The richness of the dynamics of the thin film formed between two coalescing Brownian droplets is presented. Simulations based on a previously developed model [Phys. Rev. E 81, 051404 (2010)] which solves two coupled Langevin equations in the lubrication limit were performed. The time evolution of the thickness and radius of the cylindrical film is evaluated for the first time in terms of phase space, entropy of permutation, mean-square displacements, and creeping compliance. This new perspective of analysis reveals the specific and distinctive features of the nanofilm formation and breaking: (1) a well-defined attractor-type phase space which slightly become less disperse as they approach instead of single deterministic trajectories observed in hydrodynamic dominated systems, (2) complexity values larger than the ones expected for Fractional Brownian Motion, (3) a super-diffusive behavior of the film thinning but a sub-diffusive behavior of the film growing, and (4) the effective shear forces in the film that strongly depends on its configuration. Remarkably, the major role of thermal fluctuations in to modulate the strong non linear effects of hydrodynamic interactions, forcing the film to explore configurations well delimited by the potential energy.

A large-eddy simulation of turbulent, compressible flow around a NACA 0012 airfoil at zero angle of attack and Mach number 0.115 is used to study mechanisms of trailing-edge noise. The boundary layers at both sides of the airfoil have a forced transition near the airfoil leading edge, and are turbulent near the trailing-edge. Flow-acoustic correlations and spectral (frequency-domain) proper orthogonal decomposition (SPOD) are used to evaluate turbulent structures that are relevant for the radiated sound. Homogeneity in the spanwise direction allows application of a Fourier decomposition in span prior to both correlations and SPOD. It is known that acoustic theory, based on an analysis of the tailored Green's function modeling trailing-edge scattering, shows that only spanwise wave numbers $k_z$ satisfying $k_z<k$, where $k$ is the acoustic wave number, lead to radiated sound; two-dimensional disturbances ($k_z=0$) always satisfy this criterion, and thus spanwise-coherent structures are expected to be important for trailing-edge noise. Analysis of turbulence statistics of the boundary layer close to the trailing edge shows that the well-known, dominant streaky structures have $k_z>k$ and thus should not contribute to the radiated sound. To investigate this further using simulation data, flow-acoustic correlations are obtained using either the standard two-point analysis or considering two-dimensional disturbances in velocity and pressure fields, and results show significant correlation coefficients (of about 0.5) once two-dimensional disturbances near the trailing edge are isolated. A further increase of correlation peaks (up to 0.7) is obtained once the antisymmetric parts of the fields is considered, reflecting the classical antisymmetric nature of trailing-edge scattering. SPOD is then used for frequencies around the peak radiated sound to examine the structure of two-dimensional disturbances in the trailing-edge region and their contribution to radiated sound. Leading SPOD modes show coherent hydrodynamic waves propagating from the region of boundary-layer tripping toward the trailing edge, characterizing a noncompact source akin to wave packets seen in turbulent jets. These leading SPOD modes have significant contribution to the radiated sound, as two modes lead to 50% of the acoustic intensity for the lower studied frequencies. The present results point to the scattering of spanwise-coherent boundary-layer structures as the dominant mechanism of trailing-edge noise in this flow.

The shear-induced collective or gradient diffusivity in an emulsion of viscous drops, specifically as a function of viscosity ratio, was computed using a fully resolved numerical method. An initially randomly packed layer of viscous drops spreading due to drop-drop interactions in an imposed shear has been simulated. The collective diffusivity coefficient was computed using a self-similar solution of the drop concentration profile. We also obtained the collective diffusivity (the collective diffusivity coefficient multiplied by the average drop volume fraction), computing the dynamic structure factor from the simulated drop positions—an analysis typically applied only to homogeneous systems. The two quantities computed using entirely different methods are in broad agreement, including their predictions of nonmonotonic variations with increasing capillary number and viscosity ratio. The computed values were also found to match with past experimental measurements. The collective diffusivity coefficient computed here, as expected, is 1 order of magnitude larger than the self-diffusivity coefficient for a dilute emulsion previously computed using pairwise simulation of viscous drops in shear. The collective diffusivity coefficient computed here shows a nonmonotonic variation with viscosity ratio, in contrast to self-diffusivity computed using pairwise computation. The difference might point to an intrinsic difference in physics underlying the two diffusivities. Alternatively, it also might arise from drops not reaching equilibrium deformation in the period after one interaction and before the next—an effect absent in the pairwise simulation used for the computation of self-diffusivity. We offer a qualitative explanation of the nonmonotonic variation by relating it to average nonmonotonic drop deformation with increasing viscosity ratio. We also provide empirical correlations of the collective diffusivity as a function of viscosity ratio and capillary number.

Nonlinear internal waves propagating in a quasi-two-layer stratification with a shear background current are investigated by means of high-resolution direct numerical simulations. The simulations are performed in a two-dimensional rectangular domain, representing a laboratory-scale tank of finite length. The internal waves are generated using the lock-release mechanism, while the background shear currents are induced by basin-scale internal seiches created using a tilted tank suddenly returned to the untilted configuration. Both mechanisms are readily realizable in a laboratory environment. Depending on the configuration of the initial density profile, both internal solitary-like waves (ISWs) and finite-amplitude dispersive wave trains (DWTs) may be observed in the simulations. The ISWs are observed when the pycnocline is located relatively far away from the middepth and/or the background shear is oriented against the wave-induced velocity shear. Comparison of the waveforms observed in the simulations to those described by the fully nonlinear Dubreil-Jacotin–Long (DJL) theory suggests that these waves are indeed solitary-like, implying that laboratory experiments of ISWs propagating in a shear background current can be realized in a relatively straightforward manner. As the pycnocline approaches the middepth and/or the background shear reverses its polarity and increases its magnitude, the waves tend to have a smaller amplitude and a larger half-width. When a wave's amplitude is less than approximately 3% of its half-width, the dispersive effect becomes dominant and the wave loses its solitary-like form. In this case, a finite-amplitude DWT is formed instead, in which the leading wave has a reduced propagation speed from the ISW propagation speed. Analysis based on the DJL theory suggests that exact solitary waves of similar amplitude in such a background environment have an extremely large half-width that is on the same order of magnitude as the finite length of the simulation domain, implying that their formation cannot be supported in a laboratory environment.

Horizontal convection is studied numerically in two dimensions for the case of an infinite Prandtl number and shear-free boundary conditions on all sides. Three different temperature distributions are applied at the top boundary. Steady states become unstable, giving way to unsteady flow with increasing Rayleigh number Ra. The transition to time-varying flow depends on the heating configuration. For $\text{Ra}\ge 5\times {10}^9$ simulations show a time-dependent flow behavior for a linearly varying temperature at the top, whereas other temperature distributions show this transition at even lower Rayleigh numbers. The scaling laws of heat flow and circulation strength are investigated and the results indicate that both are changed by the upcoming time-dependent flow. While in the steady boundary layer flow regime the heat flux scales as ${\text{Ra}}^{1/5}$, the exponent increases to a value of approximately $0.25$ in the unsteady regime. A similar transition is observed for the circulation strength. The observed scaling laws are in good agreement with established theories in both the steady and unsteady regimes. After the transition, the flow shows a periodic oscillation of the boundary layer circulation, which turns into a more irregular flow behavior as the Rayleigh number is increased further. We conclude that, even for an infinite Prandtl number, horizontal convection exhibits unsteady flow at sufficiently high Rayleigh numbers and that this flow resembles the turbulent circulation at low Prandtl numbers.

The transport of colloids in porous media is governed by deposition on solid surfaces and pore-scale flow variability. Classical approaches, like the colloid filtration theory (CFT), do not capture the behaviors observed experimentally, such as nonexponential steady-state deposition profiles and heavy tailed arrival time distributions. In the framework of CFT a key assumption is that the colloid attachment rate $k$ is constant and empirically estimated by a posteriori macroscopic data fitting. We propose a stochastic model that explicitly accounts for variability of the pore-scale transport and attachment properties. Colloidal motion is modeled as a sequence of displacements whose velocity and extension are statistically distributed. Colloidal depositions are modeled as random events whose statistics are determined by flow velocity, pore size, and attachment rate. The resulting deposition profiles are, in general, nonexponential and can be predicted based on the disorder distributions.

Recent discoveries have allowed a mathematical and computational foundation for understanding the rheology of particles suspended in viscoelastic fluids. We employ these new tools to understand the extensional rheology of such suspensions. We accomplish this by first calculating the renormalized particle contribution to the extensional viscosity in such a suspension in the dilute particle limit over a wide range of extensional Weissenberg number and Hencky strain. The models we use for the suspending fluids are the simplest dumbbell models—the Oldroyd-B, FENE-P, and Giesekus models—such that our results are general for polymer solutions which exhibit strong strain hardening at values of the Weissenberg number above those which engender the coil-stretch transition, $\mathrm{Wi}\ge 0.5$. We demonstrate that the effect of particles on the “extra elongational viscosity” relative to the fluid is nonmonotonic in strain (increasing for small strain and then decreasing for large strain). Thus at a fixed strain, the particle “extra viscosity” relative to the fluid may increase or decrease with Wi. We demonstrate that this interesting behavior is due to the interplay between the two contributions of the “particle-induced fluid stress” (PIFS) and the “stresslet” to the extra viscosity. The contribution of the particle-induced fluid stress to the suspension viscosity increases at small strain but plateaus and then decreases at higher values of the strain. Thus, at small strain the local velocity gradients near a particle increase the polymer stretch, while for greater strain, polymers which have undergone the coil-stretch transition collapse in the neighborhood of a given particle. On the other hand, the stresslet contribution to the viscosity relative to the fluid decreases monotonically as the polymer stretch surrounding a given particle “shields” the particle and thus reduces the local surface tractions. Beyond Hencky strain of approximately 2 the decreasing value of the stresslet coupled with the plateauing of the PIFS, causes the overall reduction in the particle-induced extra viscosity relative to that of the fluid.

The motion of finite-size particles in thermocapillary flow is investigated experimentally. Particular attention is paid to the unique phenomenon known as a particle accumulation structure (PAS), where particles gather in a solidlike structure in a half-zone liquid bridge. We study the correlation between the PAS and the flow field due to the hydrothermal wave instability by measuring the surface temperature and dynamic surface deformation. An image processing method is developed in order to verify the physical model of the particle behavior. The structures of the particle depletion zone, the PAS, and the toroidal core are observed clearly using long-term averaged images in the rotating frame of reference. The results of precise experiments reveal that the correlation between the PAS and the temperature field over the free surface is strongly dependent on the test fluid. Reconstruction of the PASs is achieved by applying particle tracking velocimetry. Time-series analysis of the particle position indicate that the ratio of the hydrothermal wave frequency to the turnover frequency of the particle is independent of the intensity of the thermocapillary-driven convection.

A Bayesian optimization framework is developed to optimize low-amplitude wall-normal blowing control of a turbulent boundary-layer flow. The Bayesian optimization framework determines the optimum blowing amplitude and blowing coverage to achieve up to a 5% net-power saving solution within 20 optimization iterations, requiring 20 direct numerical simulations (DNS). The power input required to generate the low-amplitude wall-normal blowing is measured experimentally for two different types of blowing device and is used in the simulations to assess control performance. Wall-normal blowing with amplitudes of less than 1% of the free-stream velocity generate a skin-friction drag reduction of up to 76% over the control region, with a drag reduction which persists for up to $650{\delta }_0$ downstream of actuation (where ${\delta }_0$ is the boundary-layer thickness at the start of the simulation domain). It is shown that it is the slow spatial recovery of the turbulent boundary-layer flow downstream of control which generates the net-power savings in this study. The downstream recovery of the skin-friction drag force is decomposed using the Fukagata-Iwamoto-Kasagi (FIK) identity, which shows that the generation of the net-power savings is due to changes in contributions to both the convection and streamwise development terms of the turbulent boundary-layer flow.

The journey of mammalian spermatozoa in nature is well known to be reliant on their individual motility. Often swimming in crowded microenvironments, the progress of any single swimmer is likely dependent on their interactions with other nearby swimmers. While the complex dynamics of lone spermatozoa have been well-studied, the detailed effects of hydrodynamic interactions between neighbors remain unclear, with inherent nonlinearity in the pairwise dynamics and potential dependence on the details of swimmer morphology. In this study, we will attempt to elucidate the pairwise swimming behaviors of virtual spermatozoa, forming a computational representation of an unbound swimming pair and evaluating the details of their interactions via a high-accuracy boundary element method. We have explored extensive regions of parameter space to determine the pairwise interactions of synchronized spermatozoa, with synchronized swimmers often being noted in experimental observations, and have found that two-dimensional reduced autonomous dynamical systems capture the anisotropic nature of the swimming speed and stability arising from near-field hydrodynamic interactions. Focusing on two initial configurations of spermatozoa, namely those with swimmers located side-by-side or above and below one another, we have found that side-by-side cells attract each other, and the trajectories in the phase plane are well captured by a recently proposed coarse-graining method of microswimmer dynamics via superposed regularized Stokeslets. In contrast, the above-below pair exhibit a remarkable stable pairwise swimming behavior, corresponding to a stable configuration of the plane autonomous system with swimmers lying approximately parallel to one another. At further reduced swimmer separations, we additionally observe a marked increase in swimming velocity over individual swimmers in the bulk, potentially suggesting a competitive advantage to cooperative swimming. These latter observations are not captured by the coarse-grained regularized Stokeslet modeling or simple singularity representations, emphasizing the complexity of near-field cell-cell hydrodynamic interactions and their inherent anisotropy.

Controlled gliding is one of the most energetically efficient modes of transportation for natural and human powered fliers. Here we demonstrate that gliding and landing strategies with different optimality criteria can be identified through deep-reinforcement-learning without explicit knowledge of the underlying physics. We combine a two-dimensional model of a controlled elliptical body with deep-reinforcement-learning (D-RL) to achieve gliding with either minimum energy expenditure, or fastest time of arrival, at a predetermined location. In both cases the gliding trajectories are smooth, although energy/time optimal strategies are distinguished by small/high frequency actuations. We examine the effects of the ellipse's shape and weight on the optimal policies for controlled gliding. We find that the model-free reinforcement learning leads to more robust gliding than model-based optimal control strategies with a modest additional computational cost. We also demonstrate that the gliders with D-RL can generalize their strategies to reach the target location from previously unseen starting positions. The model-free character and robustness of D-RL suggests a promising framework for developing robotic devices capable of exploiting complex flow environments.

We utilize the externally forced linearized Navier-Stokes equations to study the receptivity of pre-transitional boundary layers to persistent sources of stochastic excitation. Stochastic forcing is used to model the effect of free-stream turbulence that enters at various wall-normal locations and the fluctuation dynamics are studied via linearized models that arise from locally parallel and global perspectives. In contrast to the widely used resolvent analysis that quantifies the amplification of deterministic disturbances at a given temporal frequency, our approach examines the steady-state response to stochastic excitation that is uncorrelated in time. In addition to stochastic forcing with identity covariance, we utilize the spatial spectrum of homogeneous isotropic turbulence to model the effect of free-stream turbulence. Even though locally parallel analysis does not account for the effect of the spatially evolving base flow, we demonstrate that it captures the essential mechanisms and the prevailing length scales in stochastically forced boundary layer flows. On the other hand, global analysis, which accounts for the spatially evolving nature of the boundary layer flow, predicts the amplification of a cascade of streamwise scales throughout the streamwise domain. We show that the flow structures that are extracted from a modal decomposition of the resulting velocity covariance matrix can be closely captured by conducting locally parallel analysis at various streamwise locations and over different wall-parallel wave-number pairs. Our approach does not rely on costly stochastic simulations and it provides insight into mechanisms for perturbation growth, including the interaction of the slowly varying base flow with streaks and Tollmien-Schlichting waves.

The contraction of liquid filaments plays a central role in applications as diverse as inkjet printing and atomization where the formation of monosized primary drops is desirable, but that of smaller satellite droplets is typically undesirable. In order to avoid polydisperse drop sizes and/or minimize the deleterious consequences of satellite droplet production, contracting or retracting filaments formed during drop pinch-off and liquid atomization should either contract to spheres without further breakup or break into equally sized drops. Therefore, given the importance of understanding the fate of contracting filaments, high-accuracy numerical simulations are used to analyze the retraction of Newtonian liquid filaments in a passive outer fluid. Previously, Notz and Basaran [Notz and Basaran, J. Fluid Mech. 512, 223 (2004)] provided a comprehensive description of the dynamics of low-viscosity filaments and subsequent studies have helped to advance our understanding of the response of moderate- to high-viscosity filaments. However, as it is shown here, combining the findings of all of these separate works does not provide a complete picture of the rich physics of filament contraction. In this work, a comprehensive approach is adopted to further refine and expand upon earlier works to provide a complete phase diagram of filament retraction dynamics by considering the entire possible range of fluid properties and initial aspect ratios for filaments undergoing contraction. In doing so, a different mode of breakup, herein referred to as capillary wave breakup, is discovered and shown to be the dominant mode of pinch-off for moderately viscous filaments. The mechanism for the breakup of filaments of moderately viscous fluids is detailed along with a thorough investigation of the already established modes of breakup for nearly inviscid and highly viscous filaments.

We analyze the near-rolling motion of two-dimensional nonwetting drops down a gently inclined plane. Inspired by the scaling analysis of Mahadevan and Pomeau [Phys. Fluids 11, 2449 (1999)], we focus upon the limit of small Bond numbers, $B\ll 1$, where the drop shape is nearly circular and the internal flow is approximately a rigid-body rotation except close to the flat spot at the base of the drop. Our analysis reveals that the leading-order dissipation is contributed by both the flow in the flat-spot region and the correction to rigid-body rotation in the remaining liquid domain. The resulting leading-order approximation for the drop velocity $\mathcal{U}$ is given by$\mu \mathcal{U}/\gamma \sim \alpha /2Bln\frac{1}{B}$, wherein $\mu$ is the liquid viscosity, $\gamma$ the interfacial tension, and $\alpha$ the inclination angle.

The action of spatial modulation of the Marangoni number, corresponding to the heat source or sink at the interface on wave patterns, has been studied. The analysis is carried out in the lubrication approximation. Numerical simulations are performed by means of a finite-difference method. It is found that in the case of multistability, the spatial heat release modulation can be used for the creation of a pattern with a definite spatial structure. It is shown that the violation of the reflection symmetry of the pattern by its spatial modulation can lead to a quasiperiodic temporal evolution.

Leidenfrost drops exhibit complex shape oscillations on curved substrates. Experiments are conducted using six different liquids on three substrate geometries: curved, targeted, and toroidal, and the droplet dynamics are characterized by their oscillation amplitude. Moderate amplitude star oscillations are most common and a number of previously unreported modes are observed. Large drops exhibit constant frequency oscillations, irrespective of mode number, consistent with prior observation. Small amplitude polygonal oscillations exhibit similar dynamics to star oscillations but with twice the frequency and half the wavelength. A large amplitude $n=2$ dominates mode selection by suppressing higher-order modes over a large range of drop sizes in the most viscous liquids. Using the targeted substrate geometry allows for observation of those “dormant” modes. Two modes with unique oscillation frequencies can be simultaneously excited on the same drop with a frequency ratio that takes on a fixed value.

Flow instabilities can occur in a fluid system with two components that have significantly different diffusivities and that have opposite effects on the fluid density, as is the scenario in traditional double-diffusive convection. Here, we experimentally show that an oil-in-water emulsion exposed to salt concentration gradients generates a flowerlike pattern driven by vertical and azimuthal instabilities. We also report numerical and analytical studies to elaborate on the mechanism, the instability criteria, and the most unstable modes that determine the details of the observed patterns. We find that the instability is driven by buoyancy and stems from the differential transport between the dissolved salt and the suspended oil droplets, which have opposing effects on the density of the medium. Consequently, we identify a criterion for the development of the instability that involves the relative densities and concentrations of the salt and oil droplets. We also argue that the typical wave number of the pattern formed scales with the Péclet number of the salt, which here is equivalent to the Rayleigh number since the flow is driven by buoyancy. We find good agreement of these predictions with both experiments and numerical simulations.

This paper presents the results of a scaling study of bubble and drop plumes in a stratified ambient. Use is made of a standard integral model of the top-hat type, which can be reduced to one of the Gaussian type by a simple transformation. The focus of the work is on the effects of the dissolving material on the plume dynamics. It is pointed out that, for a drop plume, the loss of buoyancy due to dissolution can be compensated by a lightening of the ambient liquid associated with the dissolved drop material, or even aggravated if the density of the solution is greater than that of the undissolved drops. For bubbles, these effects are compounded by the volume expansion due to the falling hydrostatic pressure. This process is not important in deep water, where the peel height is smaller than the water depth, but can be significant in shallow water, where the two may be comparable. With a focus on the analysis of a point-source, three important parameters are identified. The first one compares the drop/bubble dissolution rate with the rise time to the neutral height (the level at which the plume density equals the ambient density), the second one accounts for the effect of the dissolved material on the liquid density, and the third one is the drop/bubble rise velocity compared with the characteristic plume velocity.

We study free surface gravity waves in the presence of a depth-dependent shear current with a nonzero vorticity gradient. The evolution of weakly nonlinear, narrow-band wave packets is governed by a nonlinear Schrödinger equation. When dispersion and nonlinearity are of the same (opposite) signs, modulation instability will be present (absent), and rogue waves represented by breathers can (cannot) occur, respectively. For irrotational flows, rogue waves only occur for sufficiently deep water, or more precisely, $kh>1.363$, where $k$ is the wave number of the carrier envelope and $h$ is the water depth. While the irrotational and linear shear current cases have been treated previously, the present study demonstrates the importance of a shear current with a nonzero vorticity gradient. For a concave current, the threshold for modulation instability of a wave packet moving with (or against) the shear current will reduce (or increase) the numerical bound of 1.363, respectively. The opposite will hold for the case of a convex current. The growth rate of a disturbance will also be larger for concave currents in comparison with convex and linear currents. The streamline patterns for the transient case of rogue waves will be illustrated for the simple case of a linear shear current. Shear currents near the sea surface are known to have a profound influence on the dynamics of wind-generated waves, and the present investigation bears on this.

In this paper, the electroconvective flow induced by the unipolar charge injection is extended from single-phase dielectric liquid to the solid-liquid interaction problem. The physical model with fully coupled mathematical equations is built in the liquid, solid, and interface for both the Ohmic and non-Ohmic solid models. An improved lattice Boltzmann model (LBM) is developed with three lattice Boltzmann equations for Poisson's equation, charge conservation equation, and Navier-Stokes equations, respectively. Our codes are first validated by the analytical solutions at the hydrostatic state. It is found that the LBM can well reproduce the discontinuous changes of electrical field and charge density at the interface and agrees well with the analytical results. Then, simulations are conducted under different governing parameters and interface position $f_l$. Results show that the bifurcation of electroconvection in the presence of the solid-liquid interface is still of subcritical type, but both the linear and finite amplitude stability criteria increase due to a voltage drop happening at the solid phase. Besides, the stability criterion expressed by the electrical Rayleigh number $T_c$ increases as the permittivity ratios ${\varepsilon }_r$ and the mobility ratios $K_r$ increase, but $T_c$ decreases with the increasing of dimensionless electric conductivity S and the interface position $f_l$.