Fabian Brau
Associate Professor (Chargé de cours)

Nonlinear Physical Chemistry Unit
Service de Chimie Physique et Biologie Théorique

Faculté des Sciences
Campus Plaine, C.P. 231
Université libre de Bruxelles, 1050 Brussels, Belgium

☎ +32 (0)2 650 5750
📠 +32 (0)2 650 5767

Matter can be functionalized to meet specific needs by modifying its shape or structure. This task can be relatively simple such as, for example, when a piece of glass is cut into a lens. However, manipulating objects or structuring surfaces at nano- or microscale are complex tasks, sometimes expensive and which often require advanced equipment. Parts of my research aims to develop new methods based on the theory of pattern formation and self-organization/self-assembly for manipulating matter and creating complex structures. The recent interest in these methods is fueled, among other things, by the potential impact that efficient patterning methods could have on modern surface-based technologies, such as the fabrication of optical and microfluidic devices or components for deformable electronics. Indeed, using self-organized processes to create the desired structures spontaneously, without local actions on matter, reduces significantly the production costs and the fabrication time. Understanding the mechanisms governing the emergence of these forms and identifying the parameters controlling their shapes are the main tasks in order to create tailored (meta)materials. To achieve this objective, I use various strategies involving elastic thin films or the interaction between fluids and solids (fluid-structure interaction).

Nature can also be a source of inspiration because it offers many examples of complex shapes created spontaneously. However, in a natural system, the observed structures may be due to multiple causes acting together. To understand these complex forms, it is necessary to isolate fundamental morphologies produced by a single specific and controlled stress. My approach is therefore to design simple model systems capturing the essential physics to describe some characteristics of more complex systems. From a fundamental perspective, my research contributes to highlight common formation mechanisms of structures emerging in various systems (natural or technological). These studies help to unify our understanding of the origin of forms emerging spontaneously. This allows to describe a maximum of observations with minimum basic principles. These studies are therefore intrinsically interdisciplinary, since they offer analysis tools which can be used in fields ranging from biology to geophysics. From an applied perspective, understanding these formation mechanisms allows to control the emergence and the shape of these patterns to structure materials according to the needs in a simple, inexpensive and fast way.

Recently, parts of my research focus on the physics of capture of viscous fluids by small animals such as bees. Capturing viscous fluid at small scales is a challenge that nectarivores have solved by developing various type of specialized tongues or proboscis. These appendages typically consist of a complex assembly of flexible structures having a small size compared to the capillary length. Interesting elastocapillary problems appear thus in this context. Our strategy is to carry out systematic experimental and theoretical studies of elastocapillary effects coupled with viscous flows for a series of simple model systems mimicking biological systems.

Hoa-Ai Béatrice Hua PhD student, ElastoCap project, F.R.S-FNRS
Dimitra Spanoudaki PhD student, MASNEC project, M-ERA.NET, with A. De Wit
Hadrien Bense Postdoc, CapTure project, ARC
Alessandro Comolli Postdoc, Flocon project, COFUND, with A. De Wit
Giulio Facchini Postdoc, RESOP project, F.R.S-FNRS, with A. De Wit
Emmanuel Siefert Postdoc, BioCapSoft project, Marie Sklodowska-Curie fellowship

Continuum Mechanics [Mécanique des milieux continus] (MATH-F426)
Spring 2018 [Ex]; 2019 - 2022 [Th + Ex]
These lectures introduce the basic concepts of continuum mechanics: Lagrangian and Eulerian specification of the flow field, strain and stress tensors, conservation laws. Constitutive laws (limited to Hookean solids and Newtonian fluids) are introduced to derive the Navier-Stokes and the Navier equations. Some particular phenomena are analyzed: Deep water waves, gravity-capillary waves, shallow water waves, Korteweg et de Vries and solitary waves, Kelvin wake pattern, etc. Approximate theories are also introduced: Lubrication theory, Föppl–von Kármán equations for plates, Euler elastica for elastic line. The concept of elastic instability is introduced and illustrated with classic examples: Euler buckling, wrinkling, etc.
Notes de cours

Macroscopic Physical Chemistry: From self-assembly to self-organization [Chimie physique macroscopique: de l'auto-assemblage à l'auto-organisation] (CHIM-F408)
Fall 2016 [Ex]; 2018 - 2021 [Ex]
The lectures introduce the concepts of phase transition and nonequilibrium processes to students in Chemistry. The Landau theory of second-order phase transition is introduced and illustrated with several examples. The Fick's laws of diffusion are introdced and the diffusion equation is solved in some classic cases. This process is then coupled with reaction terms to describe reaction-diffusion systems. The reaction-diffusion is also solved in some classic cases. The concept of nonequilibrium stationnary states is introduced. The stability of these states are analyzed through linear stability analyses. The influence of external forces (CSTR) on these states is also studied.

Statistical Thermodynamics (CHIM-F321)
Spring 2020 - 2022 [Ex]
This course aims to offer an introduction to statistical mechanics applied to equilibrium thermodynamics. The main goal is to succeed in connecting the macroscopic properties of matter, observable on our scale, to the intrinsic characteristics of the atoms and molecules that compose it. To this end, we introduce a probabilistic approach to the dynamics of a macroscopic system based on the notion of the Gibbs ensemble. We then apply this formalism to different examples of equilibrium systems, whether isolated, closed or open.

Asymptotic Methods in Physics (PHYS-F427)
Spring 2018 [Ex]
Various methods to analyze (algebraic or differential) equations containing one small (or large) parameter are introduced. The concepts of regular and singular asymptotic expansions are introduced and illustrated. The concept of boundary layer and matched asymptotic expansions are introduced. The WKB theory and the multiple-scale analysis are presented.

General Physics (PHYS-F110)
Spring 2017 - 2018 [TP]

Bees capture nectar by dipping their tongue into 5 times per second (see snapshots, \(\Delta t = 20\) ms). Among the 20000 bee species known to date, almost half have a long tongue of radius \(R_G\) decorated with slender papillae of length \(L\) and radius \(R\). Experiments performed under a microscope equipped with a high speed camera show that when the tongue is immersed in the nectar, the papillae open like the hairs of a brush. The presence of these outgrowths help to trap a larger quantity of nectar compared to a bare tongue. These same experiments show, however, that this morphological advantage no longer plays any role when the sugar content of the nectar exceeds a critical value. Indeed, when the sugar concentration increases, the viscosity of the nectar increases rapidly and prevents the papillae from fully opening before the tongue withdraws from the nectar (panel (a)). The motion of the papillae in a viscous liquid has been modeled using an approach based on fluid mechanics and the elasticity of rods. This theoretical model combined with measurements of the morphology of the tongue allow to explain why the ingestion rate of nectar in bees, \(Q\), drops sharply at sugar concentration above 50-60%. The theoretical predictions are confirmed by in vivo measurements of these ingestion rates carried out on several bee species (panel (b)). This study shows that the value \(c_s^*\) of the sugar concentration at which the absorption of calories is optimal is determined by the ratio \(L/R\) whereas the value \(c_s^m\) of the sugar concentration at which the ingestion rate is minimum is essentially universal and around 72% (panel (c)).
PNAS 2021


Press coverage: RTBFVivacitéLe VifLa LibreLa DH FNRS.News 122

Reaction-diffusion (RD) fronts are ubiquitous in a wide variety of phenomena ranging from population dynamics and disease spreading to image processing and nanotechnology to name a few. Among the large family of RD fronts, \(A+B \to C\) fronts are observed when initially separated reactants \(A\) and \(B\) meet by diffusion and react to produce \(C\). Depending on the nature of \(A\) and \(B\), their dynamics is representative of many problems in chemistry, geochemistry, finance and many others. The temporal evolution of the front position, \(x_f\), defined as the location of maximum  \(C\) production, of the front width \(w\) and of the local production rate, \(R(x_f)\), have long been derived theoretically and confirmed experimentally. The related scalings \(x_f \sim t^{1/2}\), \(w\sim t^{1/6}\) and \(R(x_f) \sim t^{-2/3}\), form the basis of \(A+B \to C\) RD front theory confirmed in many applications. In flows, \(A+B \to C\) processes provide another important class of dynamics, e.g. in combustion and environmental problems. The coupling between convection and reaction leads to complex dynamics when the flow, actively influenced by transported species, feedbacks on their spatio-temporal distribution. The radial advection of reacting species is currently receiving growing attention. For example, \(A+B \to C\)-type precipitation reactions in a radial flow give rise to a large variety of complex self-assembled structures, to thermodynamically unstable crystalline forms, or to microstructures significantly different from those obtained in homogeneous systems. Similarly, a suitable redefinition of distance may recast some transport phenomena into a radial spreading as done in studies of infectious disease spreading. Motivated by the broad applications of radial transport in reactive systems, the dynamics of \(A+B\to C\) fronts is analyzed theoretically in presence of passive advection when \(A\) is injected radially into \(B\) at a constant inlet flow rate \(Q\). We compute the long-time evolution of the front position, \(r_f\), of its width, \(w\), and of the local production rate \(R\) of the product \(C\) at \(r_f\). We show that, while advection does not change the well-known scaling exponents of the evolution of corresponding RD fronts, their dynamics is however significantly influenced by the injection. In particular, the total amount of product, \(n_{\text{C}}\), varies as \(n_{\text{C}} \sim Q^{-1/2}\, V\), where \(V\) is the volume of injected reactant. This control strategy paves the way to a flow control of the amount and spatial distribution of the reaction front product. It compares well with calcium carbonate precipitation experiments for which the amount of solid product generated in flow conditions at fixed concentrations of reactants can be tuned by varying the flow rate.
Physical Review Letters 2017

flow control
Precipitation reaction by radial injection in a confined quasi-2D reactor of a solution of carbonate ions (\(A=\) CO\(_3^{2-}\)) into a solution of calcium ions (\(B=\) Ca\(^{2+}\)) producing calcium carbonate (\(C=\) CaCO\(_3\)). (a) \(n_{\text{C}}\), estimated by the intensity \(I_{\text{tot}}\) of light reflected by \(C\), as a function of the volume \(V\) of the injected Na\(_2\)CO\(_3\) solution for various \(Q\) and initial concentrations \(A_0=B_0=0.2\) M. (b) Slope \(\beta\) of the asymptotic linear regime shown in (a) as a function of \(Q\) for various \(\gamma = B_0/A_0\).
Among predators using an adhesive tongue to feed, chameleons are able to capture large prey by projecting the tongue at high acceleration. Once in contact with a prey, the tongue retracts with a comparable acceleration to bring it to the mouth. A strong adhesion between the tongue tip and the prey is therefore required during the retraction phase to ensure a successful capture. To investigate the mechanism responsible for this strong bond, the viscosity of the mucus produced at the chameleon's tongue pad is measured, using the viscous drag exerted on rolling beads by a thin layer of mucus. In this Letter, we show that the viscosity of this secretion is about 400 times larger than that of human saliva. We have then incorporated this viscosity into a dynamical model for viscous adhesion, which describes the motion of the compliant tongue and the prey during the retraction phase. The variation of the maximum prey size with respect to the chameleon body length is derived, and compared with in vivo observations for various chameleon species (see figure below). Our study shows that the size of the captured prey is not limited by viscous adhesion, owing to the high mucus viscosity and large contact area between the prey and the tongue.
Nature Physics 2016


Twisted ribbons. Twisted ribbons under tension exhibit a remarkably rich morphology, from smooth and wrinkled helicoids, to cylindrical or faceted patterns. This complexity emanates from the instability of the natural, helicoidal symmetry of the system, which generates both longitudinal and transverse stresses, thereby leading to buckling of the ribbon. Here we focus on the tessellation patterns made of triangular facets. Our experimental observations are described within an "asymptotic isometry'' approach that brings together geometry and elasticity. The geometry consists of parametrized families of surfaces, isometric to the undeformed ribbon in the singular limit of vanishing thickness and tensile load. The energy, whose minimization selects the favored structure among those families, is governed by the tensile work and bending cost of the pattern. This framework describes the coexistence lines in a morphological phase diagram, and determines the domain of existence of faceted structures.
Physical Review Letters 2016
Structure stability under flows. Experiments were carried out to investigate the collapse of cylindrical shells of variable radius \(R\) and thickness \(t\) under the action of granular flows (size \(d\), density \(\rho\)). The critical heights, \(L_c\), of the granular material column beyond which the structure collapses when its content flows freely through a circular hole of given size at the base of the structure have been measured. It has been shown that the silos could withstand loads (height of the granular column) significantly higher than those obtained by an estimate based on standard theories of stability of shells and stress distribution in granular columns (Janssen model). Figure (a) shows the position of the upper layer of grains \(z\) as a function of time measured relatively to the collapse height, \(L_c\). When \(L > L_c\), irreversible deformations occur leading to a collapse of the silo. The circles indicate the time at which pictures (b)–(e) are taken. The pictures (b)–(c) show two states occurring during discharge onset for \(L > L_c\). Panel (c) shows a collapsed silo. The pictures (d)–(e) show two states occurring at the discharge onset and once the discharge is completed for \(L < L_c\). The solid arrows indicate the temporal evolution and the dashed arrow shows localized diamond dimples smoothed out during the discharge. It was also shown that the critical height decreases when the grain size \(d\) increased. The model describing the fluid-structure interaction taking into account the actual status of friction forces on the walls and deformation of the structure due to the finite size of the grains. This completely analytical model allowed to describe the experimental data of this complex problem satisfactorily, see figure (f) where each symbol shows the quantity which is varied.
Physical Review Letters 2015


Confined chemical gardens: From chemistry to structures.
Chemical gardens are formed by the spontaneous assembly of mineral precipitates generated when a metal salt is dissolved in an alkaline solution. Irregular and multicolored tubular structures rising then spontaneously by the combined action of different physical processes (osmotic pressure, gravity effects, reactions and diffusion) responsible for the irregularity and lack of reproducibility of the shapes obtained. In order to gain reproducible, we grew chemical gardens in a quasi two-dimensional confined by injecting a reagent into the other between two horizontal plates (Hele-Shaw cell). The experiments performed by F. Haudin showed that, in such conditions, a rich collection of reproducible structures can be obtained including, in particular, spiral structures. A geometric model has been developped to understand the formation of these spiral pattern. This model predicts that the spirals should be logarithmic whose polar equation is \(r = r_0 \exp(\theta/\theta_0)\). The figure below shows the evolution of radial distances \(r\), measured from the center of the spiral, as a function of the polar angle \(\theta\). The agreement between the theoretical prediction and experimental data is excellent (see figure below).
PNAS 2014


Press coverage: Physics Today — Le Monde — La Recherche n°495 — AthenaPhys.orgDailyScienceChemistry World

From fingerprints to mountain ranges. Systems involving a thin rigid sheet attached to a soft and thick substrate are ubiquitous in nature. When such a system is compressed in the plane of the sheet, wrinkles spontaneously appear and allow to explain, for example, the formation of wrinkles on the skin, mountain ranges and fingerprints. The wavelength of these undulations depends on the material properties: \(\lambda \sim h (E/K)^{1/3}\) for an elastic substrate and \(\lambda \sim h^{3/4} (E/\rho g)^{1/4}\) for a liquid substrate, where \(h\) is the sheet thickness, \(E\) is the sheet Young modulus, \(K\) and \(\rho g\) are the effective stiffness of the substrate. For an elastic substrate, if the system is further compressed, a totally new and unexpected phenomenon is observed. The folds formed are separated into two families: the amplitude of one fold increases to the detriment of its two neighbors (b,d). One fold every two folds concentrates the deformation energy to create a structure with a period twice the initial period. If the compression is increased, the same process is repeated to yield a quadrupling of the initial period. The up-down symmetry breaking induced by the elastic substrate is an essential ingredient to explain the period doubling. If the substrate is a fluid (a) or if the sheet is sandwiched between two identical soft foundations (c), the system does not break this symmetry and the period doubling instability is not observed. The experimental evolution of the pattern amplitudes \(A\) as a function of the compression ratio \(\delta = \Delta/L\), where \(\Delta\) is the horizontal displacement of the sheet and \(L\) its length, is shown in figure (e) and compares satisfactorily with the theoretical evolution.
Nature Physics 2011


Press coverage: Le SoirCNRSCNRS MagazineFNRSNewsAthenaScience & Vie n°1120

Hierarchy of folds: From graphene to curtains. Thin films are unstable when they are compressed. If a thin sheet of thickness \(h\) and Young modulus \(E\), is forced to adopt a periodic profile, with a wavelength \(\lambda_0\), along one of its edges, this constraint will be relaxed gradually as one moves away from this edge producing a cascade of folds of increasing size which progressively merge two by two. Regions where these folds merge were named "wrinklons" and represent the building blocks of the pattern. Using these wrinklons, we showed how to reconstruct the complete pattern and determine the evolution of the mean wavelength \(\lambda\) as a function of the distance to the constrained edge \(x\). When tension \(T\) in the sheet, which can be due to the weight of the sheet, is negligible then \(\lambda \sim x^{2/3}\) otherwise \(\lambda \sim x^{1/2}\). This wavelength evolution has been compared with experimental data spanning five orders of magnitude for the imposed wavelength \(\lambda_0\), the sheet thickness \(h\) or the system size (see figure below for "heavy" sheets).
Physical Review Letters 2011


Press coverage: Le SoirPhysics SynopsisPhysics WorldScience & Avenir n° 773 — Pour La Science n°405

How geometry controls fractures in thin adhesive films. One can take off a piece of adhesive film attached to a substrate by making two parallel notches on one of its edges to creat a rectangular flap which is then pulled at a constant speed \(v\) and with a constant peeling angle \(\alpha\). When the substrate is flat, it has been shown that the initially parallel cracks converge toward each other along a linear path until they meet. The shape of the flap detached from the substrate is then invariably triangular (a). The angle \(2\theta\) formed by the two cracks at the meeting point depends on the material properties: \(\sin \theta \sim (E t \tau)^{1/2}/ \gamma\), where \(E\) is the Young modulus, \(t\) is the sheet thickness, \(\tau\) is the adhesive energy per unit area and \(\gamma\) is the work of fracture of the film. We have shown that when the substrate has a finite curvature \(\kappa=1/R\), the forms adopted by the flap are much more diverse and it is even possible to obtain diverging paths for the two fractures (b, c, d). This finite curvature induces a finite geodesic curvature along the debonding front \(\bar{\kappa} = 1/\bar{R} = \kappa \cot(\alpha/2)\) which vanishes when the peeling angle is equal to \(180^{\circ}\) (in that case, we recover the same shape for the flap than the one obtained for flat substrates). The agreement between the data and the developed geometrical model is remarkable (e, f).
Physical Review Letters 2011


Press coverage: Le Soir

scopus publons

  1. E. Siéfert, H.-A. B. Hua, F. Brau
    Capillary coalescence of two partially immersed slender structures, Extreme Mech. Lett. 55, 101823 (2022).
  2. J. Wei, F. Brau, P. Damman, A. Draux, H.-A. B. Hua, Z. Wu, J. Wu
    Trade-off mechanism of honey bee sucking and lapping, Soft matter  (2022). In press.
  3. A. Draux, H.-A. B. Hua, P. Damman, F. Brau
    Relaxation dynamics of a flexible rod in a fluid, Phys. Rev. Fluids 6, 114102 (2021).
  4. A. Lechantre, A. Draux, H.-A. B. Hua, D. Michez, P. Damman, F. Brau
    Essential role of papillae flexibility in nectar capture by bees, PNAS 118, e2025513118 (2021).
    Press (see above)
  5. F. Brau & A. De Wit
    Influence of rectilinear vs radial advection on the yield of A + B → C reaction fronts: A comparison, J. Chem. Phys. 152, 054716 (2020).
  6. A. Comolli, A. De Wit & F. Brau
    Dynamics of A + B → C reaction fronts under radial advection in three dimensions, Phys. Rev. E. 100, 052213 (2019).
  7. F. Brau, S. Thouvenel-Romans, O. Steinbock, S. S. S. Cardoso & J. H. E. Cartwright
    Filiform corrosion as a pressure-driven delamination process, Soft Matter 15, 803 (2019).
  8. P. M. Reis, F. Brau & P. Damman
    The mechanics of slender structures, Nature Physics 14, 1150 (2018).
  9. F. Brau, F. Haudin, S. Thouvenel-Romans, A. De Wit, O. Steinbock, S. S. S. Cardoso & J. H. E. Cartwright
    Filament dynamics in confined chemical gardens and in filiform corrosion, Phys. Chem. Chem. Phys. 20, 784 (2018).
    2018 PCCP HOT Articles
  10. F. Brau, G. Schuszter & A. De Wit
    Flow control of A + B → C fronts by radial injection, Phys. Rev. Lett. 108, 134101 (2017).
  11. G. Schuszter, F. Brau & A. De Wit
    Flow-driven control of calcium carbonate precipitation patterns in a confined geometry, Phys. Chem. Chem. Phys. 18, 25592 (2016).
  12. H. Pham Dinh, V. Démery, B. Davidovitch, F. Brau & P. Damman
    From cylindrical to stretching ridges and wrinkles in twisted ribbons, Phys. Rev. Lett. 117, 104301 (2016).
  13. F. Brau, D. Lanterbecq, L.-N. Zghikh, V. Bels & P. Damman
    Dynamics of the prey prehension by chameleons through viscous adhesion, Nature Physics 12, 931 (2016).
    Press (see above)
  14. G. Schuszter, F. Brau & A. De Wit
    Calcium Carbonate Mineralization in a Confined Geometry, Environ. Sci. Technol. Lett. 3, 156 (2016).
  15. G. Gutiérrez, C. Colonnello, P. Boltenhagen, J. R. Darias, R. Peralta-Fabi, F. Brau & E. Clément
    Silo collapse under granular discharge, Phys. Rev. Lett. 114, 018001 (2015).
  16. O. Oshri, F. Brau & H. Diamant
    Wrinkles and folds in a fluid-supported sheet of finite size, Phys. Rev. E 91, 052408 (2015).
    Editors' Suggestion
  17. F. Haudin, V. Brasiliense, J. H. E. Cartwright, F. Brau & A. De Wit
    Genericity of confined chemical garden patterns with regard to changes in the reactants, Phys. Chem. Chem. Phys. 17, 12804 (2015).
  18. F. Haudin, J. H. E. Cartwright, F. Brau & A. De Wit
    Spiral precipitation patterns in confined chemical gardens, PNAS 111, 17363 (2014).
    See Commentary by Oliver Steinbock + Press (see above)
  19. F. Brau
    Tearing of thin sheets: Cracks interacting through an elastic ridge, Phys. Rev. E 90, 062406 (2014).
  20. F. Brau, P. Damman, H. Diamant & T. A. Witten
    Wrinkle to fold transition: Influence of the substrate response, Soft Matter 9, 8177 (2013).
  21. B. Kolaric, S. Desprez, F. Brau & P. Damman
    Design of curved photonic crystal using swelling induced instabilities, J. Materials Chem. 22, 16205 (2012).
  22. C. Zeng, F. Brau, B. Davidovitch & A. D. Dinsmore
    Capillary interactions among spherical particles at curved liquid interfaces, Soft Matter 8, 8582 (2012).
  23. A. Takei, F. Brau, J. Bico & B. Roman
    Stretch-induced wrinkles in reinforced membranes: from out-of-plane to in-plane structures, EPL 96, 64001 (2011).
  24. O. Kruglova, F. Brau, D. Villers & P. Damman
    How geometry controls the tearing of adhesive thin films on curved surfaces, Phys. Rev. Lett. 107, 164303 (2011).
    Editors' Suggestion
  25. H. Vandeparre, M. Pineirua, F. Brau, B. Roman, J. Bico, C. Gay, W. Bao, C. N. Lau, P. Reis & P. Damman
    Wrinkling hierarchy in constrained thin sheets from suspended graphene to curtains, Phys. Rev. Lett. 106, 224301 (2011).
    Editors' Suggestion + Cover + Press (see above)
  26. F. Brau, H. Vandeparre, A. Sabbah, C. Poulard, A. Boudaoud & P. Damman
    Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators, Nature Physics 7, 56 (2011).
    Press (see above)
  27. U. Ebert, F. Brau, G. Derks, W. Hundsdorfer, C.-Y. Kao, C. Li, A. Luque, B. Meulenbroek, S. Nijdam, V. Ratushnaya, L. Schäfer & S. Tanveer
    Multiple scales in streamer discharges, with an emphasis on moving boundary approximations, Nonlinearity 24, C1 (2011).
  28. H. Vandeparre, S. Gabriele, F. Brau, C. Gay, K. K. Parker & P. Damman
    Hierarchical Wrinkling Patterns, Soft Matter 6, 5751 (2010).
  29. F. Brau, B. Davidovitch & U. Ebert
    Moving boundary approximation for curved streamer ionization fronts: Solvability analysis, Phys. Rev. E 78, 056212 (2008).
  30. A. Luque, F. Brau & U. Ebert
    Saffman-Taylor streamers: Mutual finger interaction in spark formation, Phys. Rev. E 78, 016206 (2008).

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