An object moving through a plane interface into a fluid deforms the interface in such a way that fluid from one side of the interface is entrained into the other side, a phenomenon known as Darwin's drift. We investigate this phenomenon experimentally using a disk which is started exactly at the interface of two immiscible fluids, namely, oil and water. First, we observe that due to the density difference between the two fluids the deformation of the interface is influenced by gravity and show that there exists a time window of universal behavior. Second, we show by comparing with boundary integral simulations that, even though the deformation is universal, our results cannot be fully explained by potential flow solutions. We attribute this difference to the starting vortex, which is created in the wake of the disk. Besides contributing significantly to entrainment directly, the vortex also influences the interface deformation due to Darwin's drift. Universal behavior is preserved, however, because the size and strength of the vortex shows the same universality as the potential flow solution.

We demonstrate that porous nematic microfluidics is a potential route for the generation of nematic umbilic defects and regular umbilic defect lattices. By using numerical modeling we show that the mutual (backflow) coupling between the flow velocity and the orientation director field of the nematic liquid crystal leads to the formation of positive umbilic defects at local peaks and to the formation of negative umbilic defects at the local saddles in the flow profile. The number of flow peaks and the index of the flow saddles (i.e., the number of the valleys) are shown to be directly related to the strength of the umbilic defect, effectively relating the two fields at the geometrical level. The regular arrangement of the barriers in the porous channels is demonstrated to lead to the formation of regular lattices of umbilic defects, including square, triangular, and even kagome lattices. Experimental realization of such systems is discussed, with particular focus on microfluidic-tunable birefringent photonic band structures and lattices.

The lattice-Boltzmann method is used to compute the flow of a particle suspension of dilute volume fraction $\varphi \le 0.08$ around obstacles. The work focuses on the interaction of particles with the obstacle and the flow behavior in and around the recirculating wake behind the obstacle. Motivated by microfluidic experiments of dilute suspension flow over obstacles of circular, square, and other shapes showing depletion of particles in the wake behind the obstacle, we focus on the entry and exit from the wake for isolated particles, small groups of particles, and a suspension of many particles. All work is at conditions well below the Reynolds number for the transition to unsteady flow and vortex shedding. The dynamics of an isolated particle inside the recirculating wake flow exhibits motion toward a limit cycle near the outer boundary of the wake. By increasing particle density above the fluid and increasing the size of the particle, we determine the upper limits of these variables yielding a limit cycle trajectory. Hydrodynamic interaction between particles in a many-particle suspension is shown to lead to exchange of particles between the wake zone and free stream.

In the study of convective heat and mass transfer from a particle, key quantities of interest are usually the average rate of transfer and the mean distribution of the scalar (i.e., temperature or concentration) at the particle surface. Calculating these quantities using conventional equations requires detailed knowledge of the scalar field, which is available predominantly for problems involving uniform scalar and flux boundary conditions. Here we derive a reciprocal relation between two diffusing scalars that are advected by oppositely driven Stokes or potential flows whose streamline configurations are identical. This relation leads to alternative expressions for the aforementioned average quantities based on the solution of the scalar field for uniform surface conditions. We exemplify our results via two applications: (i) heat transfer from a sphere with nonuniform boundary conditions in Stokes flow at small Péclet numbers and (ii) extension of Brenner's theorem for the invariance of heat transfer rate to flow reversal.

We report the space-time characteristics of the wall-pressure fluctuations and wall shear-stress fluctuations from wall-modeled large eddy simulation (WMLES) of a turbulent channel flow at ${\text{Re}}_{\tau }=2000$. Two standard zonal wall models (equilibrium stress model and nonequilibrium model based on unsteady RANS) are employed, and it is shown that they yield similar results in predicting these quantities. The wall-pressure and wall shear-stress fields from WMLES are analyzed in terms of their r.m.s. fluctuations, spectra, two-point correlations, and convection velocities. It is demonstrated that the resolution requirement for predicting the wall-pressure fluctuations is more stringent than that for predicting the velocity. At least $\delta /\mathrm{\Delta }x>20$ and $\delta /\mathrm{\Delta }z>30$ are required to marginally resolve the integral length scales of the pressure-producing eddies near the wall. Otherwise, the pressure field is potentially aliased. Spurious high wave number modes dominate in the streamwise direction, and they contaminate the pressure spectra leading to significant overprediction of the second-order pressure statistics. When these conditions are met, the pressure statistics and spectra at low wave number or low frequency agree well with the DNS and experimental data. On the contrary, the wall shear-stress fluctuations, modeled entirely through the RANS-based wall models, are largely underpredicted and relatively insensitive to the grid resolution. The short-time, small-scale near-wall eddies, which are neither resolved nor modeled adequately in the wall models, seem to be important for accurate prediction of the wall shear-stress fluctuations.

Recent studies have shown Townsend's attached eddy hypothesis to be a promising basis for modeling the velocity statistics in the logarithmic region of turbulent wall flows. Accordingly, the attached eddy model is able to reliably estimate the functional forms of the mean velocity, second-order moments of the velocity fluctuations, and recently structure functions and higher-order moments of the velocity fluctuations. However, detailed quantitative comparisons with experimental results reveal differences, particularly for the higher-order moments. Specifically, the predicted flatness (kurtosis) is found to be invariably greater than 3 (i.e., super-Gaussian behavior) for all velocity components, while experimental results show sub-Gaussian behavior for the streamwise component of velocity. In this study, we show that this and other discrepancies can be resolved by considering the finite space occupied by each eddy. Earlier models had allowed each eddy to be perfectly randomly located, with no consideration for the locations of neighboring eddies (in other words, their locations can be described as a Poisson point process). Here we investigate the effect of mandating a minimum distance between any two eddies of the same height. We demonstrate that this spatial exclusion, when combined with an experimentally observed shape for the representative eddy, produces predictions that are now in better agreement with experimental observations. In particular, sub-Gaussian behavior is now attained for the streamwise component, while super-Gaussian behavior is maintained for the other velocity components, qualitatively matching experimental findings. Therefore, our findings infer that spatial exclusion between eddies is likely to play an important role in the laws that govern their spatial arrangement, which is likely to be more disperse than a Poisson point process.

The flow in a rapidly rotating cylinder forced to precess through a nutation angle $\alpha$ is investigated numerically, keeping all parameters constant except $\alpha$, and tuned to a triadic resonance at $\alpha =1^{\circ }$. When increasing $\alpha$, the flow undergoes a sequence of well-characterized bifurcations associated with triadic resonance, involving heteroclinic and homoclinic cycles, for $\alpha$ up to about $4^{\circ }$. For larger $\alpha$, we identify two chaotic regimes. In the first regime, with $\alpha$ between about $4^{\circ }$ and ${27}^{\circ }$, the bulk flow retains remnants of the helical structures associated with the triadic resonance, but there are strong nonlinear interactions between the various azimuthal Fourier components of the flow. For the larger $\alpha$ regime, large detuning effects lead to the triadic resonance dynamics being completely swamped by boundary layer eruptions. The azimuthal mean flow at large angles results in a large mean deviation from solid-body rotation and the flow is characterized by strong shear at the boundary layers with temporally chaotic eruptions.

In a recent Letter [Phys. Rev. Lett. 116, 138302 (2016)] Würger calculated the thermophoretic velocity of a colloidal sphere towards a solid boundary due to an imposed temperature gradient perpendicular to that wall in the limit where the particle-wall separation distance $h$ is small compared to the particle radius $a$. Würger obtained the approximation $u/u_0=3(h/a)[ln(a/h)-2.25]$ for the particle velocity $u$ towards the wall, $u_0$ being the corresponding velocity in the bulk. In deriving that approximation an ad hoc procedure was employed to circumvent the calculation of a diverging integral that emerges in the course of a lubrication analysis. We show here how to systematically address this problem, obtaining the approximation $u/u_0=3(h/a)[ln(a/h)+0.1087]$.

Transition to turbulence in pipe flow is investigated experimentally using a temporally resolved dual-plane particle image velocimetry approach, at a Reynolds number of 3440. The flow is analyzed using proper orthogonal decomposition and it is shown that the flow can be divided into two regions: a pseudolaminar region governed by the presence of azimuthally steady traveling waves, and turbulent slugs. The evolution of the structures within the slugs is identified by using the temporally resolved data along with the dual-plane velocity field. These structures are shown to be remarkably similar to the large-scale motions found in fully turbulent flows, with a streamwise and spatiotemporal extent about four pipe radii. The transition between structures is characterized by the detachment and decay of an old structure and the initiation of a new structure at the wall.

Townsend [Townsend, The Structure of Turbulent Shear Flow (Cambridge University Press, Cambridge, UK, 1976)] hypothesized that the logarithmic region in high-Reynolds-number wall-bounded flows consists of space-filling, self-similar attached eddies. Invoking this hypothesis, we express streamwise velocity fluctuations in the inertial layer in high-Reynolds-number wall-bounded flows as a hierarchical random additive process (HRAP): $u_z^{+}={\sum }_{i=1}^{N_z}{a}_i$. Here $u$ is the streamwise velocity fluctuation, $+$ indicates normalization in wall units, $z$ is the wall normal distance, and $a_i$'s are independently, identically distributed random additives, each of which is associated with an attached eddy in the wall-attached hierarchy. The number of random additives is $N_z\sim ln(\delta /z)$ where $\delta$ is the boundary layer thickness and ln is natural log. Due to its simplified structure, such a process leads to predictions of the scaling behaviors for various turbulence statistics in the logarithmic layer. Besides reproducing known logarithmic scaling of moments, structure functions, and correlation function $〈u_z\left(x\right)u_z(x+r)〉$, new logarithmic laws in two-point statistics such as ${\left[\frac{3}{2}〈u_z^2\left(x\right)u_z^2(x+r)〉-\frac{1}{2}〈u_z^4\left(x\right)〉\right]}^{1/2}, {\left[\frac{5}{2}〈u_z^3\left(x\right)u_z^3(x+r)〉-\frac{3}{2}〈u_z\left(x\right)u_z^5(x+r)〉\right]}^{1/3}$, etc. can be derived using the HRAP formalism. Supporting empirical evidence for the logarithmic scaling in such statistics is found from the Melbourne High Reynolds Number Boundary Layer Wind Tunnel measurements. We also show that, at high Reynolds numbers, the above mentioned new logarithmic laws can be derived by assuming the arrival of an attached eddy at a generic point in the flow field to be a Poisson process [Woodcock and Marusic, Phys. Fluids 27, 015104 (2015)]. Taken together, the results provide new evidence supporting the essential ingredients of the attached eddy hypothesis to describe streamwise velocity fluctuations of large, momentum transporting eddies in wall-bounded turbulence, while observed deviations suggest the need for further extensions of the model.

The effects of viscoelasticity on drop impact and spreading on a flat solid surface are studied computationally using a finite-difference–front-tracking method. The finitely extensible nonlinear elastic–Chilcott-Rallison model is used to account for the fluid viscoelasticity. It is found that viscoelasticity favors advancement of contact line during the spreading phase, leading to a slight increase in the maximum spreading, in agreement with experimental observations [Huh, Jung, Seo, and Lee, Microfluid. Nanofluid. 18, 1221 (2015)]. However, in contrast with the well-known antirebound effects of polymeric additives, the viscoelasticity is found to enhance the tendency of the drop rebound in the receding phase. These results suggest that the antirebound effects are mainly due to the polymer-induced modification of wetting properties of the substrate rather than the change in the material properties of the drop fluid. A model is proposed to test this hypothesis. It is found that the model results in good qualitative agreement with the experimental observations and the antirebound behavior can be captured by the modification of surface wetting properties in the receding phase.

Unidirectional flow is the simplest phenomenon of fluid mechanics. Its mathematical description, the Dirichlet problem for Poisson's equation in two dimensions with constant forcing, arises in many physical contexts, such as the torsion of elastic beams, first solved by de Saint-Venant for complex shapes. Here the literature is unified and extended by identifying 17 physical analogies for unidirectional flow and describing their common mathematical structure. Besides classical analogies in fluid and solid mechanics, applications are discussed in stochastic processes (first passage in two dimensions), pattern formation (river growth by erosion), and electrokinetics (ion transport in nanochannels), which also involve Poisson's equation with nonconstant forcing. Methods are given to construct approximate geometries that admit exact solutions, by adding harmonic functions to quadratic forms or by truncating eigenfunction expansions. Exact solutions for given geometries are also derived by conformal mapping. We prove that the remarkable geometrical interpretation of Poiseuille flow in an equilateral triangular pipe (the product of the distances from an interior point to the sides) is only shared by parallel plates and unbounded equilateral wedges (with the third side hidden behind the apex). We also prove Onsager reciprocity for linear electrokinetic phenomena in straight pores of arbitrary shape and surface charge, based on the mathematics of unidirectional flow.

Until recently, based on the analysis pertaining to a perfectly charge selective interface, electro-convective instability in concentration polarization was attributed to the nonequilibrium mechanism related to the extended space charge which forms next to that of the electric double layer near the limiting current. More recently it was shown that imperfect charge selectivity of the interface makes equilibrium instability possible, driven by either equilibrium electro-osmosis or bulk electro-convection, or both. In this paper we identify and analyze the major surface and bulk factors affecting the electro-convective instability. These factors, some known previously under the names of diffusio-osmosis, electro-osmosis, or bulk electro-convection, and some newly identified in this paper are manifestations of the electric force and pressure gradient, balanced by the viscous force acting in various locations in solution. The contribution of these factors to hydrodynamic stability in concentration polarization is analyzed for a varying charge selectivity of the interface.

We report experimental results on nonlinear wave coupling in surface wave turbulence on water at scales close to the crossover between surface gravity waves and capillary waves. We study three-wave correlations either in the frequency domain or in the wave-vector domain. We observe that in a weakly nonlinear regime, the dominant nonlinear interactions correspond to waves that are collinear or close to collinear. Although the resonant coupling of pure gravity waves is supposed to involve four waves, at the capillary crossover we observe a nonlocal coupling between a gravity wave and two capillary waves. Furthermore, nonlinear spectral spreading permits three-gravity wave coupling. These observations raise the question of the relevance of these processes in the oceanographic context and in particular the range of frequencies of gravity waves that may be impacted.

We study the linear stability of a laterally extended flat two-layer liquid film under the influence of external vertical vibration. The first liquid layer rests on a vibrating solid plate and is overlaid by a second layer of immiscible fluid with deformable upper surface. Surface waves, excited as the result of the Faraday instability, can be characterized by a time-dependent relative amplitude of the displacements of the liquid-liquid and the liquid-gas interfaces. The in-phase displacements are associated with a zigzag (barotropic) mode and the antiphase displacement corresponds to the varicose thinning mode. We numerically determine the stability threshold in the vibrated two-layer film and compute the dispersion relation together with the decay rates of the surface waves in the absence of vibration. The in-phase and the antiphase displacements are strongly coupled in the vibrated system. The interplay between the Faraday and the Rayleigh-Taylor instabilities in the system with heavier fluid on top of a lighter fluid is analyzed.

We present a theoretical framework to model the behavior of a concentrated colloidal dispersion confined inside a spherical cavity. Prior attempts to model such behavior were limited to a single enclosed particle and attempts to enlarge such models to two or more particles have seen limited success owing to the challenges of accurately modeling many-body and singular hydrodynamic interactions. To overcome these difficulties, we have developed a set of hydrodynamic mobility functions that couple particle motion with hydrodynamic traction moments that, when inverted and combined with near-field resistance functions, form a complete coupling tensor that accurately captures both the far-field and near-field physics and is valid for an arbitrary number of spherical particles enclosed by a spherical cavity of arbitrary relative size $a/R$, where $a$ and $R$ are the particle and cavity size, respectively. This framework is then utilized to study the effect of spherical confinement on the self- and entrained motion of the colloids, for a range of particle-to-cavity size ratios. The self-motion of a finite-size enclosed particle is studied first, recovering prior results published in the literature: The hydrodynamic mobility of the particle is greatest at the center of the cavity and decays as $(a/R)/(1-y^2)$, where $y$ is the particle distance to the cavity center. Near the cavity wall, the no-slip surfaces couple strongly and mobility along the cavity radius vanishes as $\xi \equiv R-(a+y)$, where $y$ is center-to-center distance from particle to cavity. Corresponding motion transverse to the cavity radius vanishes as ${\left[\text{ln}(1/\xi )\right]}^{-1}$. The effect of confinement on entrainment of a particle in the flow created by the motion of others is also studied, where we find that confinement exerts a qualitative effect on the strength and anisotropy of entrainment of a passive particle dragged by the flow of a forced particle. As expected, entrainment strength decays with increased distance from the forced particle. Surprisingly, however, there is a separation beyond which entrainment changes sign. For some configurations, the passive particle is dragged along with the forced particle, and at others, it is driven in the opposite direction, consistent with observations of recirculating flow and reverse particle migration in eukaryotic cells. The mobility functions presented here can be utilized to model the motion of any number of enclosed particles, making them ideal for use in dynamic simulation.

We investigate the existence of multiple turbulent states in highly turbulent Taylor-Couette flow in the range of $\mathrm{Ta}={10}^{11}$ to $9\times {10}^{12}$ by measuring the global torques and the local velocities while probing the phase space spanned by the rotation rates of the inner and outer cylinders. The multiple states are found to be very robust and are expected to persist beyond $\mathrm{Ta}={10}^{13}$. The rotation ratio is the parameter that most strongly controls the transitions between the flow states; the transitional values only weakly depend on the Taylor number. However, complex paths in the phase space are necessary to unlock the full region of multiple states. By mapping the flow structures for various rotation ratios in a Taylor-Couette setup with an equal radius ratio but a larger aspect ratio than before, multiple states are again observed. Here they are characterized by even richer roll structure phenomena, including an antisymmetrical roll state.

We report an experimental study of a magnetic liquid drop deposited on a superhydrophobic substrate and subjected to vertical vibrations in the presence of a static magnetic field. It is well known that a flattened drop of usual liquid displays oscillating lobes at its periphery when vibrated. By adding ferromagnetic nanoparticles to a water drop and varying the strength of the magnetic field, we are experimentally able to efficiently tune the resonant frequencies of the drop. By using conservation energy arguments, we show that the magnetic field contribution is equivalent to adding an effective negative surface tension to the drop. Our model is found to be in good agreement with the experiments with no fitting parameter.

The role of solid-solid contacts on the transition between reversible and irreversible dynamics occurring in periodically sheared suspensions is investigated experimentally by modifying the particle roughness. Smoother particles lead to a larger critical strain amplitude. A geometrical model based on the assumption that colliding particles produce irreversibility is derived. The model, which considers a quasiparticle having a strain- and roughness-dependent effective volume, successfully reproduces the measured values of the critical strain amplitude as functions of the volume fraction and particle roughness.

Miscible liquids commonly contact one another in natural and technological situations, often in the proximity of a solid substrate. In the scenario where a drop of one liquid finds itself on a solid surface and immersed within a second, miscible liquid, it will spread spontaneously across the surface. We show experimental findings of the spreading of sessile drops in miscible environments that have distinctly different shape evolution and power-law dynamics from sessile drops that spread in immiscible environments, which have been reported previously. We develop a characteristic time to scale radial data of the spreading sessile drops based on a drainage flow due to gravity. This time scale is effective for a homologous subset of the liquids studied. However, it has limitations when applied to significantly chemically different, yet miscible, liquid pairings; we postulate that the surface energies between each liquid and the solid surface becomes important for this other subset of the liquids studied. Initial experiments performed with pendant drops in miscible environments support the drainage flow observed in the sessile drop systems.