Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles

We construct the hydrodynamic equations for suspensions of self-propelled particles (SPPs) with spontaneous orientational order, and make a number of striking, testable predictions: (i) Nematic SPP suspensions are always absolutely unstable at long wavelengths. (ii) SPP suspensions support novel propagating modes at long wavelengths, coupling orientation, flow, and concentration. (iii) In a wave number regime accessible only in low Reynolds number systems such as bacteria, polar-ordered suspensions are invariably convectively unstable. (iv) The variance in the number N of particles, divided by the mean , diverges as (2/3 ) in polar-ordered SPP suspensions.     

Effect of acceleration by internal and external force fields on particle motion in intermediate regimes between the hydrodynamic and free-molecular limits

The kinetic theory of gases is applied to analyze slow translational motion of low-concentration particles driven by an external force in a homogeneous gas. The analysis takes into account the diffusion due to the difference in acceleration between particles and molecules in internal and external force fields. A general expression is derived for the particle drag force in hydrodynamic, free-molecular, and intermediate regimes. This expression reduces to a simple relation between the drag force and its values in the hydrodynamic and free-molecular limits and the force of intermolecular interaction between particles and gas molecules. In the case of spherically symmetric potential of interaction between the particle and molecules, the drag force is the harmonic mean of its limit values.     

Orientational order and instabilities in suspensions of self-locomoting rods

The orientational order and dynamics in suspensions of self-locomoting slender rods are investigated numerically. In agreement with previous theoretical predictions, nematic suspensions of swimming particles are found to be unstable at long wavelengths as a result of hydrodynamic fluctuations. Nevertheless, a local nematic ordering is shown to persist over short length scales and to have a significant impact on the mean swimming speed. The consequences of the large-scale orientational disorder for particle dispersion are also discussed.     

Jeans type analysis of chemotactic collapse

We perform a linear dynamical stability analysis of a general hydrodynamic model of chemotactic aggregation [P.H. Chavanis, C. Sire, Physica A 384 (2007) 199]. Specifically, we study the stability of an infinite and homogeneous distribution of cells against “chemotactic collapse”. We discuss the analogy between the chemotactic collapse of biological populations and the gravitational collapse (Jeans instability) of self-gravitating systems. Our hydrodynamic model involves a pressure force which can take into account several effects like anomalous diffusion or the fact that the organisms cannot interpenetrate. We also take into account the degradation of the chemical which leads to a shielding of the interaction like for a Yukawa potential. Finally, our hydrodynamic model involves a friction force which quantifies the importance of inertial effects. In the strong friction limit, we obtain a generalized Keller–Segel model similar to the generalized Smoluchowski–Poisson system describing self-gravitating Langevin particles. For small frictions, we obtain a hydrodynamic model of chemotaxis similar to the Euler–Poisson system describing a self-gravitating barotropic gas. We show that an infinite and homogeneous distribution of cells is unstable against chemotactic collapse when the “velocity of sound” in the medium is smaller than a critical value. We study in detail the linear development of the instability and determine the range of unstable wavelengths, the growth rate of unstable modes and the damping rate, or the pulsation frequency, of the stable modes as a function of the friction parameter and shielding length. For specific equations of state, we express the stability criterion in terms of cell density.     

An Immersed Boundary-Lattice Boltzmann Prediction for Particle Hydrodynamic Focusing in Annular Microchannels

We numerically study the dynamics of particle crystals in annular microchannels by the immersed-boundary(IB)lattice Boltzmann(LB) coupled model, analyze the fluid-particle interactions during the migration of particles,and reveal the underlying mechanism of a particle focusing on the presence of fluid flows. The results show that the Reynolds and Dean numbers are key factors influencing the hydrodynamics of particles. The particles migrate onto their equilibrium tracks by adjusting the Reynolds and Dean numbers. Elliptical tracks of particles during hydrodynamic focusing can be predicted by the IB-LB model. Both the small Dean number and the small particle can lead to a small size of the focusing track. This work would possibly facilitate the utilization of annular microchannel flows to obtain microfluidic flowing crystals for advanced applications in biomedicine and materials synthesis.     

Kinematics of the forced and overdamped sine-Gordon soliton gas

Motion of a driven and heavily damped sine-Gordon chain with a low density of kinks and tight coupling between particles is controlled by the nucleation and subsequent annihilation of pairs of kinks and antikinks. We show that in the steady state there are no spatial correlations between kinks or between kinks and antikinks. For a given number of kinks and antikinks all geometrical distributions are equally alike, as in equilibrium. A master equation for the probability distribution for the number of kinks on a finite chain is solved, and substantiates the physical reasoning in previous work. The probability distribution characterizing the spread along the direction of particle motion of a finite chain in equilibrium as well as in the driven overdamped case is derived by simple combinatorial considerations. The spatial spread of a driven chain in the thermodynamic limit does not approach a steady state; a given particle followed in time deviates as t^1/2 from its average forced motion. This result follows from the hydrodynamic equations for the dilute kink gas. Comparison is made with other recent results.     

Dimension reduction method for ODE fluid models

We develop a new dimension reduction method for large size systems of ordinary differential equations (ODEs) obtained from a discretization of partial differential equations of viscous single and multiphase fluid flow. The method is also applicable to other large-size classical particle systems with negligibly small variations of particle concentration. We propose a new computational closure for mesoscale balance equations based on numerical iterative deconvolution. To illustrate the computational advantages of the proposed reduction method, we use it to solve a system of smoothed particle hydrodynamic ODEs describing single-phase and two-phase layered Poiseuille flows driven by uniform and periodic (in space) body forces. For the single-phase Poiseuille flow driven by the uniform force, the coarse solution was obtained with the zero-order deconvolution. For the single-phase flow driven by the periodic body force and for the two-phase flows, the higher-order (the first- and second-order) deconvolutions were necessary to obtain a sufficiently accurate solution.     

Collective motion of polar active particles on a sphere

Collective motion of active particles with polar alignment is investigated on a sphere. We discussed the factors that affect particle swarm motion and define an order parameter that can show the degree of particle swarm motion. In the model, we added a polar alignment strength, along with Gaussian curvature, affecting particles swarm motion. We find that when the force exceeds a certain limit, the order parameter will decrease with the increase of the force. Combined with our definition of order parameter and observation of the model, the reason is that particles begin to move side by side under the influence of polar forces. In addition, the effects of velocity, rotational diffusion coefficient, and packing fraction on particle swarm motion are discussed. It is found that the rotational diffusion coefficient and the packing fraction have a great influence on the clustering motion of particles, while the velocity has little influence on the clustering motion of particles.     

Hydrodynamic screening and axisymmetric turbulence in the transient sedimentation of a dilute suspension at small Reynolds number

A theory of settling of a dilute suspension of identical spherical particles in a viscous incompressible fluid is developed on the basis of the equations of transient Stokesian dynamics. The equations describe hydrodynamic interactions between particles moving under the influence of a constant force, starting to act at a particular instant of time. For a dilute suspension, a monopole approximation can be used. It is argued that the growth of velocity fluctuations is bounded by a combination of two effects, destructive interference of the flow patterns of individual particles, and a rearrangement of particle positions leading to a time-dependent microstructure of the suspension. After a long time, the microstructure tends to a steady state. The corresponding structure factor is described phenomenologically. The corresponding pair correlation function and the velocity correlation functions describing axisymmetric turbulence on the length scale of the mean distance between particles are evaluated.     

Stability criterions of an oscillating tip-cantilever system in dynamic force microscopy

This work is a theoretical investigation of the stability of the non-linear behavior of an oscillating tip-cantilever system used in dynamic force microscopy. Stability criterions are derived that may help to a better understanding of the instabilities that may appear in the dynamic modes, Tapping and NC-AFM, when the tip is close to a surface. A variational principle allows to get the temporal dependence of the equations of motion of the oscillator as a function of the non-linear coupling term. These equations are the basis for the analysis of the stability. One find that the branch associated to frequencies larger than the resonance is always stable whereas the branch associated to frequencies smaller than the resonance exhibits two stable domains and one unstable. This feature allows to re-interpret the instabilities appearing in Tapping mode and may help to understand the reason why the NC-AFM mode is stable. Received 12 April 2001     

Modeling and simulation of thermocapillary flows using lattice Boltzmann method

To understand how thermocapillary forces manipulate droplet motion in microfluidic channels, we develop a lattice Boltzmann (LB) multiphase model to simulate thermocapillary flows. The complex hydrodynamic interactions are described by an improved color-fluid LB model, in which the interfacial tension forces and the Marangoni stresses are modeled in a consistent manner using the concept of a continuum surface force. An additional convection–diffusion equation is solved in the LB framework to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. A stress-free boundary condition is also introduced to treat outflow boundary, which can conserve the total mass of an incompressible system, thus improving the numerical stability for creeping flows.The model is firstly validated against the analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of three-dimensional deformable droplet at various Marangoni numbers, and its accuracy is once again verified against the theoretical prediction in the limit of zero Marangoni number. Finally, we numerically investigate how the localized heating from a laser can block the microfluidic droplet motion through the induced thermocapillary forces. The droplet motion can be completely blocked provided that the intensity of laser exceeds a threshold value, below which the droplet motion successively undergoes four stages: constant velocity, deceleration, acceleration, and constant velocity. When the droplet motion is completely blocked, four steady vortices are clearly visible, and the droplet is fully filled by two internal vortices. The external vortices diminish when the intensity of laser increases.     

Entropic splitter for particle separation

We present a particle separation mechanism which induces the motion of particles of different sizes in opposite directions. The mechanism is based on the combined action of a driving force and an entropic rectification of the Brownian fluctuations caused by the asymmetric form of the channel along which particles proceed. The entropic splitting effect shown could be controlled upon variation of the geometrical parameters of the channel and could be implemented in narrow channels and microfluidic devices.     

Giant amplification of interfacially driven transport by hydrodynamic slip: diffusio-osmosis and beyond

We demonstrate that "moderate" departures from the no-slip hydrodynamic boundary condition (hydrodynamic slip lengths in the nanometer range) can result in a very large enhancement--up to 2 orders of magnitude--of most interfacially driven transport phenomena. We study analytically and numerically the case of neutral solute diffusio-osmosis in a slab geometry to account for nontrivial couplings between interfacial structure and hydrodynamic slip. Possible outcomes are fast transport of particles in externally applied or self-generated gradient, and flow enhancement in nano- or microfluidic geometries.     

Hydrodynamic interaction between two trapped swimming model micro-organisms

We present a theoretical study of the behaviour of two active particles under the action of harmonic traps kept at a fixed distance away from each other. We classify the steady configurations the squirmers develop as a function of their self-propelling velocity and the active stresses the swimmers induce around them. We have further analyzed the stability of such configurations, and have found that the ratio between their self-propelling velocity and the apolar flow generated through active stresses determines whether collinear parallel squirmers or perpendicularly swimming particles moving away from each other are stable. Therefore, there is a close connection between the stable configurations and the active mechanisms leading to the particle self-propulsion. The trap potential does not affect the stability of the configurations; it only modifies some of their relevant time scales. We have also observed the development of characteristic frequencies which should be observable. Finally, we show that the development of the hydrodynamic flows induced by the active particles may be relevant even when its time scale orders of magnitude smaller than the other present characteristic time scales and may destabilize the stable configurations.     

Stability of circular orbits of spinning particles in Schwarzschild-like space–times

Circular orbits of spinning test particles and their stability in Schwarzschild-like backgrounds are investigated. For these space–times the equations of motion admit solutions representing circular orbits with particles spins being constant and normal to the plane of orbits. For the de Sitter background the orbits are always stable with particle velocity and momentum being co-linear along them. The world-line deviation equations for particles of the same spin-to-mass ratios are solved and the resulting deviation vectors are used to study the stability of orbits. It is shown that the orbits are stable against radial perturbations. The general criterion for stability against normal perturbations is obtained. Explicit calculations are performed in the case of the Schwarzschild space–time leading to the conclusion that the orbits are stable.     

Pulsed-field separation of particles in a microfluidic device

We demonstrate the proof-of-principle of a new separation concept for micrometer-sized particles in a structured microfluidic device. Under the action of externally applied, periodic voltage-pulses two different species of like-charged polystyrene beads are observed to simultaneously migrate into opposite directions. Based on a theoretical model of the particle motion in the microdevice that shows good agreement with the experimental measurements, the underlying separation mechanism is identified and explained. Potential biophysical applications, such as cell sorting, are briefly addressed.     

Instability of polaritons driven by infrared laser light

In the present paper the nonlinear behavior of lattice vibrations driven by a coherent electromagnetic wave on the basis of Huang's equations is discussed. Under certain conditions formulated in the text these mixed electromagnetic lattice waves (polaritons) become unstable for some regions of the parameters of the incident laser light. Unstable motion of crystal lattices may be considered as a damage mechanism of the crystal structure.     

Axial segregation in a cylindrical centrifuge

We propose a theory for axial segregation of suspensions of non-neutrally buoyant particles in a rotating cylinder. The cylinder is oriented in the horizontal plane, so that any axial forces must arise from interparticle interactions. We show that the hydrodynamic interaction between pairs of particles produces a relative motion in the axial direction, independent of the gravitational force. If the particles are denser than the suspending fluid, differential centrifuging between particles at different radial positions leads to an at-tractive interaction, inducing a rapid growth of axial density perturbations. We suggest that this mecha-nism can explain the origin of band formation in rotating suspensions of non-neutrally buoyant particles.     

Orbital and epicyclic motion of charged test particles around non-rotating Einstein-Æther black holes

The study of charged test particle dynamics in the combined black hole gravitational field and magnetic field around it could provide important theoretical insight into astrophysical processes around such compact object. We have explored the orbital and epicyclic motion of charged test particles in the background of non-rotating Einstein-Æther black holes in the presence of external uniform magnetic field. We numerically integrate the equations of motion and analyze the trajectories of the charged test particles. We examined the stability of circular orbits using effective potential technique and study the characteristics of innermost stable circular orbits. We analyze the key features of quasi-harmonic oscillations of charged test particles nearby the stable circular orbits in an equatorial plane of the black hole, and investigate the radial profiles of the frequencies of latitudinal as well as radial harmonic oscillations in dependence on the strength of magnetic field, mass of the black hole and dimensionless coupling constants of the theory. We demonstrate that the magnetic field and dimensionless parameters of the theory have strong influence on charged particle motion around Einstein-Æther black holes.