arXiv subjects

Itzhak Fouxon

Publications and source records attributed to Itzhak Fouxon.

At least 19 recordsLinked to original sources

arXiv2024

Slow linear shear flow past discrete particles adhered to a plane wall

Linear shear flow bounded by a plane wall is an idealization that occurs in microfluidic devices and many other applications. Perfect plane approximation neglects surface irregularities and discrete particles adsorbed at the surface. Here we study the disturbance to the linear shear flow due to the particle(s) rigidly attached to the surface. We first revisit the exact solution of O'Neill for a spherical particle in contact with an infinite plane boundary. While the original paper contains multiple typos and provides very few details of the derivation, we present detailed solution accompanied by an alternative and simpler derivation of the viscous force and the torque exerted on the particle. We further study the universal far-field behavior of the flow due to an arbitrary particle adhered to the surface, and demonstrate that it is controlled by the stress moment of magnitude depending on particle's volume and shape. Using the revised O'Neill solution, we compute the stress moment for a spherical particle. Using the far-field asymptotic form of the flow we estimate the net flow due to uniform and sparse layer of discrete adsorbates by superposition and demonstrate that it does not decay away from the plane.

physics.flu-dyn↗

arXiv2024

Quartz Crystal Microbalance frequency response to discrete adsorbates in liquids

Quartz Crystal Microbalance with Dissipation monitoring (QCM-D) has become a major tool in the analysis of adsorption of nanometric objects, such as proteins, viruses, liposomes and inorganic particles from the solution. While in vacuum extremely accurate mass measurements are possible, in a liquid phase the quantitative analysis is intricate due to the complex interplay of hydrodynamic and adhesion forces, varying with the physicochemical properties of adsorbent and the quartz resonator surfaces. In the present paper we dissect the role of hydrodynamics for the analytically tractable scenario of a stiff contact, whereas the adsorbed particles oscillate with the resonator as a whole without rotation. Under the assumption of the low surface coverage, we theoretically study the excess shear force exerted on the resonator due to presence of a single adsorbed particle. The excess shear force has two contributions: (i) the fluid-mediated force due to flow disturbance created by the particle and (ii) the viscous force exerted on the particle by the fluid and transmitted to the resonator via contact. We found that for small adsorbates, there is mutual cancellation of the above contributions to the excess shear force at the leading order approximation, reducing the overall effect of the hydrodynamics to the order-of-magnitude of the inertial force. However, accurate numerical solution shows that for small particles the viscous force dominates over the inertia force, rendering the standard Sauerbrey model inapplicable. These findings indicate that the accurate account of hydrodynamics in the analysis of QCM-D response is critical. The resulting dimensionless frequency and dissipation shifts and the corresponding acoustic ratio computed numerically, showing a fair agreement with previously published experimental results at low oscillation frequencies.

cond-mat.soft↗

arXiv2023

Lyapunov exponents and Lagrangian chaos suppression in compressible homogeneous isotropic turbulence

We study Lyapunov exponents of tracers in compressible homogeneous isotropic turbulence at different turbulent Mach number $M_t$ and Taylor-scale Reynolds number $Re_λ$. We demonstrate that statistics of finite-time Lyapunov exponents have the same form as in incompressible flow due to density-velocity coupling. Modulus of the smallest Lyapunov exponent $λ_3$ provides the principal Lyapunov exponent of the time-reversed flow, which usually is wrong in a compressible flow. This exponent, along with the principal Lyapunov exponent $λ_1$, determines all the exponents due to the vanishing of the sum of all Lyapunov exponents. Numerical results by high-order schemes for solving the Navier-Stokes equations and tracking particles verify these theoretical predictions. We found that: 1) The largest normalized Lyapunov exponent $λ_1 τ_η$, where $τ_η$ is the Kolmogorov time scale, is a decreasing function of $M_t$. Its dependence on $Re_λ$ is weak when the driving force is solenoidal, while it is an increasing function of $Re_λ$ when the solenoidal and compressible forces are comparable. Similar facts hold for $|λ_3|$, in contrast with well-studied short-correlated model; 2) The ratio of the first two Lyapunov exponents $λ_1/λ_2$ decreases with $Re_λ$, and is virtually independent of $M_t$ for $M_t \le 1$ in the case of solenoidal force but decreases as $M_t$ increases when solenoidal and compressible forces are comparable; 3) For purely solenoidal force, $λ_1 :λ_2 :λ_3 \approx 4:1:-5$ for $Re_λ> 80$, which is consistent with incompressible turbulence studies; 4) The ratio of dilation-to-vorticity is a more suitable parameter to characterize LEs than $M_t$.

physics.flu-dyn↗

arXiv2023

Compressible two-dimensional turbulence: cascade reversal and sensitivity to imposed magnetic field

We study the impact of compressibility on two-dimensional turbulent flows, such as those modeling astrophysical disks. We demonstrate that the direction of cascade undergoes continuous transition as the Mach number Ma increases, from inverse at zero Ma, to direct at infinite Ma. Thus, at Ma of order one comparable amounts of energy flow from the scale of the pumping to large and small scales, in accord with previous data. For supersonic turbulence with large Ma, the cascade is direct, as in three dimensions, which results in multifractal density field. For compressible flows of conducting fluids, imposing external magnetic field allows to manipulate the flow producing possibly large changes even at small Mach number. Thus Zeldovich antidynamo theorem, by which at zero Ma the magnetic field is zero in the steady state, must be used with caution. Real flows have finite Ma and, however small it is, for large magnetic flux through the disk, the magnetic field changes the flow appreciably, or rearranges it completely. For large Ma, relevant for astrophysical disks, the magnetic field energy is enhanced by a large factor as compared to estimates based on the mean field. We propose to use two-dimensional Burgers turbulence, whose three-dimensional counterpart is used for studies of the large-scale structure of the Universe, as a model for supersonic thin accretion disks.

physics.flu-dyn↗

arXiv2022

Excess shear force exerted on oscillating plate due to a nearby particle

In the present paper we theoretically study the shear force exerted on an infinite horizontal plane undergoing fast lateral oscillations in presence of a rigid particle suspended in the viscous liquid above the plate. The study is largely motivated by Quartz Crystal Microbalance (QCM-D) technique which relies on analyzing response (complex impedance) of fast oscillating (in MHz range) quartz crystal disk in the liquid medium due to small substances adsorbed at its surface. In fact, small substances suspended in the liquid medium in the vicinity of the oscillating crystal may also contribute to impedance, as they modify the local shear force the suspending liquid exerts on the quartz crystal. For a dilute suspension the contributions of individual particles are additive and, therefore, our analysis is restricted to the excess shear force due a single spherical particle located at arbitrary distance above the plane. Three distinct cases are considered: (i) limiting case of high solid inertia, whereas the heavy particle can be considered as stationary; (ii) a freely suspended particle of arbitrary mass, undergoing fluid-mediated time-periodic rotation and translation and (iii) an adsorbed particle moving with the plate as a whole without rotation. For small-amplitude plane oscillations the unsteady Stokes flow equations apply. We construct the series solution of these equations using the method of reflections, whereas its terms are written explicitly. Due to the exponential decay of the flow away from the oscillating plate, the truncated series containing only few low-order terms shows an excellent agreement with the rigorous numerical results for a wide range of particle sizes and separation distances. The present results support the notion that the hydrodynamic contribution of the suspended small substances to the measured impedance is non-negligible or even dominant.

physics.flu-dyn↗

arXiv2022

Quantitative prediction of sling events in turbulence at high Reynolds numbers

Collisional growth of droplets, such as occurring in warm clouds, is known to be significantly enhanced by turbulence. Whether particles collide depends on their flow history, in particular on their encounters with highly intermittent small-scale turbulent structures, which despite their rarity can dominate the overall collision rate. Intuitively, strong vortices may act as slings for inertial particles, leading to intersections where several streams of particles collide at large velocities. Here, we develop a quantitative criterion for sling events based on the velocity gradient history along particle paths. We demonstrate by combination of theory and simulations that the problem reduces to a one-dimensional localization problem as encountered in condensed matter physics. The reduction demonstrates that the creation of slings is completely controlled by the smallest negative eigenvalue of the velocity gradient tensor. We use fully resolved turbulence simulations to confirm our predictions and study their Stokes and Reynolds number dependence. We also discuss extrapolations to the parameter range relevant in clouds, showing that sling events at high Reynolds numbers are significantly enhanced for small Stokes numbers.

physics.flu-dyn↗

arXiv2022

Lamb-type solution and properties of unsteady Stokes equations

We derive the general solution of the unsteady Stokes equations for an unbounded fluid in spherical polar coordinates, in both time and frequency domains. The solution is an expansion in vector spherical harmonics and given as a sum of a particular solution, proportional to pressure gradient exhibiting power-law spatial dependence, and a solution of vector Helmholtz equation decaying exponentially in far field, the decomposition originally introduced by Lamb. The solution can be applied to construct the transient exterior flow induced by an arbitrary velocity distribution at the spherical boundary, such as arising in the squirmer model of a microswimmer. It can be used to construct solutions for transient flows driven by initial conditions, unbounded flows driven by volume forces or disturbance to the unsteady flow due to a stationary spherical particle. The long-time behavior of solution is controlled by the flow component corresponding to average (or collective) motion of the boundary. This conclusion is illustrated by the study of decay of transversal wave in the presence of a fixed sphere. We further show that the general representation reduces to the well-known solutions for unsteady flow around a sphere undergoing oscillatory rigid-body (translation and rotation) motion. The proposed solution representation provides an explicit form of the velocity potential far from an oscillating body ("generalized" Darcy's law) and high- and low-frequency expansions. The leading-order high-frequency expansion yields the well-known ideal (inviscid) flow approximation, and the leading-order low-frequency expansion yields the steady Stokes equations. We derive the higher-order corrections to these approximations and discuss d'Alembert paradox. Continuation of the general solution to imaginary frequency yields the general solution of the Brinkman equations describing viscous flow in porous medium.

physics.flu-dyn↗

arXiv2022

Intermittency and collisions of fast sedimenting droplets in turbulence

We study theoretically and numerically spatial distribution and collision rate of droplets that sediment in homogeneous isotropic Navier-Stokes turbulence. It is assumed that typical turbulent accelerations of fluid particles are much smaller than gravity. This was shown to imply that the particles interact weakly with individual vortices and, as a result, form a smooth flow in most of the space. In weakly intermittent turbulence with moderate Reynolds number, rare regions where the flow breaks down can be neglected in the calculation of space averaged rate of droplet collisions. However, increase of Re increases probability of rare, large quiescent vortices whose long coherent interaction with the particles destroys the flow. Thus at higher Re the space averaged collision rate forms in rare regions where the assumption of smooth flow breaks down. We describe the transition between the regimes and provide collision kernel in the case of moderate Re describable by the flow. The distribution of pairwise distances is shown to obey a separable dependence on the magnitude and the polar angle of the separation vector. Magnitude dependence obeys a power-law with a negative exponent, manifesting multifractality of the droplet's attractor. We provide the so far missing numerical confirmation of a relation between this exponent and the Lyapunov exponents and demonstrate that it holds beyond the theoretical range. The angular dependence of the RDF exhibits a maximum at small angles quantifying particle's formation of spatial columns. We derive the droplet's collision kernel using that in the considered limit the gradients of droplet's flow are Gaussian. We demonstrate that as Re increases the column's aspect ratio decreases, eventually becoming one when the isotropy is restored. We propose how the theory could be constructed at higher Re of clouds by using the example of the RDF.

physics.flu-dyn↗

arXiv2021

Linear and nonlinear hydromagnetic stability in laminar and turbulent flows

We consider the evolution of arbitrarily large perturbations of a prescribed pure hydrodynamical flow of an electrically conducting fluid. We study whether the flow perturbations as well as the generated magnetic fields decay or grow with time and constitute a dynamo process. For that purpose we derive a generalized Reynolds-Orr equation for the sum of the kinetic energy of the hydrodynamic perturbation and the magnetic energy. The flow is confined in a finite volume so the normal component of the velocity at the boundary is zero. The tangential component is left arbitrary in contrast with previous works. For the magnetic field we mostly employ the classical boundary conditions where the field extends in the whole space. We establish critical values of hydrodynamic and magnetic Reynolds numbers below which arbitrarily large initial perturbations of the hydrodynamic flow decay. This involves generalization of the Rayleigh-Faber-Krahn inequality for the smallest eigenvalue of an elliptic operator. For high Reynolds number turbulence we provide an estimate of critical magnetic Reynolds number below which arbitrarily large fluctuations of the magnetic field decay.

physics.flu-dyn↗

arXiv2021

Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles

The Navier-Stokes equations generate an infinite set of generalized Lyapunov exponents defined by different ways of measuring the distance between exponentially diverging perturbed and unperturbed solutions. This set is demonstrated to be similar, yet different, from the generalized Lyapunov exponent that provides moments of distance between two fluid particles below the Kolmogorov scale. We derive rigorous upper bounds on dimensionless Lyapunov exponent of the fluid particles that demonstrate the exponent's decay with Reynolds number $Re$ in accord with previous studies. In contrast, terms of cumulant series for exponents of the moments have power-law growth with $Re$. We demonstrate as an application that the growth of small fluctuations of magnetic field in ideal conducting turbulence is hyper-intermittent, being exponential in both time and Reynolds number. We resolve the existing contradiction between the theory, that predicts slow decrease of dimensionless Lyapunov exponent of turbulence with $Re$, and observations exhibiting quite fast growth. We demonstrate that it is highly plausible that a pointwise limit for the growth of small perturbations of the Navier-Stokes equations exists.

physics.flu-dyn↗

arXiv2020

Efficient mate finding in planktonic copepods swimming in turbulence

Zooplankton live in dynamic environments where turbulence may challenge their limited swimming abilities. How this interferes with fundamental behavioral processes remains elusive. We reconstruct simultaneously the trajectories of flow tracers and calanoid copepods and we quantify their ability to find mates when ambient flow imposes physical constrains on their motion and impairs their olfactory orientation. We show that copepods achieve high encounter rates in turbulence due to the contribution of advection and vigorous swimming. Males further convert encounters within the perception radius to contacts and then to mating via directed motion toward nearby organisms within the short time frame of the encounter. Inertial effects do not result in preferential concentration, reducing the geometric collision kernel to the clearance rate, which we model accurately by superposing turbulent velocity and organism motion. This behavioral and physical coupling mechanism may account for the ability of copepods to reproduce in turbulent environments.

physics.flu-dyn↗

arXiv2019

The effect of a wall on the interaction of two spheres in shear flow: Batchelor-Green theory revisited

The seminal Batchelor-Green's (BG) theory on the hydrodynamic interaction of two spherical particles of radii a suspended in a viscous shear flow neglects the effect of the boundaries. In the present paper we study how a plane wall modifies this interaction. Using an integral equation for the surface traction we derive the expression for the particles' relative velocity as a sum of the BG's velocity and the term due to the presence of a wall at finite distance, z_0. Our calculation is not the perturbation theory of the BG solution, so the contribution due to the wall is not necessarily small. The distance at which the wall significantly alters the particles interaction scales as z_0^{3/5}. The phase portrait of the particles' relative motion is different from the BG theory, where there are two singly-connected regions of open and closed trajectories both of infinite volume. For finite z_0, there is a new domain of closed (dancing) and open (swapping) trajectories. The width of this region behaves as 1/z_0. Along the swapping trajectories, that have been previously observed numerically, the incoming particle is turning back after the encounter with the reference particle, rather than passing it by, as in the BG theory. The region of dancing trajectories has infinite volume. We found a one-parameter family of equilibrium states, overlooked previously, whereas the pair of spheres flows as a whole without changing its configuration. These states are marginally stable and their perturbation yields a two-parameter family of the dancing trajectories, where the particle is orbiting around a fixed point in a frame co-moving with the reference particle. We suggest that the phase portrait obtained at z_0>>a is topologically stable and can be extended down to rather small z_0 of several particle diameters. We confirm this by direct numerical simulations of the Navier-Stokes equations with z_0=5a.

physics.flu-dyn↗

arXiv2019

Refined central limit theorem and infinite density tail of the Lorentz gas from Levy walk

We consider point particle that collides with a periodic array of hard-core elastic scatterers where the length of the free flights is unbounded (the infinite-horizon Lorentz gas, LG). The Bleher central limit theorem (CLT) states that the distribution of the particle displacement divided by $\sqrt{t\ln t}$ is Gaussian in the limit of infinite time $t$. However it was stressed recently that the slow convergence makes this result unobservable. Using a Lévy walk model (LW) of the LG, it was proposed that the use of a rescaled Lambert function instead of $\sqrt{t\ln t}$ provides a fast convergent, observable CLT, which was confirmed by the LG simulations. We demonstrate here that this result can simplified to a mixed CLT where the scaling factor combines normal and anomalous diffusions. For narrow infinite corridors the particle for long time obeys the usual normal diffusion, which explains the previous numerical observations. In the opposite limit of small scatterers the Bleher CLT gives a good guiding. In the intermediate cases the mixed CLT applies. The Gaussian peak determines moments of order smaller than two. In contrast, the CLT gives only half the coordinate dispersion. The missing half of the dispersion and also moments of order higher than two are described by the distribution's tail (the infinite density) which we derive here. The tail is supported along the infinite corridors and formed by anomalously long flights whose duration is comparable with the whole time of observation. The moments' calculation from the tail is confirmed by direct calculation of the fourth moment from the statistics of the backward recurrence time defined as time that elapsed since the last collision. This completes the solution of the LW model allowing full comparison with the LG.

cond-mat.stat-mech↗

arXiv2019

Fluctuations of separation of trajectories in chaos and correlation dimension

We consider the cumulant generating function of the logarithm of the distance between two infinitesimally close trajectories of a chaotic system. Its long-time behavior is given by the generalized Lyapunov exponent $γ(k)$ providing the logarithmic growth rate of the $k-$th moment of the distance. The Legendre transform of $γ(k)$ is a large deviations function that gives the probability of rare fluctuations where the logarithmic rate of change of the distance is much larger or much smaller than the mean rate defining the first Lyapunov exponent. The only non-trivial zero of $γ(k)$ is at minus the correlation dimension of the attractor which for incompressible flows reduces to the space dimension. We describe here general properties constraining the form of $γ(k)$ and the Gallavotti-Cohen type relations that hold when there is symmetry under time-reversal. This demands studying joint growth rates of infinitesimal distances and volumes. We demonstrate that quartic polynomial approximation for $γ(k)$ does not violate the Marcinkiewicz theorem on invalidity of polynomial form for the generating function. We propose that this quartic approximation will fit many experimental situations, not having the effective time-reversibility and the short correlation time properties of the quadratic Grassberger-Procaccia estimates. We take the existing $γ(k)$ for turbulent channel flow and demonstrate that the quartic fit is nearly perfect. The violation of time-reversibility for the Lagrangian trajectories of the incompressible Navier-Stokes turbulence below the viscous scale is considered. We demonstrate how the fit can be used for finding the correlation dimensions of strange attractors via easily measurable quantities. We provide a simple formula via the Lyapunov exponents, holding in quadratic approximation, and describe the construction of the quartic approximation.

cond-mat.stat-mech↗

arXiv2019

Role of the sonic scale in the growth of magnetic field in compressible turbulence

We study the growth of small fluctuations of magnetic field in supersonic turbulence, the small-scale dynamo. The growth is due to the fastest turbulent eddies above the resistive scale. We observe that for supersonic turbulence these eddies are effectively incompressible which creates a robust structure of the growth. The eddies are localised below the sonic scale $l_s$ defined as the scale where the typical velocity of the turbulent eddies equals the speed of sound. Thus the flow below $l_s$ is effectively incompressible and the field growth proceeds as in incompressible flow. At large Mach numbers $l_s$ is much smaller than the integral scale of turbulence so the fastest growing mode of the magnetic field belongs to small-scale turbulence. We derive this mode and the associated growth rate numerically in a white noise in time model of turbulence. The relevance of this model relies on considering evolution time larger than the correlation time of turbulence.

physics.flu-dyn↗

arXiv2019

Density and tracer statistics in compressible turbulence: phase transition to multifractality

We study the statistics of fluid (gas) density and concentration of passive tracer particles (dust) in compressible turbulence. We raise the question of whether the fluid density which is an active field that reacts back on the transporting flow and the passive concentration of tracers must coincide in the steady state, which we demonstrate to be crucial both theoretically and experimentally. The fields' coincidence is provable at small Mach numbers, however at finite Mach numbers the assumption of mixing is needed, not evident due to the possibility of self-organization. Irrespective of whether the fields coincide we obtain a number of rigorous conclusions on both fields. As Ma increases the fields in the inertial range go through a phase transition from a finite continuous smooth to a singular multifractal distribution. We propose a way to calculate fractal dimensions from numerical or experimental data. We derive a simple expression for the spectrum of fractal dimensions of isothermal turbulence and describe limitations of lognormality. The expression depends on a single parameter: the scaling exponent of the density spectrum. We propose a mechanism for the phase transition of concentration to multifractality. We demonstrate that the pair-correlation function is invariant under the action of the probability density function of the inter-pair distance that has the Markov property implying applicability of the Kraichnan turbulence model. We use the model to derive an explicit expression for the tracers pair correlation that demonstrates their smooth transition to multifractality and confirms the transition's mechanism. Our results are of potentially important implications on astrophysical problems such as star formation as well as on technological applications such as supersonic combustion. As an example we demonstrate strong increase of planetesimals formation rate at the transition.

physics.flu-dyn↗

arXiv2019

Inertial self-propulsion of spherical microswimmers by rotation-translation coupling

We study swimming of small spherical particles who regulate fluid flow on their surface by applying tangential squirming strokes. We derive translational and rotational velocities for any given stroke which is not restricted by axial symmetry as assumed usually. The formulation includes inertia of both the fluid and the swimmer, motivated by inertia's relevance for large Volvox colonies. We show that inertial contribution to mean speed comes from dynamic coupling between translation and rotation, which occurs only for strokes that break axial symmetry. Remarkably, this effect enables overcoming the scallop theorem on impossibility of propulsion by time-reversible stroke. We study examples of tangential strokes of axisymmetric travelling wave, and of asymmetric time-reversible flapping. In the latter case, we find that inertia-driven mean speed is optimized for flapping frequency and swimmer's size which fall well within the range of realistic physical values for Volvox colonies. We conjecture that similarly to Paramecium, large Volvox could use time-reversible strokes for inertia-driven swimming coupled with their rotations.

physics.flu-dyn↗

arXiv2017

Integral representation of channel flow with interacting particles

We construct a boundary integral representation for the low-Reynolds-number flow in a channel in the presence of freely-suspended particles (or droplets) of arbitrary size and shape. We demonstrate that lubrication theory holds away from the particles at horizontal distances exceeding the channel height and derive a multipole expansion of the flow which is dipolar to the leading approximation. We show that the dipole moment of an arbitrary particle is a weighted integral of the stress and the flow at the particle surface, which can be determined numerically. We introduce the equation of motion that describes hydrodynamic interactions between arbitrary, possibly different, distant particles, with interactions determined by the product of the mobility matrix and the dipole moment. Further, the problem of three identical interacting spheres initially aligned in the streamwise direction is considered and the experimentally observed "pair exchange" phenomenon is derived analytically and confirmed numerically. For non-aligned particles, we demonstrate the formation of a configuration with one particle separating from a stable pair. Our results suggest that in a dilute initially homogeneous particulate suspension flowing in a channel the particles will eventually separate into singlets and pairs.

physics.flu-dyn↗