The steady behavior of a low Mach number flow of a rarefied gas past a sphere is considered on the basis of the linearized Boltzmann equation with a special interest in the asymptotic behaviors of the flow velocity and temperature in the region far from the sphere. The study is motivated by the previous one [Taguchi, J. Fluid Mech. 774, 363 (2015)], in which an asymptotic analysis of the Boltzmann equation for small Mach numbers was carried out to derive the expression of the drag up to the second order of the Mach number. The derived expression contains two functions of the Knudsen number, corresponding to the leading-order drag obtained from the linearized problem and the second-order correction due to the weak nonlinear effect. This correction is also obtained through the analysis of the linearized problem; it is given by the factor of the term in the flow velocity whose magnitude is inversely proportional to the distance from the sphere and therefore vanishes at infinity. In this study, this factor (and thus the correction) is obtained for a wide range of the Knudsen number for the hard-sphere gas as well as for the ellipsoidal statistical (ES) model of the Boltzmann equation, under the conventional diffuse reflection boundary condition. The construction is based on the universal relation between the linear drag and the factor (correction). With the available data for the linear drag, this allows us to derive the latter from the former. For the ES model with the Prandtl number $\mathrm{Pr}=2/3$, a series of additional numerical computations of the linearized problem is carried out to obtain the linear drag and then the correction. It is also shown that a factor occurring in the temperature, decaying in proportion to the inverse square of the distance from the sphere, is connected to the thermal force exerted on a sphere (thermophoresis), whose numerical values on the basis of the ES model are obtained as well.

In the present study, we analytically investigate the deformation and bulk rheology of a dilute emulsion of surfactant-laden droplets suspended in linear flows. We use an asymptotic approach to determine the effect of surfactant distribution on the deformation of an isolated droplet as well as the effective shear and extensional viscosity for a dilute emulsion. The nonuniform distribution of surfactants due to the bulk flow results in the generation of a Marangoni stress, which affects both the deformation as well as the bulk rheology of the emulsion. The present analysis is done for the limiting case when the surfactant transport is dominated by the surface diffusion relative to surface convection. As an example, we have used two commonly encountered bulk flows, namely, uniaxial extensional flow and simple shear flow. With the assumption of negligible inertial forces present in either of the phases, we show that both the surfactant concentration on the droplet surface as well as the ratio of viscosity of the droplet phase with respect to the suspending fluid has a significant effect on the droplet deformation as well as the bulk rheology. It is seen that increase in the nonuniformity in surfactant distribution on the droplet surface results in a higher droplet deformation and a higher effective viscosity for either of the linear flows considered. The effect of surfactant distribution on effective viscosity is insignificant for highly viscous droplets. For the case of simple shear flow, surfactant distribution is found to have no effect on the inclination angle. However, a higher viscosity ratio predicts the droplet to be more aligned toward the direction of flow. First and second normal stress differences are present for the case of a simple shear flow, of which the former is found to increase with nonuniformity in surfactant distribution, whereas the later remains unaffected.

A numerical method is presented that allows to compute time-periodic flow states, even in the presence of hydrodynamic instabilities. The method is based on filtering nonharmonic components by way of delayed feedback control, as introduced by Pyragas [Phys. Lett. A 170, 421 (1992)]. Its use in flow problems is demonstrated here for the case of a periodically forced laminar jet, subject to a subharmonic instability that gives rise to vortex pairing. The optimal choice of the filter gain, which is a free parameter in the stabilization procedure, is investigated in the context of a low-dimensional model problem, and it is shown that this model predicts well the filter performance in the high-dimensional flow system. Vortex pairing in the jet is efficiently suppressed, so that the unstable periodic flow state in response to harmonic forcing is accurately retrieved. The procedure is straightforward to implement inside any standard flow solver. Memory requirements for the delayed feedback control can be significantly reduced by means of time interpolation between checkpoints. Finally, the method is extended for the treatment of periodic problems where the frequency is not known a priori. This procedure is demonstrated for a three-dimensional cubic lid-driven cavity in supercritical conditions.

Weak-turbulence theory is a statistical framework to describe a large ensemble of nonlinearly interacting waves. The archetypal example of such system is the ocean surface that is made of interacting surface gravity waves. Here we describe a laboratory experiment dedicated to probe the statistical properties of turbulent gravity waves. We set up an isotropic state of interacting gravity waves in the Coriolis facility (13-m-diam circular wave tank) by exciting waves at 1 Hz by wedge wave makers. We implement a stereoscopic technique to obtain a measurement of the surface elevation that is resolved in both space and time. Fourier analysis shows that the laboratory spectra are systematically steeper than the theoretical predictions and the field observations in the Black Sea by Leckler et al. [F. Leckler et al., J. Phys. Oceanogr. 45, 2484 (2015)]. We identify a strong impact of surface dissipation on the scaling of the Fourier spectrum at the scales that are accessible in the experiments. We use bicoherence and tricoherence statistical tools in frequency and/or wave-vector space to identify the active nonlinear coupling. These analyses are also performed on the field data by Leckler et al. for comparison with the laboratory data. Three-wave coupling is characterized by and shown to involve mostly quasiresonances of waves with second- or higher-order harmonics. Four-wave coupling is not observed in the laboratory but is evidenced in the field data. We discuss temporal scale separation to explain our observations.

As a necessary preliminary step toward geophysically significant extrapolations, we study the scale effects in internal wave attractors in the linear and nonlinear regimes. We use two geometrically similar experimental setups, scaled to factor 3, and numerical simulations (a spectral element method, based on the Nek5000 open solver) for a range of parameters that is typically accessible in laboratory. In the linear regime, we recover the classical viscous scaling for the beam width, which is not affected by variations of the amplitude of the input perturbation. In the nonlinear regime, we show that the scaling of the width-to-length ratio of the attractor branches is intimately related with the energy cascade from large-scale energy input to dissipation. We present results for the wavelength, amplitude, and width of the beam as a function of time and as a function of the amplitude of the forcing.

We investigate how drops merging on a nonwettable surface jump off this surface, for both symmetric and asymmetric coalescences. For this purpose, we design and build a microelectromechanical system sensor able to quantify forces down to the micro-Newton scale at a high acquisition rate (200 kHz). Using this device, we perform direct force measurements of self-propelled droplets coupled to high-speed imaging. Experimental data show that the total momentum of the drop after coalescence mainly depends on the size of the smaller drop. Exploiting this finding, we quantitatively predict the takeoff speed of jumping drop pairs and show how to correct the usual argument based on energy conservation.

Microbial flagellates typically inhabit complex suspensions of polymeric material which can impact the swimming speed of motile microbes, filter feeding of sessile cells, and the generation of biofilms. There is currently a need to better understand how the fundamental dynamics of polymers near active cells or flagella impacts these various phenomena, in particular, the hydrodynamic and steric influence of a rotating helical filament on suspended polymers. Our Stokesian dynamics simulations show that as a stationary rotating helix pumps fluid along its long axis, polymers migrate radially inward while being elongated. We observe that the actuation of the helix tends to increase the probability of finding polymeric material within its pervaded volume. This accumulation of polymers within the vicinity of the helix is stronger for longer polymers. We further analyze the stochastic work performed by the helix on the polymers and show that this quantity is positive on average and increases with polymer contour length.

Lyapunov exponents measure the average exponential growth rate of typical linear perturbations in a chaotic system, and the inverse of the largest exponent is a measure of the time horizon over which the evolution of the system can be predicted. Here, Lyapunov exponents are determined in forced homogeneous isotropic turbulence for a range of Reynolds numbers. Results show that the maximum exponent increases with Reynolds number faster than the inverse Kolmogorov time scale, suggesting that the instability processes may be acting on length and time scales smaller than Kolmogorov scales. Analysis of the linear disturbance used to compute the Lyapunov exponent, and its instantaneous growth, show that the instabilities do, as expected, act on the smallest eddies and that at any time, there are many sites of local instabilities.

We report the existence of a self-similar scaling in the vortices of freely evolving two-dimensional turbulence: The vortex number density $n(A,t)$ compensated by the mean vortex intensity $\overline{{\omega }_{\text{v}}^2}(A,t)$, where $A$ is vortex area and $t$ is time, follows the self-similar form $n(A,t)/\overline{{\omega }_{\text{v}}^2}(A,t)\sim t^{-2/3}{A}^{-1}$. This extended scale-invariant behavior holds for different initial conditions, despite very different scaling of $n(A,t)$ and $\overline{{\omega }_{\text{v}}^2}(A,t)$ taken separately, and ensures that the number of turnaround times ${\left[\overline{{\omega }_{\text{v}}^2}(A,t)\right]}^{-1/2}$ taken to cross the characteristic intervortex distance is independent of scale.

To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation including the effects of fluctuations has to be solved. In Shishkina et al. [Phys. Rev. Lett. 114, 114302 (2015)], the thermal BL equation with the fluctuations taken into account as an eddy thermal diffusivity has been solved for large Prandtl-number fluids for which the eddy thermal diffusivity and the velocity field can be approximated, respectively, as a cubic and a linear function of the distance from the plate. In the present work, we make use of the idea of Prandtl's mixing length model and relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal BL equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters for fluids with a general Prandtl number. With a proper choice of the parameters, our predictions of the temperature profiles are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers from 0.01 to 2547.9 and Rayleigh numbers from ${10}^7$ to ${10}^9$.

It is well known that a spherical drop charged beyond its Rayleigh limit becomes unstable (when subjected to a small shape perturbation) and ejects a significant fraction of the original charge in the form of a jet. To understand the detailed mechanism of this process, we model a charged drop of a perfectly conducting viscous liquid numerically using the boundary-element method. It is found from the simulations that the drop progressively deforms into a conical shape with pointed ends just prior to breakup. For ethylene glycol droplets suspended in air, the drop attains an aspect ratio of 3.86 at its critical shape, close to the observed value of 3.85 in the experimental images available in the literature. As the numerical approach fails to predict charge ejection due to the occurrence of a singularity at this point, the charge loss fraction is estimated by determining how much minimum charge must be removed if the drop is to relax back to a spherical shape. In this manner, the charge loss due to Rayleigh fission (the difference between the original charge and the minimum charge removed) is estimated to be about 39%, which falls within the range of experimental data lying between 20% and 50%. The results on scaling laws, time scales of the process, and the role of various stresses on the dynamics of the drop are discussed.

We investigate the stability of streaks in the buffer layer of turbulent channel flows with temperature-dependent density and viscosity by means of linear theory. The adopted framework consists of an extended set of the Orr-Sommerfeld-Squire equations that accounts for density and viscosity nonuniformity in the direction normal to the walls. The base flow profiles for density, viscosity, and velocity are averaged from direct numerical simulations (DNSs) of fully developed turbulent channel flows. We find that the inner scaling based on semilocal quantities provides an effective parametrization of the effect of variable properties on the linearized flow. The spanwise spacing of optimal buffer layer streaks scales to ${\lambda }_{z,\mathrm{opt}}^{★}\approx 90$ for all cases considered and the maximum energy amplification decreases, compared to the one for a flow with constant properties, if the semilocal Reynolds number ${\text{Re}}_{\tau }^{★}$ increases away from the walls, consistently with less energetic streaks observed in DNSs of turbulent channels. A secondary stability analysis of the two-dimensional velocity profile formed by the mean turbulent velocity and the nonlinearly saturated optimal streaks predicts a streamwise instability mode with wavelength ${\lambda }_{x,\mathrm{cr}}^{★}\approx 230$ in semilocal units, regardless of the fluid property distribution across the channel. The threshold for the onset of the secondary instability is reduced, compared to a constant property flow, if ${\text{Re}}_{\tau }^{★}$ increases away from the walls, which explains the more intense ejection events reported in DNSs. The opposite behavior is predicted by the linear theory for decreasing ${\text{Re}}_{\tau }^{★}$, in accord with DNS observations. We finally show that the phase velocity of the critical mode of secondary instability agrees well with the convection velocity calculated by DNSs in the near-wall region for both constant and variable viscosity flows.

We present the results of a numerical investigation of droplets walking in a harmonic potential on a vibrating fluid bath. The droplet's trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. We produce a regime diagram that summarizes the dependence of the walker's behavior on the system parameters for a droplet of fixed size. At relatively low vibrational forcing, a number of periodic and quasiperiodic trajectories emerge. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but the resulting trajectories can be decomposed into portions of unstable quasiperiodic states.

A sophisticated type of electrohydrodynamic instability, the thermoelectrokinetic instability, in an electrolyte solution near ion-selective surfaces in an external electric field was discovered and investigated theoretically. The investigation used parallel computing on the SKIF MSU Lomonosov supercomputer and it was based on the direct numerical simulation of the Nernst-Planck-Poisson-Navier-Stokes system, along with the energy equation and corresponding boundary conditions. Although the physical mechanism of the instability is connected with Joule heating, it dramatically differs from the well-known Raleigh-Bénard convection: The instability is caused by nonuniformity of the electric current and electric conductivity and, in contrast to the Raleigh-Bénard instability, it occurs when the heating is from above. The thermoelectrokinetic instability prevails in long microchannels and a good enough thermal insulation of the system and it can be an additional factor for the overlimiting current mode. Initial unstable small random disturbances of the unknowns evolve towards rather unusual coherent structures. For the cation-exchange membranes, the salt concentration was eventually localized in long narrow fingers that are similar to stalactites and that stretched from the anode in the direction of the cathode. Outside the stalactites, the salt concentration was practically zero, while inside the stalactites it could reach hundreds from the initial concentration. Consequently, it is possible to talk about its superconcentration. The electric current propagates in a narrow needle along the stalactite with maximal electric conductivity and this gives rise to Joule heating in this region. The space charge forms crownlike micropatterns near the cathode. Note that the regime of chaotic motion that is characteristic of the electrokinetic instability was not observed for the thermoelectrokinetic instability.

Recent numerical simulations showed that mean flow is generated in the inhomogeneous turbulence of an incompressible fluid that is accompanied by helicity and system rotation. In order to investigate the mechanism of the phenomenon, we perform a numerical simulation of inhomogeneous turbulence in a rotating system. In the simulation, an external force is applied to inject inhomogeneous turbulent helicity and the rotation axis is perpendicular to the inhomogeneous direction. The mean velocity is set to zero in the initial condition of the simulation. The simulation results show that the mean flow directed to the rotation axis is generated and sustained only in the case with both the helical forcing and the system rotation. We investigate the physical origin of this flow-generation phenomenon by considering the budget of the Reynolds-stress transport equation. The results indicate that the pressure diffusion term significantly contributes to the Reynolds-stress equation and supports the generated mean flow. The results also reveal that a model expression for the pressure diffusion is expressed by the turbulent helicity gradient coupled with the angular velocity of the system rotation. This implies that inhomogeneous helicity plays a significant role in the generation of the large-scale velocity distribution in incompressible turbulent flows.

The relationship between lateral static stability derivative, $C_{l_{\beta }}$, lift coefficient, $C_L$, and angle of attack was investigated for rectangular wings of aspect ratio $A\mathrm{R}$ $=0.75,1,1.5,$ and 3 using Stereo-Digital Particle Image Velocimetry (S-DPIV) and direct force and moment measurements. When the product $C_{l_{\beta }}A\mathrm{R}$ is plotted with respect to $C_L$, the lateral stability curves of each wing collapse to a single line for $C_L<0.7$. For $C_L>0.7$, the linearity and scaling of $C_{l_{\beta }}$ with respect to $C_L$ is lost. S-DPIV is used to elucidate the flow physics in this nonlinear regime. At $\alpha ={10}^{\circ }$, the leading-edge separation region emerges on the leeward portion of the sideslipped wing by means of vortex shedding. For the $A\mathrm{R}$ $\le 1.5$ wings at $\alpha >{15}^{\circ }$, the tip vortex downwash is sufficient to restrict the shedding of leading-edge vorticity thereby sustaining the lift of the leading-edge separation region at high angles of attack. Concurrently, the windward tip vortex grows in size and strength with increasing angle of attack, displacing the leading-edge separation region further toward the leeward wing. This reorganization of lift-generating vorticity results in the initial nonlinearities between $C_{l_{\beta }}$ and $C_L$ at angles of attack for which $C_L$ is still increasing. At angles of attack near that of maximum lift for the $A\mathrm{R}$ $\le 1$ wings, the windward tip vortex lifts off the wing, decreasing the lateral static stability of the wing prior to lift stall. For the $A\mathrm{R}=3$ wing at $\alpha >{10}^{\circ }$, nonlinear trends in $C_{l_{\beta }}$ versus $C_L$ occur due to the spanwise evolution of stalled flow.

A computational framework is proposed for the linear modal and nonmodal analysis of fluid systems consisting of a periodic array of $n$ identical units. A formulation in either time or frequency domain is sought and the resulting block-circulant global system matrix is analyzed using roots-of-unity techniques, which reduce the computational effort to only one unit while still accounting for the coupling to linked components. Modal characteristics as well as nonmodal features are treated within the same framework, as are initial-value problems and direct-adjoint looping. The simple and efficient formalism is demonstrated on selected applications, ranging from a Ginzburg-Landau equation with an $n$-periodic growth function to interacting wakes to incompressible flow through a linear cascade consisting of 54 blades. The techniques showcased here are readily applicable to large-scale flow configurations consisting of $n$-periodic arrays of identical and coupled fluid components, as can be found, for example, in turbomachinery, ring flame holders, or nozzle exit corrugations. Only minor corrections to existing solvers have to be implemented to allow this present type of analysis.

Results are presented of direct numerical simulations of incompressible, homogeneous magnetohydrodynamic turbulence without a mean magnetic field, subject to different mechanical forcing functions commonly used in the literature. Specifically, the forces are negative damping (which uses the large-scale velocity field as a forcing function), a nonhelical random force, and a nonhelical static sinusoidal force (analogous to helical ABC forcing). The time evolution of the three ideal invariants (energy, magnetic helicity, and cross helicity), the time-averaged energy spectra, the energy ratios, and the dissipation ratios are examined. All three forcing functions produce qualitatively similar steady states with regard to the time evolution of the energy and magnetic helicity. However, differences in the cross-helicity evolution are observed, particularly in the case of the static sinusoidal method of energy injection. Indeed, an ensemble of sinusoidally forced simulations with identical parameters shows significant variations in the cross helicity over long time periods, casting some doubt on the validity of the principle of ergodicity in systems in which the injection of helicity cannot be controlled. Cross helicity can unexpectedly enter the system through the forcing function and must be carefully monitored.

The role of absolute viscosities on the dynamics of droplet spreading on solids and droplet-droplet coalescence in liquid-liquid systems is experimentally characterized for a broad range of fluid parameters. We show in particular the existence of a viscous function based on both inner and outer fluid viscosities that allows for the determination of the critical wetting velocity and the evolution of contact diameters during immersed spreading and coalescence of droplets. This work quantifies the cooperative effects of fluid viscosities on droplet dynamics as well as their overall reduced influence from initial wetting to spreading and coalescence, which provides insights into the role of wetting contact lines on spontaneous capillary phenomena.

Fluid-resistance limited transport of vesicles through narrow constrictions is a recurring theme in many biological and engineering applications. Inspired by the motor-driven movement of soft membrane-bound vesicles into closed neuronal dendritic spines, here we study this problem using a combination of passive three-dimensional simulations and a simplified semianalytical theory for the active transport of vesicles forced through constrictions by molecular motors. We show that the motion of these objects is characterized by two dimensionless quantities related to the geometry and to the strength of forcing relative to the vesicle elasticity. We use numerical simulations to characterize the transit time for a vesicle forced by fluid pressure through a constriction in a channel and find that relative to an open channel, transport into a blind end leads to the formation of a smaller forward-flowing lubrication layer that strongly impedes motion. When the fluid pressure forcing is complemented by forces due to molecular motors that are responsible for vesicle trafficking into dendritic spines, we find that the competition between motor forcing and fluid drag results in multistable dynamics reminiscent of the real system. Our study highlights the role of nonlocal hydrodynamic effects in determining the kinetics of vesicular transport in constricted geometries.