A numerical model was employed to investigate the vortex instability in a two-dimensional inkjet print-zone. The simulation models the entrainment effect of the droplets on the airflow via a dispersed-phase continuum method that, due to the separation of length scales, treats the force exerted by the main droplets as a continuum smooth field. The trajectory and speed of the main droplets are also assumed to be unaffected by the airflow. The results indicate the existence of a dimensionless droplet density threshold ($N_c$) at which the vortex shifts from steady to oscillatory. This demonstrates that the two-dimensional instability has a supercritical Hopf type of bifurcation, i.e., the shift from stable to unstable is continuous but not smooth and the amplitude of oscillation follows the asymptotic square-root behavior characteristic of this type of bifurcation. Further, as shown by tests with stationary paper and no induced cross-flow, the mechanism of instability cannot be attributed to the interaction between the incoming cross-flow and the entrained airflow. Characterizing the two-dimensional airflow instabilities and their mechanism provides a better understanding of the airflow dynamics in the print gap of inkjet printers.

The conceptual picture underlying resolvent analysis is that the nonlinear term in the Navier-Stokes equations acts as an intrinsic forcing to the linear dynamics, a description inspired by control theory. The inverse of the linear operator, defined as the resolvent, is interpreted as a transfer function between the forcing and the velocity response. From a theoretical point of view this is an attractive approach since it allows for the vast mathematical machinery of control theory to be brought to bear on the problem. However, from a practical point of view, this is not always advantageous. The inversion of the linear operator inherent in the control theoretic definition obscures the physical interpretation of the governing equations and is prohibitive to analytical manipulation, and for large systems it leads to significant computational cost and memory requirements. In this work we suggest an alternative, inverse-free, definition of the resolvent basis based on an extension of the Courant–Fischer–Weyl min-max principle in which resolvent modes are defined as stationary points of a constrained variational problem. This definition leads to a straightforward approach to approximate the resolvent (response) modes of complex flows as expansions in any arbitrary basis. The proposed method avoids matrix inversions and requires only the spectral decomposition of a matrix of significantly reduced size as compared to the original system. To illustrate this method and the advantages of the variational formulation we present three examples. First, we consider streamwise constant fluctuations in turbulent channel flow where an asymptotic analysis allows us to derive closed form expressions for the optimal resolvent mode. Second, to illustrate the cost-saving potential and investigate the limits of the proposed method, we apply our method to both a two-dimensional, three-component equilibrium solution in Couette flow and, finally, to a streamwise developing turbulent boundary layer. For these larger systems we achieve a model reduction of up to two orders of magnitude. Such savings have the potential to open up RA to the investigation of larger domains and more complex flow configurations.

We investigate the coherent structures by low-Stokes-number particles suspended in half-zone liquid bridges of high Prandtl number via microgravity and terrestrial experiments. We especially focus on the structure by the suspended particle accumulation, or, the particle accumulation structure (PAS), of $m=1$ in azimuthal wave number emerged in the traveling-wave-type time-dependent “oscillatory” convection. The particles form the closed path with a spiral structure in tall or high-aspect-ratio liquid bridges under both micro- and normal gravity conditions. Their formation process is evaluated by applying the accumulation measure, and it is indicated that the formation time of the structure is of the order of a thermal diffusion time regardless of the difference of Prandtl number as well as the gravitational acceleration. The spatial correlation between the coherent structure and the thermal wave over the free surface is illustrated in the rotating frame of reference. By the terrestrial experiments, we indicate how the PAS is conformed by the individual particle motions in the laboratory frame and the rotating frame of reference via two-dimensional particle tracking. We also indicate their occurring condition as a function of the slenderness of the liquid bridges in terms of the volume ratio, and we evaluate the morphological characteristics of the spiral structure.

Unlike inertialess two-layer shear flows, three-layer ones can become unstable to long-wave interfacial instabilities due to a resonance mechanism between the interfaces. This interaction is codified in this paper through a set of coupled nonlinear evolution equations derived here in the limit of strong surface tension. A number of parameters are employed to cover a fairly general range of three-layer shear flows driven by a constant pressure gradient. The equations are analyzed using a combination of linear and computational techniques, identifying two linear instability mechanisms noted in the literature previously. The first is a kinematic instability due to the viscosity jumps across fluid phases and the second is a counterintuitive diffusion-derived instability, known in the literature as the Majda-Pego instability and mostly studied for second order diffusion. In the present work it is fourth order, due to surface tension, making the problem mathematically much more challenging. Three unstable parameter regimes of interest are identified linearly and are explored nonlinearly via pseudospectral numerical simulations. For thin middle layers we find steady-state traveling waves or states with asymptotically thinning regions leading to interfacial contact. However, for thin upper or lower layers, complex spatiotemporal dynamics emerge at large times that are characterized by fast time oscillations of the near-wall interface and slow oscillations of that farther away. Data analysis suggests that the dynamics is quasiperiodic in time and additionally coarsening phenomena are observed for large domain sizes leading to modulated traveling wave trains. The kinematic instability mechanism is shown to be triggered nonlinearly via the Majda-Pego mechanism. It can also be triggered by sufficiently large amplitude initial disturbances where linear instabilities are absent, although the transition is not necessarily self-sustaining in all cases.

Using absorbance measurements through a Couette cell containing an emulsion of buoyant droplets, volume fraction profiles are measured at various shear rates. These viscous resuspension experiments allow a direct determination of the normal stress in the vorticity direction in connection with the suspension balance model that has been developed for suspensions of solid particles. The results unambigously show that the normal viscosity responsible for the shear-induced migration of the droplets is independent on the capillary number, implying that particle deformation does not play a major role. It is similar to that of rigid particles at volume fractions below 40% but much smaller at higher ones.

In steady-state Rayleigh-Bénard convection, heat is transported by turbulent thermal convection from the bottom, hot surface to the top, cold surface, leading to a height-independent sensible heat flux. When water vapor is present and cloud formation occurs, there is also an additional latent heat flux. Heat transport in cloudy Rayleigh-Bénard convection depends on turbulent flow as well as the microphysical state of the clouds: specifically, whether substantial supersaturations exist and whether cloud liquid water is removed through sedimentation/precipitation. In this article we bridge between the Rayleigh-Bénard convection literature and the atmospheric literature. We express the governing equations for cloudy convection in dimensionless form, thereby explicitly identifying the governing parameters relevant to the cloudy case, including Schmidt, Damköhler, supersaturation, and sedimentation numbers. We further connect to the atmospheric literature by obtaining a Nusselt number (dimensionless heat flux) for a cloud-convection system, directly from the conservation equations for temperature and water vapor. This flux has the same form as that identified by Zhang et al. [L. Zhang, K. L. Chong, and K.-Q. Xia, J. Fluid Mech. 874, 1041 (2019)] for convection with water vapor, but is extended to the cloudy case. For equal thermal and water vapor diffusivities, the flux corresponds to the widely used atmospheric quantities equivalent temperature and moist static energy. Using large eddy simulation (LES) of an idealized cloudy Rayleigh-Bénard convection system with fixed boundary conditions, we find that the equivalent heat flux (Nusselt number) is only weakly dependent on the microphysical details of the system, such as liquid water mixing ratio and cloud droplet number concentration. From the results, we show the vertical profiles of sensible and latent heat fluxes depend on the liquid water content, whereas the equivalent heat flux remains a constant throughout the height of the chamber.

Many plankton species undergo daily vertical migration to large depths in the turbulent ocean. To do this efficiently, the plankton can use a gyrotactic mechanism, aligning them with gravity to swim downwards or against gravity to swim upwards. Many species show passive mechanisms for gyrotactic stability. For example, bottom-heavy plankton tend to align upwards. This is efficient for upward migration in quiescent flows, but it is often sensitive to turbulence which upsets the alignment. Here we suggest a simple, robust active mechanism for gyrotactic stability, which is only lightly affected by turbulence and allows alignment both along and against gravity. We use a model for a plankton that swims with a constant speed and can actively steer in response to hydrodynamic signals encountered in simulations of a turbulent flow. Using reinforcement learning, we identify the optimal steering strategy. By using its setae to sense its settling velocity transversal to its swimming direction, the swimmer can deduce information about the direction of gravity, allowing it to actively align upwards. The mechanism leads to a rate of upward migration in a turbulent flow that is of the same order as in quiescent flows, unless the turbulence is very vigorous. In contrast, passive swimmers with typical parameters of copepods show much smaller upward velocity in turbulence. Settling may even cause them to migrate downwards in vigorous turbulence.

Recently, Kumar et al. analyzed the Taylor dispersion of prolate spheroidal particles in a steady two-dimensional Poiseuille flow between parallel rigid walls [A. H. Kumar et al., Taylor dispersion of elongated rods, Phys. Rev. Fluids 6, 094501 (2021)]. They showed that the dispersivity of a spheroid is enhanced relative to that of an equivalent sphere whose translational diffusivity is equal to the orientation-averaged translational diffusivity of the spheroid. The largest enhancement in dispersivity occurs for elongated spheroids at large rotational Péclet numbers. The main purpose of this paper is to derive asymptotic approximations of the dispersivity and mean particle advection speed in these limits. Asymptotic approximations of these quantities at small rotational Péclet numbers are also presented.

The electrohydrodynamics of a concentric compound drop migrating and deforming in a plane Poiseuille flow under the influence of an arbitrarily orientated uniform electric field is investigated using a double asymptotic approach with the electric Reynolds number and capillary number as small perturbation parameters. The effect of viscosity, conductivity, and permittivity ratios, the orientation of the applied electric field, and radius ratio is thoroughly investigated, and the underlying physics is examined in terms of surface charge distribution and shape deformation of the shell and core of the compound drop. For an undeformable compound drop, we found that as the radius ratio increases, the magnitude of lateral velocity due to charge convection increases for both the shell and core, while the longitudinal velocity decreases. The intensity of the drop to lag behind the imposed flow increases as the electric field strength increases. For deformable compound drops, it is observed that the influence of the tilt angle of the applied electric field in altering the direction of motion gets dampened out or minimized when the size of the core increases. We also find that under the combined action of charge convection and shape deformation, the increase in electric Reynolds number enhances the lateral velocity of both the shell and the core drop while the longitudinal velocity decreases. However, it is found that the magnitude of the lateral and longitudinal velocity of the shell and core drop increases with an increase in the capillary number. Finally, by solving for the velocity field of an eccentric compound drop under plane Poiseuille flow and subjected to an applied electric field, we show that there is a critical eccentricity limit and critical time limit within which the concentric and eccentric compound drop configurations produce similar results and beyond which the increment or decrement in shell and core drop velocity is dictated by the value of eccentricity.

Pipe flow often traverses a regime where laminar and turbulent flow coexist. Prandtl and Tietjens explained this intermittency as a feedback between the fluctuations of the internal flow resistance and the constant pressure drop driving the flow. However, because the focus has moved towards studying intermittency without flow fluctuations near the universal critical Reynolds number, their explanation has largely disappeared. Here, we refine the mechanism, and put it to a quantitative test, to develop a model that agrees with experiments at higher Reynolds numbers, enabling us to demonstrate that Prandtl and Tietjens' mechanism is, in fact, intrinsic to flows where both the pressure gradient and perturbation are constant.

To investigate the interaction between two equal-sized immiscible droplets in a confined shear flow, a two-dimensional finite-element analysis is carried out by employing a ternary diffuse-interface model. Particularly, the present numerical model allows wetting between the immiscible droplets, and three different types of interaction could be identified, namely, (i) reverse motion, (ii) pass-over motion, and (iii) rotary Janus motion after wetting. A reversing flow that is generated near the droplets can entrain them, resulting in a reverse motion. The rotary Janus motion is observed when the droplets are wetted on each other as the fluid film between them is completely drained. More details of the rotary Janus motion are presented in terms of the droplet trajectory and deformation, and the effect of the initial droplet position, capillary number, and confinement is studied.

The electric conductance characterizes the response of a nanochannel or nanopore to an electrical potential drop. It has long been known that at low concentrations the electric conductance is independent of the bulk electrolyte concentration but depends on the surface-charge density. In recent years, several works have demonstrated that the low-concentration conductance is concentration dependent. This dependence is implicit through the mechanism known as surface-charge regulation which causes the surface-charge density to depend on the concentration, ${\tilde{c}}_0$. As a result, the conductance, $\tilde{G}$, has power-law dependency such that $\tilde{G}\sim {\tilde{c}}_0^{\alpha }$ with a slope of $\alpha$. It is typically assumed the slope takes on the particular values $\alpha =0,\frac{1}{3},\frac{1}{2}$. Here, we will analytically show that slope varies continuously from 0 to $\frac{1}{2}$ as the system parameters vary. Thereafter, we will account for convection and the effects of a slip length at the channel surface. We will show that convection without slip increases the conductance but does not vary the slope $\alpha$. The inclusion of slip not only increases the conductance but also increases the slope to $2\alpha$. Direct numerical simulations confirm our theoretical predictions. The consequences of these findings are important in the design of any electrokinetically driven nanofluidic system insofar as they provide the experimentalist an accurate prediction of the system response as a function of the material properties.

Microswimmers (planktonic microorganisms or artificial active particles) immersed in a fluid interact with the ambient flow, altering their trajectories. In surface gravity waves, a common goal for microswimmers is vertical migration (e.g., to reach the free surface or to dive to deeper depths). By modeling microswimmers as spheroidal bodies with an intrinsic swimming velocity that supplements advection and reorientation by the flow, we investigate how shape and swimming affect vertical transport of microswimmers in waves. We find that it is possible for microswimmers to be initially swimming downwards but to recover and head back to the surface and vice versa. This is because the coupling between swimming and flow-induced reorientations introduces a shape dependency in the vertical transport. From a wave-averaged analysis of microswimmer trajectories, we show that each trajectory is bounded by critical planes in the position-orientation phase space that depend only on the shape. We also give explicit solutions to these trajectories and determine the fraction of microswimmers that begin within the water column and eventually reach the surface. For microswimmers that are initially randomly oriented, the fraction that reach the surface increases monotonically as the starting depth decreases, as expected, but also varies with shape and swimming speed. In the limit of small swimming speed, the fraction of highly prolate microswimmers reaching the surface is 0.5, suggesting that these swimmers would be able to choose direction of vertical transport with small changes in swimming behavior.

Small-scale intermittency is an omnipresent feature of turbulent flows, and the velocity gradient tensor in three dimensions provides a rich characterization of its salient statistical and geometric features. A recent model for isotropic turbulence involving a set of stochastic differential equations was derived for the Lagrangian time evolution of the velocity gradient tensor at multiple scales. Dominant terms in the model could be derived directly from the Navier-Stokes equations while also requiring phenomenological models for unclosed pressure and viscous terms. In this work we show that instead of having to integrate this stochastic model in time at multiple levels and scales, the same predictions can be obtained based on a single level that recursively modulates the model terms interpreted at arbitrarily higher levels. The equivalence is explored based on actual realizations of the modeled stochastic process and then extended to probability densities and moments at all levels and scales. Statistical properties compare very well to direct numerical simulation and experimental data at different Reynolds numbers. The proposed formalism represents a new type of multifractal intermittency model that can be connected directly to terms in the Navier-Stokes equation and is capable of predicting many intricate statistical and geometric features of turbulent flows over a wide range of Reynolds numbers.

Tracer particles immersed in suspensions of biological microswimmers such as E. coli or C. reinhardtii display phenomena unseen in conventional equilibrium systems, including strongly enhanced diffusivity relative to the Brownian value and non-Gaussian displacement statistics. In dilute, three-dimensional suspensions, these phenomena have typically been explained by the hydrodynamic advection of point tracers by isolated microswimmers, while, at higher concentrations, correlations between pusher microswimmers such as E. coli can increase the effective diffusivity even further. Anisotropic tracers in active suspensions can be expected to exhibit even more complex behavior than spherical ones due to the presence of a nontrivial translation-rotation coupling. Using large-scale lattice Boltzmann simulations of model microswimmers described by extended force dipoles, we study the motion of ellipsoidal point tracers immersed in three-dimensional microswimmer suspensions. We find that the rotational diffusivity of tracers is much less affected by swimmer-swimmer correlations than the translational diffusivity. We furthermore study the anisotropic translational diffusion in the particle frame and find that, in pusher suspensions, the diffusivity along the ellipsoid major axis is higher than in the direction perpendicular to it, albeit with a smaller ratio than for Brownian diffusion. Thus, we find that far-field hydrodynamics cannot account for the anomalous coupling between the translation and rotation observed in experiments, as was recently proposed. Finally, we study the probability distributions (PDFs) of translational and rotational displacements. In accordance with experimental observations, for short observation times we observe strongly non-Gaussian PDFs that collapse when rescaled with their variance, which we attribute to the ballistic nature of tracer motion at short times.

The unsteady, lineal translation of a solid spherical particle through viscoelastic fluids described by the Johnson-Segalman and Giesekus models is studied analytically. Solutions for the pressure and velocity fields as well as the force on the particle are expanded as a power series in the Weissenberg number. The momentum balance and constitutive equation are solved asymptotically for a steadily translating particle up to second order in the particle velocity, and rescaling of the pressure and velocity in the frequency domain is used to relate the solutions for steady lineal translation to those for unsteady lineal translation. The unsteady force at third order in the particle velocity is then calculated through application of the Lorentz reciprocal theorem, and it is shown that this weakly nonlinear contribution to the force can be expressed as part of a Volterra series. Through a series of examples, it is shown that a truncated representation of this Volterra series, which can be manipulated to describe the velocity in terms of an imposed force, is useful for analyzing specific time-dependent particle motions. Two examples studied using this relationship are the force on a particle suddenly set into motion and the velocity of a particle in response to a suddenly imposed steady force. Additionally, the weakly nonlinear response of a particle captured by a harmonic trap moving lineally through the fluid is computed. This is an analog to active microrheology experiments and can be used to explain how weakly nonlinear responses manifest in active microrheology experiments with spherical probes.

Recent experimental results have shown that binary granular materials fluidized by combined vibration and gas flow exhibit Rayleigh-Taylor-like instabilities that manifest themselves in rising plumes, rising bubbles, and the sinking and splitting of granular droplets. This work explores the physics behind the splitting of a granular droplet that is composed of smaller and denser particles in a bed of larger and lighter particles. During its sinking motion, a granular droplet undergoes a series of binary splits resembling the fragmentation of a liquid droplet falling in a miscible fluid. However, different physical mechanisms cause a granular droplet to split. By applying particle image velocimetry and numerical simulations, we demonstrate that the droplet of high-density particles causes the formation of an immobilized zone underneath the droplet. This zone obstructs the downwards motion of the droplet and causes the droplet to spread and ultimately to split. The resulting fragments sink at inclined trajectories around the immobilized zone until another splitting event is initiated. The occurrence of consecutive splitting events is explained by the reformation of an immobilized zone underneath the droplet fragments. Our investigations identified three requirements for a granular droplet to split: (1) frictional interparticle contacts, (2) a higher density of the particles composing the granular droplet compared to the bulk particles, and (3) a minimal granular droplet diameter.

A practical application of universal wall scalings is near-wall turbulence modeling. In this paper, we exploit the semilocal scaling [Patel, Boersma, and Pecnik, Phys. Rev. Fluids, 2, 084604 (2017)] and derive an eddy conductivity closure for wall-modeled large-eddy simulation of high-speed flows. We show that while the semilocal scaling does not collapse high-speed direct numerical simulation (DNS) data, the resulting eddy conductivity and the wall model work fairly well. The paper attempts to answer the following outstanding question: why the semilocal scaling fails but the resulting eddy conductivity works well. We conduct DNSs of Couette flows at Mach numbers from $M=1.4$ to 6. We add a source term in the energy equation to get a cold wall, a close-to-adiabatic wall, and a hot wall. Detailed analysis of the flows' energy budgets shows that aerodynamic heating is the answer to our question: Aerodynamic heating is not accounted for in Patel et al.'s semilocal scaling but is modeled in the equilibrium wall model. We incorporate aerodynamic heating in the semilocal scaling and show that the new scaling successfully collapses the high-speed DNS data. We also show that incorporating aerodynamic heating or not, the semilocal scaling gives rise to the exact same eddy conductivity, thereby answering the outstanding question.

Synchronous single point measurements of the instantaneous surface elevation and of two components of the surface slope in a wind-wave tank are carried out using a high-speed camera and a laser slope gauge. The applied technique allows study of joint statistics between the temporal variation of surface elevation and the slope components at a point, as well as computation of directional wave energy spectra. It is demonstrated that while significant coherence exists between surface elevation and along-wind slope component, the coherence between crosswind slope and surface elevation is almost negligible. Evolution of the directional spectra along the test section is presented for different wind velocities and compared with results obtained in other relevant field and laboratory studies.

We report on an experimental investigation of the propagation of gravity-capillary waves in a narrow channel with a pinned contact line. By using Fourier transform profilometry we measure the static curved meniscus as well as the surface perturbation. By varying the channel width, between 7 and 15 times the capillary length, we show how edge constraints modify the surface curvature and therefore the dispersion relation. From the space-time-resolved field, we obtain a decomposition of the linear mode onto transverse modes satisfying the condition of pinned contact line. This approach, in which we complement the theoretical model with experimental analysis, allows computations of wave numbers and natural frequencies with a robust statistics. We verify experimentally the convergence of the model and the pertinence of the linear approximation. In addition, we analyze the relative contribution of the experimentally measured static meniscus. An excellent agreement between the computed natural frequencies and the forcing frequency confirms the contribution of the actual space-time-resolved measured surface. These experimental results are an accurate estimation of the influence of the additional restoring force exerted by the pinned contact line on the deformed surface which increases the wave celerity. The local character of this effect is evidenced by the decrease of the shift of the dispersion relation as a function of the channel width.