Finite-time collapse and soliton-like states in the dynamics of dissipative gases

A study of the gas dynamics of a dilute collection of the inelastically colliding hard spheres is presented. When diffusive processes are neglected the gas density blows up in a finite time. The blowup is the mathematical expression for one of the possible mechanisms for cluster formation in dissipative gases. The way diffusive processes smoothen the singularity has been studied. Exact localized soliton-type solutions of the gas dynamics when heat diffusion balances non-linear cooling are obtained. The presented results generalize previous findings for planar flows.Comment: 4 pages, 1 figur

The impact of hydrodynamic interactions on the preferential concentration of inertial particles in turbulence

We consider a dilute gas of inertial particles transported by the turbulent flow. Due to inertia the particles concentrate preferentially outside vortices. The pair-correlation function of the particles' concentration is known to obey at small separations a power-law with a negative exponent, if the hydrodynamic interactions between the particles are neglected. The divergence at zero separation is the signature of the random attractor asymptoted by the particles' trajectories at large times. However the hydrodynamic interactions produce a repulsion between the particles that is non-negligible at small separations. We introduce equations governing the repulsion and show it smoothens the singular attractor near the particles where the pair correlation function saturates. The effect is most essential at the Stokes number of order one, where the correlations decrease by a factor of a few.Comment: 4 pages, 1 figur

Convective stability of turbulent Boussinesq flow in the dissipative range and flow around small particles

We consider arbitrary, possibly turbulent, Boussinesq flow which is smooth below a dissipative scale $l_d$. It is demonstrated that the stability of the flow with respect to growth of fluctuations with scale smaller than $l_d$ leads to a non-trivial constraint. That involves the dimensionless strength $Fl$ of fluctuations of the gradients of the scalar in the direction of gravity and the Rayleigh scale $L$ depending on the Rayleigh number $Ra$, the Nusselt number $Nu$ and $l_d$. The constraint implies that the stratified fluid at rest, which is linearly stable, develops instability in the limit of large $Ra$. This limits observability of solution for the flow around small swimmer in quiescent stratified fluid that has closed streamlines at scale $L$ [A. M. Ardekani and R. Stocker, Phys. Rev. Lett. 105, 084502 (2010)]. Correspondingly to study the flow at scale $L$ one has to take turbulence into account. We demonstrate that the resulting turbulent flow around small particles or swimmers can be described by scalar integro-differential advection-diffusion equation. Describing the solutions we show that closed streamlines persist with finite probability. Our results seem to be the necessary basis in understanding flows around small swimmers.Comment: 15 pages, 1 figure. arXiv admin note: text overlap with arXiv:1301.635

Solvable continuous time random walk model of the motion of tracer particles through porous media

We consider the continuous time random walk model (CTRW) of tracer's motion in porous medium flows based on the experimentally determined distributions of pore velocity and pore size reported in Holzner et al. Phys. Rev. E 92, 013015 (2015). The particle's passing through one channel is modelled as one step of the walk. The step's (channel) length is random and the walker's velocity at consecutive steps of the walk is conserved with finite probability mimicking that at the turning point there could be no abrupt change of velocity. We provide the Laplace transform of the characteristic function of the walker's position and reductions for different cases of independence of the CTRW's step's duration \tau, length l and velocity v. We solve our model with independent l and v. The model incorporates different forms of the tail of the probability density of small velocities that vary with the model parameter \alpha. Depending on that parameter all types of anomalous diffusion can hold, from super- to subdiffusion. In a finite interval of \alpha, ballistic behavior with logarithmic corrections holds that was observed in a previously introduced CTRW model with independent l and \tau. Universality of tracer's diffusion in the porous medium is considered.Comment: 15 page

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.Comment: Final version. Accepted to Physical Review Fluids on January 201

Impact of turbulence on the stratified flow around small particles

We study the turbulent flow of the density-stratified fluid around a small translating (either passively or self-propelled) particle. It was found recently [A. M. Ardekani and R. Stocker, Phys. Rev. Lett. vol. 105, 084502 (2010)] that without turbulence, the familiar Stokes flow is dramatically altered by the stratification. Stratification-induced inhomogeneity "turns on" the buoyancy introducing a new "cutoff" or "screening" length scale for the flow, yielding closed streamlines and a faster (exponential-like) decay of velocity. This result, however, did not account for the potential role of the background turbulence, intrinsically present in many aquatic environments. Turbulence mixes the density opposing the effect. Here we derive and solve the advection-diffusion equation that describes the interplay of turbulent mixing, diffusion of the stratifying agent and buoyancy. We derive an exact expression for fluctuations due to weak background turbulence and show that stronger turbulence can completely change the flow around the particle, canceling the effect of stratification and restoring the unstratified Stokes flow.Comment: the problem was re-visited following the critique of the anonymous Referee to take into account the effect of turbulence on background density stratificatio

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.Comment: 41 pages; revised versio

Distribution of Brownian particles in turbulence

We consider Brownian particles immersed in the fluid which flow is turbulent. We study the limit where the particles' inertia is weak and their velocity relaxes fast to the velocity of the flow. The trajectories of the particles in this case have a strange attractor in the physical space, if the particles' diffusion is neglected. Under the latter condition the singular density of the particles was recently described completely. The analysis was done for real turbulence and did not involve the flow modeling. Here we take the diffusion into account showing how it modifies the statistics. The analysis is performed also for real turbulence. Experimentally testable predictions are made.Comment: 4 page

Construction and description of the stationary measure of weakly dissipative dynamical systems

We consider the stationary measure of the dissipative dynamical system in a finite volume. A finite dissipation, however small, generally makes the measure singular, while at zero dissipation the measure is constant. Thus dissipative part of the dynamics is a singular perturbation producing an infinite change in the measure. This is a result of the infinite time of evolution that enhances the small effects of dissipation to form singularities. We show how to deal with the singularity of the perturbation and describe the statistics of the measure. We derive all the correlation functions and the statistics of "mass" contained in a small ball. The spectrum of dimensions of the attractor is obtained. The fractal dimension is equal to the space dimension, while the information dimension is equal to the Kaplan-Yorke dimension.Comment: 9 page

Turbulence - "motion of multitude": a multi-agent spin model for complex flows

We propose a new paradigm for emergence of macroscopic flows. The latter are considered as a collective phenomenon created by many agents that exchange abstract information. The information exchange causes agents to change their relative positions which results in a flow. This paradigm, aimed at the study of the nature of turbulence, appeals to the original meaning of the word: "turbulence" siginifies "disordered motions of crowds". We give a preliminary discussion on a model of multi-agent dynamics that realizes the paradigm. This dynamics is reminiscent of spin glasses and neural networks. The model is relational, i. e. it assumes no spatio-temporal background and may serve as a basis for an approach to quantum gravity via spin models.Comment: 4 page