium system with unipolar charge injectionand an external electric field. We first extract dominant coherent structures from DNS of thethree-way coupled system, using POD. Strong symmetries are observed in the time evolution ofthe POD modes, and these symmetries are used as constraints to identify a sparse nonlinear modelwith SINDy. Surprisingly, we find that the resulting model captures the dominant dynamics usingdata from the charge density field alone. According to Lumley [57] and Holmes [41], POD was introduced independently at various times by severalresearchers, including Karhunen [59], Loeve [60], Pugachev [61], Obukhov [62], and Lorenz [63]. lectroconvection data POD modes Mode CoefficientsCharge density fields Sparse Model Constrained SINDy Data POD SINDytime a a a a a a a a a a a a a a min Ξ k ˙ X − Θ ( X ) Ξ k + λ k Ξ k subject to C ξ = d Figure 1:
Summary of nonlinear reduced-order modeling framework for electroconvection.
Elec-troconvection data are taken from the high-fidelity TRT LBM simulation. POD is performed on the chargedensity field data to extract the dominant spatial coherent structures (modes) and time series (coefficients)for how the mode amplitudes evolve in time. A strong symmetry is observed in the POD coefficient timeseries. We enforce symmetries in the first three coefficients as constraints on the sparse identification ofnonlinear dynamics (SINDy) algorithm. SINDy discovers a sparse nonlinear model to describe the evolutionof the mode coefficients, and this model may be used to reconstruct the full charge density field as a linearcombination of the corresponding POD modes.
The governing equations for EHD driven flow with unipolar charge injection include the Navier-Stokes equations (NSE) with the electric forcing term F e = − ρ c ∇ φ in the momentum equation,the charge transport equation, and the Poisson equation for electric potential: ∇ · u ∗ = 0 (1a) ρ D u ∗ Dt ∗ = −∇ P ∗ + µ ∇ u ∗ − ρ ∗ c ∇ φ ∗ (1b) ∂ρ ∗ c ∂t ∗ = −∇ · [( u ∗ − µ b ∇ φ ∗ ) ρ ∗ c − D c ∇ ρ ∗ c ] (1c) ∇ φ ∗ = − ρ ∗ c ǫ (1d)where the asterisk denotes dimensional variables: the 2D velocity field u ∗ = ( u ∗ x , u ∗ y ), the fluid den-sity ρ ∗ , the static pressure P ∗ , the charge density ρ ∗ c , and the electric potential φ ∗ . The parametersare given by: the dynamic viscosity µ , the ion mobility µ b , the ion diffusivity D c , and the electricpermittivity ǫ . The electric force acts as a source term in the momentum equation (1b) [71, 72]. Thesystem can be non-dimensionalized [40] by introducing four dimensionless parameters [33, 34, 39]: M = ( ǫ/ρ ) / µ b , T = ǫφ µµ b , C = ρ H ǫφ , F e = µ b φ D c (2)where H is the distance between the electrodes (two plates infinite in x), ρ is the injected chargedensity at the anode, and φ is the voltage difference between the electrodes. In turn, the time t ∗ can be non-dimensionalized by H / ( µ b φ ), u ∗ – by the ion drift velocity u drift = µ b φ /H , P ∗ – by3able 1: Boundary conditions for the numerical simulations.Boundary Macro-variables Conditions Meso-variables Conditionsx-direction Periodic PeriodicUpper plate u = 0, φ = 0, ∂ρ c /∂y = 0 LBM FBB scheme for f i ; ∂g i /∂y = 0Lower plate u = 0, φ = φ , ρ c = ρ LBM FBB for both f i and g i ρ ( µ b φ ) /H , φ ∗ – by φ , and the charge density ρ ∗ c – by ρ . The physical interpretation of theseparameters are as follows: M is the ratio between hydrodynamic mobility and the ionic mobility, T is the ratio between electric force to the viscous force, C is the charge injection level, and F e isthe reciprocal of the charge diffusivity coefficient [33, 34].The nondimensionalized form of the governing equations (1) are: ∇ · u = 0 (3a) D u Dt = −∇ P + M T ∇ u − CM ρ c ∇ φ (3b) ∂ρ c ∂t = −∇ · [( u − ∇ φ ) ρ c − F e ∇ ρ c ] (3c) ∇ φ = − Cρ c (3d)where variables without asterisk are dimensionless. Several numerical approaches have been developed to study EHD, including finite-difference [29],particle-in-cell [37], and finite-volume methods with the total variation diminishing scheme [31, 32].More recently, the lattice Boltzmann method (LBM) was used to predict the linear and finite-amplitude stability criteria of the subcritical bifurcation in the EC flow [34, 38]. A segregatedsolver was proposed that combines a two-relaxation time (TRT) LBM modeling of the fluid andcharge transport, and a Fast Fourier Transform (FFT) Poisson solver for the electrical field[39, 40].Due to its computational efficiency, direct numerical simulations in this study are performedusing the TRT LBM approach to solve the transport equations for fluid flow and charge density,coupled to a fast Poisson solver for the electric potential [39, 40, 73]. The numerical method isimplemented in C++ using CUDA GPU computing. A spatial resolution of 122 ×
100 is used,balancing accuracy with efficiency; the number of threads in the x-direction in each GPU blockis equal to the grid resolution in x, and the number of GPU blocks in the y-direction is equal tothe grid resolution in y. FFT and IFFT operations are performed using the cuFFT library [74].All variables are computed with double precision to reduce the truncation error. The numericalmethod was shown to be 2 nd order accurate in both time and space.The non-dimensional parameters used in this study are C = 10, M = 10, T = 312 .
5, and
F e = 4000. C = 10 corresponds to a strong charge injection; M = 10 and F e = 4000 correspondto the typical values of mobility and charge diffusivity of a dielectric liquid [33, 75]. T governs theviscosity of the flow, where T = 312 . ∇ ρ c = 0 at the cathode (upper plate) representsan outflowing current. A constant electric potential is applied at the anode; the cathode is grounded( φ = 0). At mesoscale (LBM scale), the discrete distribution function of velocity f i ( x , t ) and chargedensity g i ( x , t ) are used. The details on the transformations between macro-variables ( u , ρ c ) and4hargeVoidRegion Vortex pair(a) Illustration of electroconvection(b) Snapshots ofcharge density field (c) POD modes (d) AmplitudeTime Time, t PC0PC1PC2PC3PC4PC5Figure 2:
Electroconvection simulation data and proper orthogonal decmoposition. (a): Illus-tration of electroconvection of a dielectric fluid between parallel electrodes under strong unipolar injection;(b): Snapshots of charge density field used for POD analysis; (c,d): Mean charge density field and first 5POD modes and amplitudes in time. (PC - principal component; PC0 is the mean field) meso-variables ( f i , g i ) can be found in recent publications [39, 40]. The LBM full-way bounce-back(FBB) scheme is used for the Dirichlet (no-slip) boundary conditions for the fluid flow and chargedensity at the lower plate [34, 40]. The g i Neumann boundary condition is set as a current outletboundary condition for charge density transport [34, 39].Fig. 2(a) shows an illustration of the charge density from the the electroconvection simulation,and Fig. 2(b) shows the unsteady nature of the ionic convection leading to electric field variation andconsequently the unsteady flow patterns. This is apparent by examining the momentum equation,Eq. (3b), as the variance in the source term drives the instability in the flow if the viscosity termcannot dampen the flow fluctuations. For the chosen parameters C = 10, M = 10, T = 312 .
5, and
F e = 4000, the flow is driven by a strong charge injection with a high electric force to viscous forceratio ( T ). The irregular profiles of charge density represent states with an imbalance of electricforce and viscous force. 5 a a a a a a a a
3D phase portraits( a , a , a ) 2D projection Symmetries( a , − a , a ) ( a , a , − a )( a , − a , − a ) ( − a , a , a )( − a , − a , − a )Figure 3: Symmetries in POD mode coefficients.
Left: 3D phase portraits of the first 3 POD modes.Middle: 2D projection of the 3D phase portraits onto 2D planes; Right: Symmetry transformations of the3D phase portraits, which imply symmetry in the underlying dynamical system.
We now extract dominant coherent structures and symmetries from the time resolved DNS datafrom above. In particular, we use POD to extract an orthogonal set of spatial modes from thecharge density field, along with time series for how these mode amplitudes evolve in time; thesemodes and amplitudes are shown in Fig. 2(c,d). A spatially and temporally resolved data set isused with a time step of ∆ t = 0 .
005 from t = 0 to t = 1000. The 0 th POD mode (PC0) representsthe charge density mean-field and does not vary with time. The 1 st POD mode (PC1) is stronglyperiodic in both space and time, with periodic structures corresponding to the two up-driftingion channels between charge void regions, as shown in Fig. 2(a). During the simulation, the ionchannels oscillate and can merge and separate, as shown in Fig. 2(b). The 2 nd and 3 rd POD modes(PC2,3) are a phase-shifted mode pair that contribute to the oscillation of the ion channels. Highermodes are not obviously paired, although they do correspond to spatial and temporal harmonics.POD was also computed on the other spatial fields, and structures were qualitatively similar.Fig. 3 shows the trajectories of the first 3 POD coefficients ( a , a , and a ) and their projectionsonto 2D planes. These trajectories exhibit several strong symmetries, so that the system may beviewed as invariant with respect to the following transformations:[ a , a , a ] ↔ [ a , − a , a ] ↔ [ a , a , − a ] ↔ [ a , − a , − a ] ↔ [ − a , a , a ] ↔ [ − a , − a , − a ] . (4)These symmetries in the original system will be enforced in our sparse nonlinear models.Note that POD and Galerkin projection have been widely applied to RBC, although analysesare limited for EC. Sirovich et al. applied POD to RBC to investigate the scaling of POD modesto Rayleigh number [46] and to specify the chaotic RBC with Liapunov and Karhunen-Loevedimensions [47–49]. Low-dimensional dynamics and data compression were also considered [50].To further investigate the low-dimensional model of RBC, Bailon-Cuda et al. performed Galerkinprojection onto POD modes from DNS of turbulent RBC and found that a few hundred POD modesare required to qualitatively reproduce the large-scale turbulent convection [51–53]. Recently, Caiet al. developed a closure scheme for 2D RBC at high Ra number with POD-Galerkin models [54].6ataModel a a a FP-AFP-BPath CPath DPath EPath F
Data POD SINDy Orbits
Loop 1Loop 2Loop 3Loop 4
Figure 4:
Comparison of full simulation and SINDy model for electroconvection.
Left: 3D phaseportraits of data from numerical simulation and from SINDy model; Middle: Charge density fields and theirreconstructions by POD modes and SINDy model. The flow is charactereized by two fixed points, FP-A andFP-B, and there are four paths that are taken between these fixed points, labeled C-F. Right: Four differentloops are formed by the four paths (C-F).
We now demonstrate the identification of a parsimonious nonlinear model for EC using the sparseidentification of nonlinear dynamics (SINDy) [67] approach; results are summarized in Fig. 4.SINDy has been widely applied for model identification in a variety of applications, includingchemical reaction dynamics [76], nonlinear optics [77], fluid dynamics [68–70, 78, 79] and turbu-lence modeling [80, 81], plasma convection [82], numerical algorithms [83], and structural model-ing [84], among others [85–87]. Of particular note are its uses in identifying Lorenz-like dynamicsfrom a thermosyphon simulation by Loiseau [70] and to identify a model for a nonlinear magne-tohydrodynamic plasma system by Kaptanoglu et al. [88]. It has also been extended to handlemore complex modeling scenarios such as partial differential equations [89, 90], systems with in-puts or control [91], to enforce physical constraints [68], to identify models from corrupt or limiteddata [92, 93] and ensembles of initial conditions [94], and extending the formulation to includeintegral terms [95, 96], tensor representations [97, 98], deep autoencoders [99], and stochastic forc-ing [100, 101]; an open-source software package, PySINDy, has been developed to integrate anumber of these innovations [102]. We will enforce the symmetries observed above as constraints,as in Loiseau and Brunton [68].We apply SINDy to develop a sparse reduced-order model of electroconvection, in particularthe evolution of the POD coefficients a ( t ) , a ( t ) , and a ( t ). First, we construct a data matrix X whose columns are time-series of a , a , and a , along with the corresponding matrix of time7erivative, ˙X : X = a ( t ) a ( t ) a ( t ) a ( t ) a ( t ) a ( t )... ... ... a ( t m ) a ( t m ) a ( t m ) ˙X = ˙ a ( t ) ˙ a ( t ) ˙ a ( t )˙ a ( t ) ˙ a ( t ) ˙ a ( t )... ... ...˙ a ( t m ) ˙ a ( t m ) ˙ a ( t m ) . (5)Based on the data in X , a library of candidate nonlinear functions Θ ( X ) is constructed, contain-ing functions that may describe the observed dynamics. This library is only limited by the user’simagination, and may be guided by terms present in the overarching partial differential equation.For fluid systems, polynomials have been quite effective: Θ ( X ) = [ · · · X d ] (6)where the matrix X d denotes a matrix with column vectors given by all possible time series of d th degree polynomials in the state x = h a a a i .SINDy develops reduced-order models by selecting the fewest columns of Θ ( X ) that add up ina linear combination to describe the derivatives in ˙X :˙ X = Θ ( X ) Ξ (7)where Ξ is the sparse coefficient matrix that denotes which columns of Θ are active, and hence,which terms are active in the dynamics. A parsimonious model will provide an accurate fit with asfew terms as possible in Ξ . Such a model can be developed using a convex regression: Ξ k = argmin Ξ ′ k k ˙ X k − Θ ( X ) Ξ ′ k k + λ k Ξ ′ k k (8)where Ξ k denotes the k th column of Ξ . There are several algorithms to obtain a sparse model Ξ , andwe use the sequential-thresholded least squares (STLS) algorithm [67]. Loiseau and Brunton [68]showed that it is also possible to incorporate known linear constraint equations in this regression:min Ξ k ˙ X − Θ ( X ) Ξ k + λ k Ξ k (9a)subject to C ξ = d (9b)where ξ = Ξ (:) is the vectorized form of the sparse matrix of coefficients, and C ξ = d are linearequality constraints, which can be used to enforce symmetries and other known constraints, suchas energy conservation. We perform SINDy on the POD coefficients a i for i = 1 , , As shown in Fig. 3, the trajectories remain nearly unchanged under the symmetry transformationsin Eq. (4). These symmetries imply various constraints on the coefficients in Ξ , resulting in severalterms that are zero and several other terms that must be related across the dynamics; theseconstraints are summarized in Table. 2.The symmetry constraints are determined as follows. First we rewrite the unknown system as:˙ a = Θ ( a , a , a ) Ξ , (10a)˙ a = Θ ( a , a , a ) Ξ , (10b)˙ a = Θ ( a , a , a ) Ξ . (10c)8able 2: Structure of the Ξ matrix. a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a ˙ a ξ ξ ∗ − ξ ξ ξ ξ ˙ a ξ − ξ ∗ ξ ξ ξ ˙ a ξ ξ ξ ξ ξ For the first row, Eq. (10a), the equation is invariant to switching the sign of a and/or a .Thus, the coefficients of the terms with odd powers of either a or a must vanish ( ξ ij correspondingto a l a m a n where at least one of m and n is odd). From the observed symmetry [ a , a , a ] ↔ [ − a , a , a ], the coefficients ξ ij corresponding to a l a m a n should be set to zero where l is even andnon-constrained where l is odd. The rest of the ξ ij correspond to a l a m a n where both m and n areeven numbers, namely a a , a a , a a a , a a a . When a is replaced by − a , a and a must beswapped for the equation to remain unchanged, so the coefficients ξ ij for a a and a a should beequal and opposite and those for a a a and a a a should be identical.The coefficients in the second and third rows, Eqs. (10b) and (10c), should be consideredtogether. From the symmetry transformations a ↔ − a and a ↔ − a , the coefficients ξ ij corresponding to a l a m a n in the equation of ˙ a and those corresponding to a l a n a m in the equationof ˙ a vanish when m is even or n is odd. The rest of the coefficients ξ ij in Eq. (10b) correspondto a l a m a n where m is odd and n is even, namely a , a a , a a a , a a a , a a a . Similarly, therest of the coefficients ξ ij in Eq. (10c) correspond to a l a n a m where n is even and m is odd,namely a , a a , a a a , a a a , a a a . From the observed symmetry [ a , a , a ] ↔ [ − a , a , a ],the coefficients ξij corresponding to a l a m a n in Eq. (10b) and those corresponding to a l a n a m inEq. (10c) should be identical where l is even, and equal and opposite where l is odd.We explicitly enforce these constraints in the SINDy regression. Each row of Table 2 representseach row of the Eq. (7). The blank entries represent zero’s and the each ξ i represent one value,i.e. the entries with the same ξ i should have the same values. To enforce the energy constraints asdescribed by Loiseau [68], we set the coefficients with asterisk equal and opposite in Table 2.The system is therefore:˙ a = ξ a + ξ a − ξ a + ξ a + ξ a a + ξ a a (11a)˙ a = ξ a − ξ a a + ξ a a + ξ a + ξ a a (11b)˙ a = ξ a + ξ a a + ξ a a + ξ a a + ξ a . (11c)To further take advantage of the symmetry, we augment the data with all symmetry-transformedcopies of the data: X = [ a , a , a , a , − a , − a , a , − a , a , − a , a , − a , a , a , − a , − a , a , − a ].For the mildly chaotic EC system in a dielectric liquid driven by a strong charge injection, specifiedby C = 10, M = 10, T = 312 .
5, and
F e = 4000, the resulting sparse nonlinear model is:˙ a = 0 . a + 0 . a − . a − . a + 0 . a a + 0 . a a (12a)˙ a = − . a − . a a + 0 . a a − . a + 0 . a a (12b)˙ a = − . a + 0 . a a + 0 . a a + 0 . a a − . a . (12c) The results from the SINDy model are depicted in Fig. 4. The reduced-order model accuratelycaptures the qualitative behavior of the system, characterized by four paths connecting two fixedpoints, denoted A and B in the figure. The system pseudo-randomly chooses each of the two pathsstarting at a given fixed point, resulting in the four orbits shown in Fig. 4 (right); this behavior9losely matches the behavior of the full high-dimensional system. Further, the SINDy model maybe used to recombine POD modes to approximate the high-dimensional dynamics around thesefixed points and connecting paths. The SINDy reconstruction faithfully captures the truncatedPOD approximation of the true system; thus, a better model cannot be obtained without includingmore POD modes in the expansion.
In this work, we developed a data-driven reduced-order model for chaotic electroconvection. Ourapproach is reminiscent of that employed by Lorenz to model Rayleigh-Benard convection, althoughdimensionality reduction and nonlinear model identification procedures are automated and do notrequire recourse to the governing equations. In particular, we investigate spatiotemporal data fromdirect numerical simulations of the three-way coupling between the fluid, charge density, and electricfield that give rise to chaotic electroconvection. We extract coherent structures from the chargedensity field using POD, and we develop a nonlinear model for how the POD mode amplitudesevolve in time using the SINDy modeling framework. Critically, we identify several symmetries inthe time-series data, which we later enforce as constraints in the SINDy algorithm, dramaticallyreducing the search space of candidate models. The resulting model faithfully captures the dominantbehavior of the leading POD modes, including chaotic switching in the time series that correspondsto chaotic convection in the full field. Because the nonlinear model is sparse and is formulatedin terms of a few key spatial modes, it is interpretable by construction and may be less proneto overfitting compared to deep learning. Thus, we demonstrate the ability of automated modeldiscovery techniques to uncover interpretable dynamical systems models for convective phenomenawith more complex three-way coupling than the original Lorenz model of RBC.There are a number of natural extensions and future directions suggested by this work. Three-dimensional flows and experimental measurements would provide challenging tests to the method-ology and may point out limitations that require further development. Although the present workonly considered data from the charge density, incorporating information from the three fluid, elec-tric, and charge density fields is the subject of ongoing research. This will require developing adimensionally consistent POD, as has been done for compressible flows [103] and magnetohydrody-namics [88]. This modeling framework is also amenable to including the effect of varying parame-ters, which could result in a parameterized nonlinear model that would enable more sophisticateddesign optimization and the prediction of bifurcation events; however, incorporating a parameterinto the model increases the size of the library of candidate functions and requires more trainingdata. Increasing the number of POD modes in the model will also likely increase the fidelity of theprediction, for example enabling the modeling of subtle secondary effects in the data. However,when additional POD modes are added, it will be necessary to automatically extract symmetriesand determine what constraints these symmetries impose on the candidate library. This challengefits into the broader literature of symmetry reduction [104]. Finally, using these models for design,optimization, and control will be a true test of the model.
SLB acknowledges funding from the Army Research Office (ARO W911NF-19-1-0045). The authorswould also like to acknowledge fruitful discussions with Jared Callaham, Alan Kaptanoglu, NathanKutz, and Jean-Christophe Loiseau. This work was partially supported by the National Institutesof Health (grant numbers: NIBIB U01 EB021923, NIBIB R42ES026532 subcontract to UW).10 eferences [1] H. B´enard, “Les tourbillons cellulaires dans une nappe liquide,”
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