Quartz Crystal Microbalance Frequency Response to Discrete Adsorbates in Liquids.

Quartz crystal microbalance with dissipation monitoring (QCM-D) has become a major tool enabling accurate investigation of the adsorption kinetics of nanometric objects such as DNA fragments, polypeptides, proteins, viruses, liposomes, polymer, and metal nanoparticles. However, in liquids, a quantitative analysis of the experimental results is often intricate because of the complex interplay of hydrodynamic and adhesion forces varying with the physicochemical properties of adsorbates and functionalized QCM-D sensors. In the present paper, we dissect the role of hydrodynamics for the analytically tractable case of stiff contact, whereas the adsorbed rigid particles oscillate with the resonator without rotation. Under the assumption of the low surface coverage, we theoretically study the excess shear force exerted on the resonator, which has two contributions: (i) the fluid-mediated force due to flow disturbance created by the particle and (ii) the force exerted on the particle by the fluid and transmitted to the sensor via contact. The theoretical analysis enables an accurate interpretation of the QCM-D impedance measurements. It is demonstrated inter alia that for particles of the size comparable with protein molecules, the hydrodynamic force dominates over the inertial force and that the apparent mass derived from QCM independently of the overtone is about 10 times the Sauerbrey (inertial) mass. The theoretical results show excellent agreement with the results of experiments and advanced numerical simulations for a wide range of particle sizes and oscillation frequencies.

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. Here, we develop a quantitative criterion for sling events based on the velocity gradient history along particle paths. We show by a 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 controlled by the minimal real 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 for typical cloud droplets, showing that sling events at high Reynolds numbers are enhanced by an order of magnitude for small Stokes numbers. Thus, intermittency could be a significant ingredient in the collisional growth of rain droplets.

Fluid flow induced by helical microswimmers in bulk and near walls.

Magnetic nano- and microswimmers provide a powerful platform to study driven colloidal systems in fluidic media and are relevant to futuristic medical technologies requiring precise yet minimally invasive motion control at small scales. Upon the action of a rotating magnetic field, the helical microswimmers rotate and translate, generating flow in the surrounding fluid. In this paper, we study the fluid flow induced by the rotating helices using a combination of experiments, numerical simulations, and theory. The microhelices are actuated either in a fluid bulk or in proximity to the bottom wall using typical microfluidic device setup. We conclude that the mean hydrodynamic flow due to the helix actuation can be closely approximated by a system of rotlets line distributed along the helical axis (i.e., representing the flow due to rotating cylinder) which gets modified close to a wall through appropriate contributions from image multipoles. As the mean flow can be obtained in closed form, this study can be further applied towards modeling of the dynamics in a swarm of driven microswimmers interacting hydrodynamically near a bounding surface.

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.

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 hyperintermittent, 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.

Planetary, inertia-gravity and Kelvin waves on the f-plane and ß -plane in the presence of a uniform zonal flow.

A linear wave theory of the Rotating Shallow-Water Equations (RSWE) is developed in a channel on the midlatitude f-plane or ß -plane in the presence of a uniform mean zonal flow that is balanced geostrophically by a meridional gradient of the fluid surface height. Here we show that this surface height gradient is a potential vorticity (PV) source that generates Rossby waves even on the f-plane similar to the generation of these waves by PV sources such as the ß -effect, shear of the mean flow and bottom topography. Numerical solutions of the RSWE show that the resulting Rossby, Poincaré and "Kelvin-like" waves differ from their counterparts without mean flow in both their phase speeds and meridional structures. Doppler shifting of the "no mean-flow" phase speeds does not account for the difference in phase speeds, and the meridional structure is often trapped near one of the channel's boundaries and does not oscillate across the channel. A comparison between the phase speeds of Rossby waves of the present theory and those of the Quasi-Geostrophic Shallow-Water (QG-SW) theory shows that the former can be 2.5 times faster than those of the QG-SW theory. The phase speed of "Kelvin-like" waves is modified by the presence of a mean flow compared to the classical gravity wave speed, and furthermore their meridional velocity does not vanish. The gaps between the dispersion curves of adjacent Poincaré modes are not uniform but change with the zonal wave number, and the convexity of the dispersion curves also changes with the zonal wave number. These results have implications for the propagation of Rossby wave packets: QG theory overestimates the zonal group velocity.

Occupational structure of bearers of Jewish rabbinical, occupational and generic surnames.

We study choice of profession in three groups of Russian-speaking Jewish families with different occupational distributions of the ancestors. This study continues exploration of the persistence of social status of families over centuries that was initiated in recent years. It was found previously that in some cases professions remain associated with the same surnames for many generations. Here the studied groups are defined by a class of the surname of individuals composing them. The class serves as a label that indicates a professional bias of the ancestors of the individual. One group are the bearers of the class of surnames which were used by rabbinical dynasties. The other group is constituted by occupational surnames, mostly connected to crafts. Finally, the last group are generic Jewish names defined as surnames belonging to neither of the above groups. We use the database that consists of 858 and 1057 of the first two groups, respectively, and 7471 generic Jewish surnames. The statistics of the database are those of individuals drawn at random from the considered groups. We determine shares of members of the groups working in a given type of occupations together with the confidence interval. The occupational type's definition agrees with International Standard Classification of Occupations. It is demonstrated that there is a statistically significant difference in the occupational structure of the three groups that holds beyond the uncertainty allowed by 95% confidence interval. We quantify the difference with a numerical measure of the overlap of professional preferences of different groups. We conclude that in our study the occupational bias of different population groups is preserved at least for two centuries that passed since the considered surnames appeared.

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.

Fluid-Mediated Force on a Particle Due to an Oscillating Plate and Its Effect on Deposition Measurements by a Quartz Crystal Microbalance.

Interaction of particles with boundaries is a fundamental problem in many fields of physics. In this Letter, we theoretically examine the fluid-mediated interaction between a horizontally oscillating plate and a spherical particle, revealing emergence of the novel nonlinear vertical force exerted on the particle. Although we demonstrate that the phenomenon only slightly alters deposition of colloidal (sub-)µm-sized particles measured by quartz crystal microbalance, it can result in levitation of larger particles above the plate, considerably hindering their deposition.

Large deviations, singularity, and lognormality of energy dissipation in turbulence.

We study implications of the assumption of power-law dependence of moments of energy dissipation in turbulence on the Reynolds number Re, holding due to intermittency. We demonstrate that at Re→∞ the dissipation's logarithm divided by lnRe converges with probability one to a negative constant. This implies that the dissipation is singular in the limit, as is known phenomenologically. The proof uses a large deviations function, whose existence is implied by the power-law assumption, and which provides the general asymptotic form of the dissipation's distribution. A similar function exists for vorticity and for velocity differences where it proves the moments representation of the multifractal model (MF). Then we observe that derivative of the scaling exponents of the dissipation, considered as a function of the order of the moment, is small at the origin. Thus the variation with the order is slow and can be described by a quadratic function. Indeed, the quadratic function, which corresponds to log-normal statistics, fits the data. Moreover, combining the lognormal scaling with the MF we derive a formula for the anomalous scaling exponents of turbulence which also fits the data. Thus lognormality, not to be confused with the Kolmogorov (1962) assumption of lognormal dissipation in the inertial range, when used in conjunction with the MF provides a concise way to get all scaling exponents of turbulence available at present.

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. As Ma increases from small or moderate values, the density and the concentration in the inertial range go through a phase transition from a finite continuous smooth distribution to a singular multifractal spatial distribution. Multifractality is associated with scaling, which would not hold if the solenoidal and the potential components of the flow scaled differently, producing transport which is not self-similar. Thus, we propose that the transition occurs when the difference of the scaling exponents of the components, decreasing with Ma, becomes small. Under the smallness assumption, the particles' volumes obey a power-law evolution. That, by the use of conservation of the total volume of the flow, entails the volumes' shrinking to zero with probability 1 and formation of a singular distribution. We discuss various concepts of multifractality and 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 demonstrate that the pair-correlation function of the tracer concentration has the Markov property. This implies applicability of the compressible version of the Kraichnan turbulence model. We use the model to derive an explicit expression for the tracer pair correlation that demonstrates their smooth transition to multifractality and confirms the transition's mechanism. The obtained fractal dimension explains previous numerical observations. Our results have potentially important implications for astrophysical problems such as star formation as well as technological applications such as supersonic combustion. As an example, we demonstrate the strong increase of planetesimal formation rate at the transition. We prove that finiteness of internal energy implies vanishing of the sum of Lyapunov exponents in the dissipation range. Our study leads to 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. This is demonstrated 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, which we demonstrate to be not self-evident because of the possibility of self-organization.

Dispersion of particles in an infinite-horizon Lorentz gas.

We consider a two-dimensional Lorentz gas with infinite horizon. This paradigmatic model consists of pointlike particles undergoing elastic collisions with fixed scatterers arranged on a periodic lattice. It was rigorously shown that when t→∞, the distribution of particles is Gaussian. However, the convergence to this limit is ultraslow, hence it is practically unattainable. Here, we obtain an analytical solution for the Lorentz gas' kinetics on physically relevant timescales, and find that the density in its far tails decays as a universal power law of exponent -3. We also show that the arrangement of scatterers is imprinted in the shape of the distribution.

The mixed Rossby-gravity wave on the spherical Earth.

This work revisits the theory of the mixed Rossby-gravity (MRG) wave on a sphere. Three analytic methods are employed in this study: (a) derivation of a simple ad hoc solution corresponding to the MRG wave that reproduces the solutions of Longuet-Higgins and Matsuno in the limits of zero and infinite Lamb's parameter, respectively, while remaining accurate for moderate values of Lamb's parameter, (b) demonstration that westward-propagating waves with phase speed equalling the negative of the gravity-wave speed exist, unlike the equatorial ß-plane, where the zonal velocity associated with such waves is infinite, and (c) approximation of the governing second-order system by Schrödinger eigenvalue equations, which show that the MRG wave corresponds to the branch of the ground-state solutions that connects Rossby waves with zonally symmetric waves. The analytic conclusions are confirmed by comparing them with numerical solutions of the associated second-order equation for zonally propagating waves of the shallow-water equations. We find that the asymptotic solutions obtained by Longuet-Higgins in the limit of infinite Lamb's parameter are not suitable for describing the MRG wave even when Lamb's parameter equals 104. On the other hand, the dispersion relation obtained by Matsuno for the MRG wave on the equatorial ß-plane is accurate for values of Lamb's parameter as small as 16, even though the equatorial ß-plane formally provides an asymptotic limit of the equations on the sphere only in the limit of infinite Lamb's parameter.

Zooplankton can actively adjust their motility to turbulent flow.

Calanoid copepods are among the most abundant metazoans in the ocean and constitute a vital trophic link within marine food webs. They possess relatively narrow swimming capabilities, yet are capable of significant self-locomotion under strong hydrodynamic conditions. Here we provide evidence for an active adaptation that allows these small organisms to adjust their motility in response to background flow. We track simultaneously and in three dimensions the motion of flow tracers and planktonic copepods swimming freely at several intensities of quasi-homogeneous, isotropic turbulence. We show that copepods synchronize the frequency of their relocation jumps with the frequency of small-scale turbulence by performing frequent relocation jumps of low amplitude that seem unrelated to localized hydrodynamic signals. We develop a model of plankton motion in turbulence that shows excellent quantitative agreement with our measurements when turbulence is significant. We find that, compared with passive tracers, active motion enhances the diffusion of organisms at low turbulence intensity whereas it dampens diffusion at higher turbulence levels. The existence of frequent jumps in a motion that is otherwise dominated by turbulent transport allows for the possibility of active locomotion and hence to transition from being passively advected to being capable of controlling diffusion. This behavioral response provides zooplankton with the capability to retain the benefits of self-locomotion despite turbulence advection and may help these organisms to actively control their distribution in dynamic environments. Our study reveals an active adaptation that carries strong fitness advantages and provides a realistic model of plankton motion in turbulence.

Classification of eastward propagating waves on the spherical Earth.

Observational evidence for an equatorial non-dispersive mode propagating at the speed of gravity waves is strong, and while the structure and dispersion relation of such a mode can be accurately described by a wave theory on the equatorial ß-plane, prior theories on the sphere were unable to find such a mode except for particular asymptotic limits of gravity wave phase speeds and/or certain zonal wave numbers. Here, an ad hoc solution of the linearized rotating shallow-water equations (LRSWE) on a sphere is developed, which propagates eastward with phase speed that nearly equals the speed of gravity waves at all zonal wave numbers. The physical interpretation of this mode in the context of other modes that solve the LRSWE is clarified through numerical calculations and through eigenvalue analysis of a Schrödinger eigenvalue equation that approximates the LRSWE. By comparing the meridional amplitude structure and phase speed of the ad hoc mode with those of the lowest gravity mode on a non-rotating sphere we show that at large zonal wave number the former is a rotation-modified counterpart of the latter. We also find that the dispersion relation of the ad hoc mode is identical to the n = 0 eastward propagating inertia-gravity (EIG0) wave on a rotating sphere which is also nearly non-dispersive, so this solution could be classified as both a Kelvin wave and as the EIG0 wave. This is in contrast to Cartesian coordinates where Kelvin waves are a distinct wave solution that supplements the EIG0 mode. Furthermore, the eigenvalue equation for the meridional velocity on the ß-plane can be formally derived as an asymptotic limit (for small (Lamb Number)-1/4) of the corresponding second order equation on a sphere, but this expansion is invalid when the phase speed equals that of gravity waves i.e. for Kelvin waves. Various expressions found in the literature for both Kelvin waves and inertia-gravity waves and which are valid only in certain asymptotic limits (e.g. slow and fast rotation) are compared with the expressions found here for the two wave types.

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 nonaligned particles, we demonstrate the formation of a configuration with one particle separating from a stable pair. Our results suggest that in a dilute initially homogenous particulate suspension flowing in a channel the particles will eventually separate into singlets and pairs.

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

We consider the continuous-time random walk (CTRW) model of tracer motion in porous medium flows based on the experimentally determined distributions of pore velocity and pore size reported by Holzner et al. [M. Holzner et al., Phys. Rev. E 92, 013015 (2015)PLEEE81539-375510.1103/PhysRevE.92.013015]. The particle's passing through one channel is modeled as one step of the walk. The step (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 duration τ, 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 α. Depending on that parameter, all types of anomalous diffusion can hold, from super- to subdiffusion. In a finite interval of α, ballistic behavior with logarithmic corrections holds, which was observed in a previously introduced CTRW model with independent l and τ. Universality of tracer diffusion in the porous medium is considered.

Clustering of particles in turbulence due to phoresis.

We demonstrate that diffusiophoretic, thermophoretic, and chemotactic phenomena in turbulence lead to clustering of particles on multifractal sets that can be described using one single framework, valid when the particle size is much smaller than the smallest length scale of turbulence l_{0}. To quantify the clustering, we derive positive pair correlations and fractal dimensions that hold for scales smaller than l_{0}. For scales larger than l_{0} the pair-correlation function is predicted to show a stretched exponential decay towards 1. In the case of inhomogeneous turbulence we find that the fractal dimension depends on the direction of inhomogeneity. By performing experiments with particles in a turbulent gravity current we demonstrate clustering induced by salinity gradients in conformity to the theory. The particle size in the experiment is comparable to l_{0}, outside the strict validity region of the theory, suggesting that the theoretical predictions transfer to this practically relevant regime. This clustering mechanism may provide the key to the understanding of a multitude of processes such as formation of marine snow in the ocean and population dynamics of chemotactic bacteria.

Inhomogeneous distribution of water droplets in cloud turbulence.

We consider sedimentation of small particles in the turbulent flow where fluid accelerations are much smaller than acceleration of gravity g. The particles are dragged by the flow by linear friction force. We demonstrate that the pair-correlation function of particles' concentration diverges with decreasing separation as a power law with negative exponent. This manifests fractal distribution of particles in space. We find that the exponent is proportional to ratio of integral of energy spectrum of turbulence times the wave number over g. The proportionality coefficient is a universal number independent of particle size. We derive the spectrum of Lyapunov exponents that describes the evolution of small patches of particles. It is demonstrated that particles separate dominantly in the horizontal plane. This provides a theory for the recently observed vertical columns formed by the particles. We confirm the predictions by direct numerical simulations of Navier-Stokes turbulence. The predictions include conditions that hold for water droplets in warm clouds thus providing a tool for the prediction of rain formation.

Phytoplankton's motion in turbulent ocean.

We study the influence of turbulence on upward motion of phytoplankton. Interaction with the flow is described by the Pedley-Kessler model considering spherical microorganisms. We find a range of parameters when the upward drift is only weakly perturbed or when turbulence completely randomizes the drift direction. When the perturbation is small, the drift is either determined by the local vorticity or is Gaussian. We find a range of parameters where the phytoplankton interaction with the flow can be described consistently as diffusion of orientation in effective potential. By solving the corresponding Fokker-Planck equation we find exponential steady-state distribution of phytoplankton's propulsion orientation. We further identify the range of parameters where phytoplankton's drift velocity with respect to the flow is determined uniquely by its position. In this case, one can describe phytoplankton's motion by a smooth flow and phytoplankton concentrates on fractal. We find fractal dimensions and demonstrate that phytoplankton forms vertical stripes in space with a nonisotropic pair-correlation function of concentration increased in the vertical direction. The probability density function of the distance between two particles obeys power law with the negative exponent given by the ratio of integrals of the turbulent energy spectrum. We find the regime of strong clustering where the exponent is of order one so that turbulence increases the rate of collisions by a large factor. The predictions hold for Navier-Stokes turbulence and stand for testing.