We derive a minimal continuum model to investigate the hydrodynamic mechanism behind the fingering instability recently discovered in a suspension of microrollers near a floor [M. Driscoll et al., Nat. Phys. 13, 375 (2017)]. Our model, consisting of two continuous lines of rotlets, exhibits a linear instability driven only by hydrodynamic interactions and reproduces the length-scale selection observed in large-scale particle simulations and in experiments. By adjusting only one parameter, the distance between the two lines, our dispersion relation exhibits quantitative agreement with the simulations and qualitative agreement with experimental measurements. Our linear stability analysis indicates that this instability is caused by the combination of the advective and transverse flows generated by the microrollers near a no-slip surface. Our simple model offers an interesting formalism to characterize other hydrodynamic instabilities that have not been well understood, such as size scale selection in suspensions of particles sedimenting adjacent to a wall, or the recently observed formations of traveling phonons in systems of confined driven particles.

It has been known for a long time that fully developed turbulence is sustained in a precessing container. The aim of the present study is to reveal the sustaining mechanism of turbulence in a precessing sphere by means of laboratory experiments. We conduct experiments using a Newtonian fluid (water) and viscoelastic fluids (dilute solutions of surfactant, cetyltrimethyl ammonium chloride, and polymers, polyethylene oxide) to understand the sustaining mechanism of turbulence of Newtonian fluids by examining turbulence modifications due to the surfactant and polymer additives. When the Reynolds number based on the spin angular velocity and radius of the sphere is fixed, the most developed turbulence is sustained with the Poincaré number (the precession rate) being about 0.1. The key ingredient of the developed turbulence is a pair of large-scale vortex tubes which robustly exists in the flow. Assuming that these vortex tubes sustain small-scale turbulent eddies through an energy cascading process, we can explain all our experimental observations. Concerning the turbulence modification by the additives, the time-scale criteria by Lumley [J. Polymer Sci.: Macromol. Rev. 7, 263 (1973)] and the refined theory by Tabor and de Gennes [Europhys. Lett. 2, 519 (1986)] explain the experimental result that the pair of large-scale vortex tubes survives even when small-scale turbulent eddies are drastically suppressed by the surfactant additive.

We use boundary element simulations to study the interaction of model microswimmers with a neutrally buoyant spherical particle. The ratio of the size of the particle to that of the swimmer is varied from $R^{\mathrm{P}}/R^{\mathrm{S}}\ll 1$, corresponding to swimmer-tracer scattering, to $R^{\mathrm{P}}/R^{\mathrm{S}}\gg 1$, approximately equivalent to the swimmer interacting with a fixed, flat surface. We find that details of the swimmer and particle trajectories vary for different swimmers. However, the overall characteristics of the scattering event fall into two regimes, depending on the relative magnitudes of the impact parameter, $\rho$, and the collision radius, $R^{\mathrm{coll}}=R^{\mathrm{P}}+R^{\mathrm{S}}$. The range of particle motion, defined as the maximum distance between two points on the trajectory, has only a weak dependence on the impact parameter when $\rho <R^{\mathrm{coll}}$ and decreases with the radius of the particle. In contrast, when $\rho >R^{\mathrm{coll}}$ the range decreases as a power law in $\rho$ and is insensitive to the size of the particle. We also demonstrate that large particles can cause swimmers to be deflected through large angles. In some instances, this swimmer deflection can lead to larger net displacements of the particle. Based on these results, we estimate the effective diffusivity of a particle in a dilute bath of swimmers and show that there is a nonmonotonic dependence on particle radius. Similarly, we show that the effective diffusivity of a swimmer scattering in a suspension of particles varies nonmonotonically with particle radius.

The phenomenon of incomplete mixing reduces bulk effective reaction rates in reactive transport. Many existing models do not account for these effects, resulting in the overestimation of reaction rates in laboratory and field settings. To date, most studies on incomplete mixing have focused on diffusive systems; here, we extend these to explore the role that flow heterogeneity has on incomplete mixing. To do this, we examine reactive transport using a Lagrangian reactive particle tracking algorithm in two-dimensional idealized heterogeneous porous media. Contingent on the nondimensional Peclet and Damköhler numbers in the system, it was found that near well-mixed behavior could be observed at late times in the heterogeneous flow field simulations. We look at three common flow deformation metrics that describe the enhancement of mixing in the flow due to velocity gradients: the Okubo-Weiss parameter $\left(\theta \right)$, the largest eigenvalue of the Cauchy-Green strain tensor $\left({\lambda }_C\right)$, and the finite-time Lyapunov exponent $\left(\mathrm{\Lambda }\right)$. Strong mixing regions in the heterogeneous flow field identified by these metrics were found to correspond to regions with higher numbers of reactions, but the infrequency of these regions compared to the large numbers of reactions occurring elsewhere in the domain imply that these strong mixing regions are insufficient in explaining the observed near well-mixed behavior. Since it was found that reactive transport in these heterogeneous flows could overcome the effects of incomplete mixing, we also search for a closure for the mean concentration. The conservative quantity $\overline{u^2}$, where $u=C_A-C_B$, was found to predict the late time scaling of the mean concentration, i.e., $\overline{C_i}\sim \overline{u^2}$.

Oscillating bodies in stratified fluids may emit higher harmonics in addition to fundamental waves. In the present experimental study, we consider higher harmonics of an internal wave field generated by a horizontally oscillating spheroid in a linearly stratified fluid for moderate to high oscillation amplitudes, i.e., scaled oscillation amplitude $A/a\ge 0.5$, with $a$ the minor radius of the spheroid. Three different spheroid shapes are tested. The results are discussed in the context of the different theories on the generation of higher harmonics. Higher harmonics are observed at the intersections of fundamental wave beams, and at the critical points of the topography where the topographic slope equals the wave slope. The velocity amplitudes of the fundamental, second, and third harmonic waves grow respectively linearly, quadratically, and with the third power of the scaled oscillation amplitude $A/a$. Though these amplitudes are generally higher when the object's slope is larger, the increase in amplitude above and below the axisymmetric oscillating objects is found to be due to the effect of focusing. In order to discern the relative importance of the harmonics to the fundamental wave, the horizontal structure of the wave amplitude is measured. The results suggest that the $n\text{th}$ harmonic of the internal wave field is associated with a radiation diagram corresponding to a multipole of order $2^n$, with $2n$ directions of propagation.

Solutions of the forced Navier-Stokes equation have been conjectured to thermalize at scales larger than the forcing scale, similar to an absolute equilibrium obtained for the spectrally truncated Euler equation. Using direct numeric simulations of Taylor-Green flows and general-periodic helical flows, we present results on the probability density function, energy spectrum, autocorrelation function, and correlation time that compare the two systems. In the case of highly helical flows, we derive an analytic expression describing the correlation time for the absolute equilibrium of helical flows that is different from the $E^{-1/2}{k}^{-1}$ scaling law of weakly helical flows. This model predicts a new helicity-based scaling law for the correlation time as $\tau \left(k\right)\sim H^{-1/2}{k}^{-1/2}$. This scaling law is verified in simulations of the truncated Euler equation. In simulations of the Navier-Stokes equations the large-scale modes of forced Taylor-Green symmetric flows (with zero total helicity and large separation of scales) follow the same properties as absolute equilibrium including a $\tau \left(k\right)\sim E^{-1/2}{k}^{-1}$ scaling for the correlation time. General-periodic helical flows also show similarities between the two systems; however, the largest scales of the forced flows deviate from the absolute equilibrium solutions.

Analytical expressions are obtained for the change in speed of translation of the von Kármán point vortex streets of given aspect ratios due to the effects of weak compressibility in subsonic flow of an isentropic fluid. We also clarify the nature of the force-free condition on a weakly compressible point vortex in equilibrium. For staggered streets, it is found that the speed of a compressible point vortex street can both increase and decrease relative to its incompressible counterpart of the same aspect ratio. Compressibility increases the speeds of streets with aspect ratios less than the critical value of 0.38187, at which no change in speed occurs to first order in the (squared) Mach number. In particular, the compressible counterpart to the neutrally stable incompressible point vortex street of aspect ratio 0.28056 is found to propagate with increased speed. Streets with aspect ratios larger than 0.38187 slow down under the effects of compressibility, with the slowdown becoming maximal at a street aspect ratio of 0.52630. On the other hand, the speed of unstaggered streets always increases, with the first-order correction in speed relative to its incompressible value increasing maximally at aspect ratios around $\kappa =0.36216$.

The turbulent convection of low-Prandtl-number fluids ($\text{Pr}=0.0083$) in a long cylindrical cell, heated at one end face and cooled at the other, inclined to the vertical at angle $\beta , 0\le \beta \le \pi /2$ with step $\pi /20$, is studied numerically by solving the Oberbeck-Boussinesq equations with the large-eddy-simulation approach for small-scale turbulence. The cylinder length is $L=5D$, where $D$ is the diameter. The Rayleigh number, determined by the cylinder diameter, is of the order of $5\times {10}^6$. We show that the structure of the flow strongly depends on the inclination angle. A stable large-scale circulation (LSC) slightly disturbed by small-scale turbulence exists in the horizontal cylinder. The deviation from a horizontal position provides strong amplification of both LSC and small-scale turbulence. The energy of turbulent pulsations increases monotonically with decreasing inclination angle $\beta$, matching the energy of the LSC at $\beta \approx \pi /5$. The intensity of the LSC has a wide, almost flat, maximum for an inclined cylinder and slumps approaching the vertical position, in which the LSC vanishes. The dependence of the Nusselt number on the inclination angle has a maximum at $\beta \approx 7\pi /20$ and generally follows the dependence of the intensity of LSC on the inclination. This indicates that the total heat transport is highly determined by LSC. We examine the applicability of idealized thermal boundary conditions (BCs) for modeling a real experiment with liquid sodium flows. Therefore, the simulations are done with two types of temperature BCs: fixed face temperature and fixed heat flux. The intensity of the LSC is slightly higher in the latter case and leads to a corresponding increase of the Nusselt number and enhancement of temperature pulsations.

The decay of Taylor-Couette turbulence, i.e., the flow between two coaxial and independently rotating cylinders, is numerically studied by instantaneously stopping the forcing from an initially statistically stationary flow field at a Reynolds number of $\text{Re}=3.5\times {10}^4$. The effect of wall friction is analyzed by comparing three separate cases, in which the cylinders are either suddenly made no-slip or stress-free. Different life stages are observed during the decay. In the first stage, the decay is dominated by large-scale rolls. Counterintuitively, when these rolls fade away, if the flow inertia is small a redistribution of energy occurs and the energy of the azimuthal velocity behaves nonmonotonically, first decreasing by almost two orders of magnitude and then increasing during the redistribution. The second stage is dominated by non-normal transient growth of perturbations in the axial (spanwise) direction. Once this mechanism is exhausted, the flow enters the final life stage, viscous decay, which is dominated by wall friction. We show that this stage can be modeled by a one-dimensional heat equation, and that self-similar velocity profiles collapse onto the theoretical solution.

Motivated by observations of large effective slip over nanostructured surfaces, we analyze here the hydrodynamic resistance to the rigid-body motion of a cylinder near a slippery wall—perhaps the simplest anisotropic configuration which nonetheless exhibits a nontrivial translation-rotation coupling. Focusing upon the lubrication limit, the properly scaled resistance-matrix coefficients depend only upon the ratio of the slip length to the cylinder-wall clearance. A distinct feature of a lubrication analysis with slippage is that the pressure field cannot be determined in closed form; nonetheless, the dependence upon the slip length of the resistance coefficients is explicitly obtained. The limit of large slip length, where the wall behaves as a free surface, is elucidated. Considering the companion problem of a slippery cylinder, we observe a symmetry relating the two problems.

Active nematics are orientationally ordered but apolar fluids composed of interacting constituents individually powered by an internal source of energy. When activity exceeds a system-size-dependent threshold, spatially uniform active apolar fluids undergo a hydrodynamic instability leading to spontaneous macroscopic fluid flow. Here we show that a special class of spatially nonuniform configurations of such active apolar fluids display laminar (i.e., time-independent) flow even for arbitrarily small activity. We also show that two-dimensional active nematics confined on a surface of nonvanishing Gaussian curvature must necessarily experience a nonvanishing active force. This general conclusion follows from a key result of differential geometry: Geodesics must converge or diverge on surfaces with nonzero Gaussian curvature. We derive the conditions under which such curvature-induced active forces generate thresholdless flow for two-dimensional curved shells. We then extend our analysis to bulk systems and show how to induce thresholdless active flow by controlling the curvature of confining surfaces, external fields, or both. The resulting laminar flow fields are determined analytically in three experimentally realizable configurations that exemplify this general phenomenon: (i) toroidal shells with planar alignment, (ii) a cylinder with nonplanar boundary conditions, and (iii) a Frederiks cell that functions like a pump without moving parts. Our work suggests a robust design strategy for active microfluidic chips and could be tested with the recently discovered living liquid crystals.

Optical tweezers accomplished with fast (100 µs) particle tracking are employed to determine the frequency (1–1000 Hz) dependence of the electrophoretic mobility µ of a single colloid in an external oscillating electric (ac) field. The colloid under study is held in a microfluidic channel filled with monovalent or divalent salt (KCl and $\mathrm{CaC}{\mathrm{l}}_2$) solutions of varying concentrations (${10}^{-4}$ to ${10}^{-1}\mathrm{molar}$), and its response to an external ac field is measured. A pronounced steplike frequency dependence of the single colloid electrophoretic mobility (SCE-µ) is observed with a plateau on the low-frequency side (1–100 Hz) and a corner frequency, $f_c$, which shifts with increasing concentration of the salt solution. The results are explained by considering the characteristic time needed for polarizing the electric double layer surrounding the colloidal particle under study.

Turbulence generated by an oscillating grid in a two-layer stably stratified system is a classical flow utilized to study various aspects of turbulence in presence of stratification without mean shear. This flow evolves in a quasisteady state, in which the layer thickness and density difference evolves in a quasisteady manner due to the large separation of timescales between the turbulence and the setup. We present an extension of the classical setup that enables full steady state conditions and in which the entrainment velocity can be prescribed separately from the Richardson number. We develop a theoretical box-model and show that the model is in good agreement with the experiments. The model allows to predict the transient response of the system for a variety of initial conditions and the imposed steady state. The new setup is necessary to obtain the steady position of the density interface, for example, when using advanced optical techniques to measure the small-scale features of turbulence near the interface.

The entrainment process in stably stratified mixing layers is studied in relation to the turbulent-nonturbulent interface (TNTI) using direct numerical simulations. The statistics are calculated with the interface coordinate in an Eulerian frame as well as with the Lagrangian fluid particles entrained from the nonturbulent to the turbulent regions. The characteristics of entrainment change as the buoyancy Reynolds number ${\text{Re}}_b$ decreases and the flow begins to layer. The baroclinic torque delays the enstrophy growth of the entrained fluids at small ${\text{Re}}_b$, while this effect is less efficient for large ${\text{Re}}_b$. The entrained particle movement within the TNTI layer is dominated by the small dissipative scales, and the rapid decay of the kinetic energy dissipation rate due to buoyancy causes the entrained particle movement relative to the interface location to become slower. Although the Eulerian statistics confirm that there exists turbulent fluid with strong vorticity or with large buoyancy frequency near the TNTI, the entrained fluid particles circumvent these regions by passing through the TNTI in strain-dominant regions or in regions with small buoyancy frequency. The multiparticle statistics show that once the nonturbulent fluid volumes are entrained, they are deformed into flattened shapes in the vertical direction and diffuse in the horizontal direction. When ${\text{Re}}_b$ is large enough for small-scale turbulence to exist, the entrained fluid is able to penetrate into the turbulent core region. Once the flow begins to layer with decreasing ${\text{Re}}_b$, however, the entrained fluid volume remains near the outer edge of the turbulent region and forms a stably stratified layer without vertical overturning.

Most work on the mechanical behavior of red blood cells (RBCs) in flow has focused on simple shear flows. Relatively little work has examined RBC deformations in the physiologically important extensional flow that occurs at the entrance to a constriction. In particular, previous work suggests that RBCs rapidly stretch out and then retract upon entering the constriction, but to date no model predicts this behavior for the extremely high strain rates typically experienced there. In this Rapid Communication, we use high speed video to perform systematic measurements of the dynamic stretching behavior of RBCs as they enter a microfluidic constriction. We demonstrate that both the Kelvin-Voigt and Skalak viscoelastic models capture the observed stretching dynamics, up to strain rates as high as 2000 ${\text{s}}^{-1}$. The results indicate that the effective elastic modulus of the RBC membrane at these strain rates is an order of magnitude larger than moduli measured by micropipette aspiration or other low strain rate techniques.

We report measurements of global dissipated power within a turbulent flow homogeneously forced at small scale by an original forcing technique. The forcing is random in both time and space within the fluid by using magnetic particles in an alternating magnetic field. By measuring the growth rate of the fluid temperature, we show how the dissipated power is governed by the external control parameters (magnetic field and number $N$ of particles). We experimentally found that the mean dissipated power scales linearly with these parameters, as expected from the magnetic injected power scalings. These experimental results are well described by simple scaling arguments showing that the main origins of the energy dissipation are due to viscous turbulent friction of particles within the fluid and to the inelasticity of collisions. Finally, by measuring the particle collision statistics, we also show that the particle velocity is independent of $N$ and is only fixed by the magnetic “thermostat.”

We study the energy transfer properties of three-dimensional homogeneous and isotropic turbulence where the nonlinear transfer is altered in a way that helicity is made sign-definite, say, positive. In this framework, known as homochiral turbulence, an adapted eddy-damped quasinormal Markovian closure is derived to analyze the dynamics at very large Reynolds numbers, of order ${10}^5\le {\text{Re}}_{\lambda }\le {10}^6$. In agreement with previous findings, an inverse cascade of energy with a kinetic energy spectrum such as $\propto k^{-5/3}$ is found for scales larger than the forcing one. Conjointly, a forward cascade of helicity towards larger wave numbers is obtained, where the kinetic energy spectrum scales as $\propto k^{-7/3}$. By following the evolution of the closed spectral equations for a very long time and over a huge extension of scales, we found the development of a nonmonotonic shape for the front of the inverse energy flux. The asymptotic temporal scaling laws for the kinetic energy, helicity, and integral scales in both the forced and unforced cases are also determined.

A cross-sectionally averaged model is used to study several aspects of the physics of a bubble plume rising in a stratified quiescent liquid. Scaling analyses for the peel height, at which the plume momentum vanishes, and the neutral height, at which its average density equals the ambient density, are presented. Contrary to a widespread practice in the literature, it is argued that the neutral height cannot be identified with the experimentally reported intrusion height. Recognizing this difference provides an explanation of the reason why the intrusion height is found so frequently to be much above predictions and brings the theoretical results in line with observations. The mathematical model depends on three dimensionless parameters, some of which are related to the inlet conditions at the plume source. Their influence on the peel and neutral heights is illustrated by means of physical considerations, scaling analyses, and numerical results.

We present an analysis of the Navier-Stokes equations based on a spatial filtering technique to elucidate the multiscale nature of fully developed turbulence. In particular, the advection of a band-pass-filtered small-scale contribution by larger scales is considered, and rigorous upper bounds are established for the various dynamically active scales. The analytical predictions are confirmed with direct numerical simulation data. The results are discussed with respect to the establishment of effective large-scale equations valid for turbulent flows.

The irreversible turbulent energy cascade epitomizes strongly nonequilibrium systems. At the level of single fluid particles, time irreversibility is revealed by the asymmetry of the rate of kinetic energy change, the Lagrangian power, whose moments display a power-law dependence on the Reynolds number, as recently shown by Xu et al. [H. Xu et al., Proc. Natl. Acad. Sci. USA 111, 7558 (2014)]. Here Lagrangian power statistics are rationalized within the multifractal model of turbulence, whose predictions are shown to agree with numerical and empirical data. Multifractal predictions are also tested, for very large Reynolds numbers, in dynamical models of the turbulent cascade, obtaining remarkably good agreement for statistical quantities insensitive to the asymmetry and, remarkably, deviations for those probing the asymmetry. These findings raise fundamental questions concerning time irreversibility in the infinite-Reynolds-number limit of the Navier-Stokes equations.