Amplification of velocity gradients, a key feature of turbulent flows, is affected by the non-loc... more Amplification of velocity gradients, a key feature of turbulent flows, is affected by the non-local character of the incompressible fluid equations expressed by the second derivative (Hessian) of the pressure field. By analysing the structure of the flow in regions where the vorticity is the highest, we propose an approximate expression for the pressure Hessian in terms of the local vorticity, consistent with the existence of intense vortex tubes. Contrary to the often used simplification of an isotropic form for the pressure Hessian, which in effect inhibits vortex stretching, the proposed approximate form of the pressure Hessian enables much stronger vortex stretching. The prediction of the approximation proposed here is validated with results of direct numerical simulations of turbulent flows.
Bulletin of the American Physical Society, Nov 21, 2021
Turbulent fluid flows are ubiquitous in nature and technology, and are mathematically described b... more Turbulent fluid flows are ubiquitous in nature and technology, and are mathematically described by the incompressible Navier-Stokes equations. A hallmark of turbulence is spontaneous generation of intense whirls, resulting from amplification of the fluid rotation-rate (vorticity) by its deformationrate (strain). This interaction, encoded in the non-linearity of Navier-Stokes equations, is non-local, i.e., depends on the entire state of the flow, constituting a serious hindrance in turbulence theory and even establishing regularity of the equations. Here, we unveil a novel aspect of this interaction, by separating strain into local and non-local contributions utilizing the Biot-Savart integral of vorticity in a sphere of radius R. Analyzing highly-resolved numerical turbulent solutions to Navier-Stokes equations, we find that when vorticity becomes very large, the local strain over small R surprisingly counteracts further amplification. This uncovered self-attenuation mechanism is further shown to be connected to local Beltramization of the flow, and could provide a direction in establishing the regularity of Navier-Stokes equations.
Bulletin of the American Physical Society, Nov 22, 2020
Submitted for the DFD20 Meeting of The American Physical Society Effect of particle inertia on th... more Submitted for the DFD20 Meeting of The American Physical Society Effect of particle inertia on the alignment of small ice crystals in turbulent clouds BERNHARD MEHLIG, K. GUSTAVSSON, U Gothenburg,
We investigate the large-scale intermittency of vertical velocity and temperature, and the mixing... more We investigate the large-scale intermittency of vertical velocity and temperature, and the mixing properties of stably stratified turbulent flows using both Lagrangian and Eulerian fields from direct numerical simulations, in a parameter space relevant for the atmosphere and the oceans. Over a range of Froude numbers of geophysical interest (≈ 0.05 − 0.3) we observe very large fluctuations of the vertical components of the velocity and the potential temperature, localized in space and time, with a sharp transition leading to non-Gaussian wings of the probability distribution functions. This behavior is captured by a simple model representing the competition between gravity waves on a fast timescale and nonlinear steepening on a slower timescale. The existence of a resonant regime characterized by enhanced large-scale intermittency, as understood within the framework of the proposed model, is then linked to the emergence of structures in the velocity and potential temperature fields, localized overturning and mixing. Finally, in the same regime we observe a linear scaling of the mixing efficiency with the Froude number and an increase of its value of roughly one order of magnitude.
Collisions between particles suspended in a fluid play an important role in many physical process... more Collisions between particles suspended in a fluid play an important role in many physical processes. As an example, collisions of microscopic water droplets in clouds are a necessary step in the production of macroscopic raindrops. Collisions of dust grains are also conjectured to be important for planet formation in the gas surrounding young stars and to play a role in the dynamics of sand storms. In these processes, collisions are favored by fast turbulent motions. Here we review recent advances in the understanding of collisional aggregation due to turbulence. We discuss the role of fractal clustering of particles and caustic singularities of their velocities. We also discuss limitations of the Smoluchowski equation for modeling such processes. These advances lead to a semiquantitative understanding on the influence of turbulence on collision rates and point to deficiencies in the current understanding of rainfall and planet formation.
HAL (Le Centre pour la Communication Scientifique Directe), 2005
Artificial biological pacemakers were developed and tested in canine ventricles. Next steps will ... more Artificial biological pacemakers were developed and tested in canine ventricles. Next steps will require obtaining oscillations sensitive to external regulations, and robust with respect to long term drifts of expression levels of pacemaker currents and gap junctions. We introduce mathematical models intended to be used in parallel with the experiments. The models describe human mesenchymal stem cells (hMSC) transfected with HCN2 genes and connected to myocytes. They are intended to mimic experiments with oscillation induction in a cell pair, in cell culture and in the cardiac tissue. We give examples of oscillations in a cell pair, in a one dimensional cell culture and dependence of oscillation on number of pacemaker channels per cell and number of gap junctions. The models permit to mimic experiments with levels of gene expressions that are not achieved yet and to predict if the work to achieve this levels will significantly increase the quality of oscillations. This give arguments for selecting the directions of the experimental work.
Bulletin of the American Physical Society, Mar 15, 2016
chemotaxis. Here we report experimental results on the self-oscillation mechanism and the role of... more chemotaxis. Here we report experimental results on the self-oscillation mechanism and the role of regulatory proteins and myosin II. We stimulate cells rapidly and periodically by using photo un-caging of the chemoattractant in a micro-fluidic device and measured the cellular responses. We found that the response amplitude grows with stimulation strength only in a very narrow region of stimulation, after which the response amplitude reaches a plateau. Moreover, the frequency-response is not constant but rather varies with the strength of external stimuli. To understand the underlying mechanism, we analyzed the polymerization and de-polymerization time in the single cell level. Despite of the large cell-to-cell variability, we found that the polymerization time is independent of external stimuli and the de-polymerization time is prolonged as the stimulation strength increases. Our conclusions will be summarized and the role of noise in the signaling network will be discussed.
The serious study of singularities in the equations of three dimensional hydrodynamics goes back ... more The serious study of singularities in the equations of three dimensional hydrodynamics goes back at least to 1934, and a proof still has not been given that solutions of the Navier-Stokes equations remain smooth.1 Far less is known rigorously about solutions to the Euler equations. Although on physical grounds one might expect a singularity here, the numerical evidence is unpersuasive.
ABSTRACT Fast self sustained waves (autowaves) associated with chemical or phase transformations ... more ABSTRACT Fast self sustained waves (autowaves) associated with chemical or phase transformations are observed in many situations in condensed matter. They are governed neither by diffusion of matter or heat (as in combustion processes) nor by a travelling shock wave (as in gaseous detonation). Instead, they result from a coupling between phase transformation and the stress field, and may be classified as gasless detonation autowaves in solids. We propose a simple model to describe these regimes. The model rests on the classical equations of elastic deformations in a 1-dimensional solid bar, with the extra assumption that the phase (chemical) transformation induces a change of the sound velocity. The transformations are assumed to occur through a chain branched mechanism, which starts when the mechanical stress exceeds a given threshold. Our investigation shows that supersonic autowaves exist in this model. In the absence of diffusion (dissipation factor, losses), a continuum of travelling wave solutions is found. In the presence of diffusion, a steady state supersonic wave solution is found, along with a slower wave controlled by diffusion. PACS. 82.20.Mj Nonequilibrium kinetics - 05.70.Ln Nonequilibrium and irreversible thermodynamics
We investigate the role of pressure, via its Hessian tensor H, on amplification of vorticity and ... more We investigate the role of pressure, via its Hessian tensor H, on amplification of vorticity and strain-rate and contrast it with other inviscid nonlinear mechanisms. Results are obtained from direct numerical simulations of isotropic turbulence with Taylor-scale Reynolds number in the range 140 − 1300. Decomposing H into local isotropic (H I) and nonlocal deviatoric (H D) components reveals that H I depletes vortex stretching (VS), whereas H D enables it, with the former slightly stronger. The resulting inhibition is significantly weaker than the nonlinear mechanism which always enables VS. However, in regions of intense vorticity, identified using conditional statistics, contribution from H dominates over nonlinearity, leading to overall depletion of VS. We also observe near-perfect alignment between vorticity and the eigenvector of H corresponding to the smallest eigenvalue, which conforms with well-known vortex-tubes. We discuss the connection between this depletion, essentially due to (local) H I , and recently identified self-attenuation mechanism [Buaria et al. Nat. Commun. 11:5852 (2020)], whereby intense vorticity is locally attenuated through inviscid effects. In contrast, the influence of H on strain-amplification is weak. It opposes strain self-amplification, together with VS, but its effect is much weaker than VS. Correspondingly, the eigenvectors of strain and H do not exhibit any strong alignments. For all results, the dependence on Reynolds number is very weak. In addition to the fundamental insights, our work provides useful data and validation benchmarks for future modeling endeavors, for instance in Lagrangian modeling of velocity gradient dynamics, where conditional H is explicitly modeled.
Bulletin of the American Physical Society, Nov 20, 2017
Nucleation of droplets and ice-crystals, starting from aerosols, leads in clouds to the formation... more Nucleation of droplets and ice-crystals, starting from aerosols, leads in clouds to the formation of very small particles, whose size has to grow by orders of magnitude to become rain drops or hail particles. The growth of droplets in warm clouds (in the absence of ice crystals) has recently received much attention. Here, I will discuss the problem of settling and collision of ice crystals in a turbulent environment. In the temperature range-20C <T <-10C, the crystals shapes can be approximated as very thing oblate ellipsoids, with a size smaller than the size of the smallest eddies (Kolmogorov scale). Neglecting completely the inertia of the fluid, I will show that the crystals tend to settle predominantly with their face first. The orientation bias of the crystals is particularly strong when turbulence intensity is low. When the flow Reynolds number is moderate, the collisions between crystals results essentially from the difference in the settling velocities, due to the different orientation (the settling velocity depends on the crystal orientation). The role of turbulence in determining the collision rate becomes prevalent when the Reynolds number increases.
In 3D turbulent flows, the direct cascade of energy, characterized by a flux through scales, ε, i... more In 3D turbulent flows, the direct cascade of energy, characterized by a flux through scales, ε, is a strong source of irreversibility. This irreversibility manifests itself in the asymmetry between negative and positive kinetic energy differences along particle trajectories. In particular, it is observed that the odd moments of the power of the forces acting on a fluid particle, p ≡ a • V are negative, and that − p 3 /ε 3 ∝ R 2 λ. This property rests on subtle correlations between acceleration and velocity. I will discuss two representations of the acceleration a, which shed light on the irreversibility of the flow.
We consider a particle moving with equation of motionẋ = f (t), where f (t) is a random function ... more We consider a particle moving with equation of motionẋ = f (t), where f (t) is a random function with statistics which are independent of x and t, with a finite drift velocity v = f and in the presence of a reflecting wall. Far away from the wall, translational invariance implies that the stationary probability distribution is P (x) ∼ exp(αx). A classical example of a problem of this type is sedimentation equilibrium, where α is determined by temperature. In this work we do not introduce a thermal reservoir and α is determined from the equation of motion. We consider a general approach to determining α which is not always in agreement with Einstein's relation between the mean velocity and the diffusion coefficient. We illustrate our results with a model inspired by the Boltzmann equation.
ABSTRACT : Fronts of weakly exothermal chemical reaction may propagate in solids at very low temp... more ABSTRACT : Fronts of weakly exothermal chemical reaction may propagate in solids at very low temperatures ( 4K≤T≤77K) thanks to a quite unusual mechanism, involving a feedback between the heat produced by the reaction and the disruption of the solid matrix. In this class of phenomena, the reaction may be induced by mechanical constraints, without a large elevation of temperature. On the basis of a simple phenomenological model, we investigate ignition of a propagating front by initially (i) disrupting a localized zone of the solid matrix, or by (ii) introducing a temperature jump, leading to a thermal shock with strong temperature gradients. In particular, we show that reaction can be initiated by disrupting only a very small fraction of the sample. Applications to the problem of initiation of solid explosives by friction or shocks is briefly discussed.
Steady, nonpropagating, fronts in reaction diffusion systems usually exist only for special sets ... more Steady, nonpropagating, fronts in reaction diffusion systems usually exist only for special sets of control parameters. When varying one control parameter, the front velocity may become zero only at isolated values ͑where the Maxwell condition is satisfied, for potential systems͒. The experimental observation of fronts with a zero velocity over a finite interval of parameters, e.g., in catalytic experiments ͓Barelko et al., Chem. Eng. Sci., 33, 805 ͑1978͔͒, therefore, seems paradoxical. We show that the velocity dependence on the control parameter may be such that velocity is very small over a finite interval, and much larger outside. This happens in a class of reaction diffusion systems with two components, with the extra assumptions that ͑i͒ the two diffusion coefficients are very different, and that ͑ii͒ the slowly diffusing variables has two stable states over a control parameter range. The ratio of the two velocity scales vanishes when the smallest diffusion coefficient goes to zero. A complete study of the effect is carried out in a model of catalytic reaction.
Amplification of velocity gradients, a key feature of turbulent flows, is affected by the non-loc... more Amplification of velocity gradients, a key feature of turbulent flows, is affected by the non-local character of the incompressible fluid equations expressed by the second derivative (Hessian) of the pressure field. By analysing the structure of the flow in regions where the vorticity is the highest, we propose an approximate expression for the pressure Hessian in terms of the local vorticity, consistent with the existence of intense vortex tubes. Contrary to the often used simplification of an isotropic form for the pressure Hessian, which in effect inhibits vortex stretching, the proposed approximate form of the pressure Hessian enables much stronger vortex stretching. The prediction of the approximation proposed here is validated with results of direct numerical simulations of turbulent flows.
Bulletin of the American Physical Society, Nov 21, 2021
Turbulent fluid flows are ubiquitous in nature and technology, and are mathematically described b... more Turbulent fluid flows are ubiquitous in nature and technology, and are mathematically described by the incompressible Navier-Stokes equations. A hallmark of turbulence is spontaneous generation of intense whirls, resulting from amplification of the fluid rotation-rate (vorticity) by its deformationrate (strain). This interaction, encoded in the non-linearity of Navier-Stokes equations, is non-local, i.e., depends on the entire state of the flow, constituting a serious hindrance in turbulence theory and even establishing regularity of the equations. Here, we unveil a novel aspect of this interaction, by separating strain into local and non-local contributions utilizing the Biot-Savart integral of vorticity in a sphere of radius R. Analyzing highly-resolved numerical turbulent solutions to Navier-Stokes equations, we find that when vorticity becomes very large, the local strain over small R surprisingly counteracts further amplification. This uncovered self-attenuation mechanism is further shown to be connected to local Beltramization of the flow, and could provide a direction in establishing the regularity of Navier-Stokes equations.
Bulletin of the American Physical Society, Nov 22, 2020
Submitted for the DFD20 Meeting of The American Physical Society Effect of particle inertia on th... more Submitted for the DFD20 Meeting of The American Physical Society Effect of particle inertia on the alignment of small ice crystals in turbulent clouds BERNHARD MEHLIG, K. GUSTAVSSON, U Gothenburg,
We investigate the large-scale intermittency of vertical velocity and temperature, and the mixing... more We investigate the large-scale intermittency of vertical velocity and temperature, and the mixing properties of stably stratified turbulent flows using both Lagrangian and Eulerian fields from direct numerical simulations, in a parameter space relevant for the atmosphere and the oceans. Over a range of Froude numbers of geophysical interest (≈ 0.05 − 0.3) we observe very large fluctuations of the vertical components of the velocity and the potential temperature, localized in space and time, with a sharp transition leading to non-Gaussian wings of the probability distribution functions. This behavior is captured by a simple model representing the competition between gravity waves on a fast timescale and nonlinear steepening on a slower timescale. The existence of a resonant regime characterized by enhanced large-scale intermittency, as understood within the framework of the proposed model, is then linked to the emergence of structures in the velocity and potential temperature fields, localized overturning and mixing. Finally, in the same regime we observe a linear scaling of the mixing efficiency with the Froude number and an increase of its value of roughly one order of magnitude.
Collisions between particles suspended in a fluid play an important role in many physical process... more Collisions between particles suspended in a fluid play an important role in many physical processes. As an example, collisions of microscopic water droplets in clouds are a necessary step in the production of macroscopic raindrops. Collisions of dust grains are also conjectured to be important for planet formation in the gas surrounding young stars and to play a role in the dynamics of sand storms. In these processes, collisions are favored by fast turbulent motions. Here we review recent advances in the understanding of collisional aggregation due to turbulence. We discuss the role of fractal clustering of particles and caustic singularities of their velocities. We also discuss limitations of the Smoluchowski equation for modeling such processes. These advances lead to a semiquantitative understanding on the influence of turbulence on collision rates and point to deficiencies in the current understanding of rainfall and planet formation.
HAL (Le Centre pour la Communication Scientifique Directe), 2005
Artificial biological pacemakers were developed and tested in canine ventricles. Next steps will ... more Artificial biological pacemakers were developed and tested in canine ventricles. Next steps will require obtaining oscillations sensitive to external regulations, and robust with respect to long term drifts of expression levels of pacemaker currents and gap junctions. We introduce mathematical models intended to be used in parallel with the experiments. The models describe human mesenchymal stem cells (hMSC) transfected with HCN2 genes and connected to myocytes. They are intended to mimic experiments with oscillation induction in a cell pair, in cell culture and in the cardiac tissue. We give examples of oscillations in a cell pair, in a one dimensional cell culture and dependence of oscillation on number of pacemaker channels per cell and number of gap junctions. The models permit to mimic experiments with levels of gene expressions that are not achieved yet and to predict if the work to achieve this levels will significantly increase the quality of oscillations. This give arguments for selecting the directions of the experimental work.
Bulletin of the American Physical Society, Mar 15, 2016
chemotaxis. Here we report experimental results on the self-oscillation mechanism and the role of... more chemotaxis. Here we report experimental results on the self-oscillation mechanism and the role of regulatory proteins and myosin II. We stimulate cells rapidly and periodically by using photo un-caging of the chemoattractant in a micro-fluidic device and measured the cellular responses. We found that the response amplitude grows with stimulation strength only in a very narrow region of stimulation, after which the response amplitude reaches a plateau. Moreover, the frequency-response is not constant but rather varies with the strength of external stimuli. To understand the underlying mechanism, we analyzed the polymerization and de-polymerization time in the single cell level. Despite of the large cell-to-cell variability, we found that the polymerization time is independent of external stimuli and the de-polymerization time is prolonged as the stimulation strength increases. Our conclusions will be summarized and the role of noise in the signaling network will be discussed.
The serious study of singularities in the equations of three dimensional hydrodynamics goes back ... more The serious study of singularities in the equations of three dimensional hydrodynamics goes back at least to 1934, and a proof still has not been given that solutions of the Navier-Stokes equations remain smooth.1 Far less is known rigorously about solutions to the Euler equations. Although on physical grounds one might expect a singularity here, the numerical evidence is unpersuasive.
ABSTRACT Fast self sustained waves (autowaves) associated with chemical or phase transformations ... more ABSTRACT Fast self sustained waves (autowaves) associated with chemical or phase transformations are observed in many situations in condensed matter. They are governed neither by diffusion of matter or heat (as in combustion processes) nor by a travelling shock wave (as in gaseous detonation). Instead, they result from a coupling between phase transformation and the stress field, and may be classified as gasless detonation autowaves in solids. We propose a simple model to describe these regimes. The model rests on the classical equations of elastic deformations in a 1-dimensional solid bar, with the extra assumption that the phase (chemical) transformation induces a change of the sound velocity. The transformations are assumed to occur through a chain branched mechanism, which starts when the mechanical stress exceeds a given threshold. Our investigation shows that supersonic autowaves exist in this model. In the absence of diffusion (dissipation factor, losses), a continuum of travelling wave solutions is found. In the presence of diffusion, a steady state supersonic wave solution is found, along with a slower wave controlled by diffusion. PACS. 82.20.Mj Nonequilibrium kinetics - 05.70.Ln Nonequilibrium and irreversible thermodynamics
We investigate the role of pressure, via its Hessian tensor H, on amplification of vorticity and ... more We investigate the role of pressure, via its Hessian tensor H, on amplification of vorticity and strain-rate and contrast it with other inviscid nonlinear mechanisms. Results are obtained from direct numerical simulations of isotropic turbulence with Taylor-scale Reynolds number in the range 140 − 1300. Decomposing H into local isotropic (H I) and nonlocal deviatoric (H D) components reveals that H I depletes vortex stretching (VS), whereas H D enables it, with the former slightly stronger. The resulting inhibition is significantly weaker than the nonlinear mechanism which always enables VS. However, in regions of intense vorticity, identified using conditional statistics, contribution from H dominates over nonlinearity, leading to overall depletion of VS. We also observe near-perfect alignment between vorticity and the eigenvector of H corresponding to the smallest eigenvalue, which conforms with well-known vortex-tubes. We discuss the connection between this depletion, essentially due to (local) H I , and recently identified self-attenuation mechanism [Buaria et al. Nat. Commun. 11:5852 (2020)], whereby intense vorticity is locally attenuated through inviscid effects. In contrast, the influence of H on strain-amplification is weak. It opposes strain self-amplification, together with VS, but its effect is much weaker than VS. Correspondingly, the eigenvectors of strain and H do not exhibit any strong alignments. For all results, the dependence on Reynolds number is very weak. In addition to the fundamental insights, our work provides useful data and validation benchmarks for future modeling endeavors, for instance in Lagrangian modeling of velocity gradient dynamics, where conditional H is explicitly modeled.
Bulletin of the American Physical Society, Nov 20, 2017
Nucleation of droplets and ice-crystals, starting from aerosols, leads in clouds to the formation... more Nucleation of droplets and ice-crystals, starting from aerosols, leads in clouds to the formation of very small particles, whose size has to grow by orders of magnitude to become rain drops or hail particles. The growth of droplets in warm clouds (in the absence of ice crystals) has recently received much attention. Here, I will discuss the problem of settling and collision of ice crystals in a turbulent environment. In the temperature range-20C <T <-10C, the crystals shapes can be approximated as very thing oblate ellipsoids, with a size smaller than the size of the smallest eddies (Kolmogorov scale). Neglecting completely the inertia of the fluid, I will show that the crystals tend to settle predominantly with their face first. The orientation bias of the crystals is particularly strong when turbulence intensity is low. When the flow Reynolds number is moderate, the collisions between crystals results essentially from the difference in the settling velocities, due to the different orientation (the settling velocity depends on the crystal orientation). The role of turbulence in determining the collision rate becomes prevalent when the Reynolds number increases.
In 3D turbulent flows, the direct cascade of energy, characterized by a flux through scales, ε, i... more In 3D turbulent flows, the direct cascade of energy, characterized by a flux through scales, ε, is a strong source of irreversibility. This irreversibility manifests itself in the asymmetry between negative and positive kinetic energy differences along particle trajectories. In particular, it is observed that the odd moments of the power of the forces acting on a fluid particle, p ≡ a • V are negative, and that − p 3 /ε 3 ∝ R 2 λ. This property rests on subtle correlations between acceleration and velocity. I will discuss two representations of the acceleration a, which shed light on the irreversibility of the flow.
We consider a particle moving with equation of motionẋ = f (t), where f (t) is a random function ... more We consider a particle moving with equation of motionẋ = f (t), where f (t) is a random function with statistics which are independent of x and t, with a finite drift velocity v = f and in the presence of a reflecting wall. Far away from the wall, translational invariance implies that the stationary probability distribution is P (x) ∼ exp(αx). A classical example of a problem of this type is sedimentation equilibrium, where α is determined by temperature. In this work we do not introduce a thermal reservoir and α is determined from the equation of motion. We consider a general approach to determining α which is not always in agreement with Einstein's relation between the mean velocity and the diffusion coefficient. We illustrate our results with a model inspired by the Boltzmann equation.
ABSTRACT : Fronts of weakly exothermal chemical reaction may propagate in solids at very low temp... more ABSTRACT : Fronts of weakly exothermal chemical reaction may propagate in solids at very low temperatures ( 4K≤T≤77K) thanks to a quite unusual mechanism, involving a feedback between the heat produced by the reaction and the disruption of the solid matrix. In this class of phenomena, the reaction may be induced by mechanical constraints, without a large elevation of temperature. On the basis of a simple phenomenological model, we investigate ignition of a propagating front by initially (i) disrupting a localized zone of the solid matrix, or by (ii) introducing a temperature jump, leading to a thermal shock with strong temperature gradients. In particular, we show that reaction can be initiated by disrupting only a very small fraction of the sample. Applications to the problem of initiation of solid explosives by friction or shocks is briefly discussed.
Steady, nonpropagating, fronts in reaction diffusion systems usually exist only for special sets ... more Steady, nonpropagating, fronts in reaction diffusion systems usually exist only for special sets of control parameters. When varying one control parameter, the front velocity may become zero only at isolated values ͑where the Maxwell condition is satisfied, for potential systems͒. The experimental observation of fronts with a zero velocity over a finite interval of parameters, e.g., in catalytic experiments ͓Barelko et al., Chem. Eng. Sci., 33, 805 ͑1978͔͒, therefore, seems paradoxical. We show that the velocity dependence on the control parameter may be such that velocity is very small over a finite interval, and much larger outside. This happens in a class of reaction diffusion systems with two components, with the extra assumptions that ͑i͒ the two diffusion coefficients are very different, and that ͑ii͒ the slowly diffusing variables has two stable states over a control parameter range. The ratio of the two velocity scales vanishes when the smallest diffusion coefficient goes to zero. A complete study of the effect is carried out in a model of catalytic reaction.
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Papers by Alain Pumir