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.

We have uncovered a drop rebound regime, characteristic of highly viscous liquids impacting tilted sublimating surfaces. Here the drops, rather than showing a slide, spread, recoil, and rebound behavior, exhibit a prompt tumbling rebound. As a result, glycerol surprisingly rebounds faster than three orders of magnitude less viscous water. When a viscous drop impacts a sublimating surface, part of its initial linear momentum is converted into angular momentum: Lattice Boltzmann simulations confirmed that tumbling owes its appearance to the rapid transition of the internal angular velocity prior to rebound to a constant value, as in a tumbling solid body.

By introducing the notion of an ideal large-scale filter, a formal statement is given of the hypothesis of the quasisteady quasihomogeneous nature of the interaction between the large and small scales in the near-wall part of turbulent flows. This makes the derivations easier and more rigorous. A method is proposed to find the optimal large-scale filter by multiobjective optimization, with the first objective being a large correlation between large-scale fluctuations near the wall and in the layer at a certain finite distance from the wall and the second objective being a small correlation between the small scales in the same layers. The filter is demonstrated to give good results. Within the quasisteady quasihomogeneous theory, expansions for various quantities are found with respect to the amplitude of the large-scale fluctuations. Including the higher-order terms improves the agreement with numerical data. Interestingly, it turns out that the quasisteady quasihomogeneous theory implies a dependence of the mean profile logarithmic-law constants on the Reynolds number. The main overall result of the present work is the demonstration of the relevance of the quasisteady quasihomogeneous theory for near-wall turbulent flows.

We study the motion of a hot particle in a viscous liquid at low Reynolds numbers, which is inspired by recent experiments with Brownian particles heated by a laser. The difference in temperature between a particle and the ambient fluid causes a spatial variation of the viscosity in the vicinity of the solid body. We derive a general analytical expression determining the force and the torque on a particle for low Péclet numbers by exploiting the Lorentz reciprocal theorem. For small temperature and viscosity variations, a perturbation analysis is implemented to evaluate the leading-order correction to the hydrodynamic force and torque on the particle. The results are applied to describe dynamics of a uniformly hot spherical particle and to spherical particles with a nonuniform surface temperature described by dipole and quadrupole moments. Among other results, we find for dipolar thermal fields that there is coupling of the translational and rotational motions when there are local viscosity variations; such coupling is absent in an isothermal fluid.

We explore the processes by which a dense set of small liquid droplets (a spray) evaporates in a dry, stirred gas phase. A dense spray of micron-sized liquid (water or ethanol) droplets is formed in air by a pneumatic atomizer in a closed chamber. The spray is conveyed in ambient air as a plume whose extension depends on the relative humidity of the diluting medium. Standard shear instabilities develop at the plume edge, forming the stretched lamellar structures familiar with passive scalars. Unlike passive scalars however, these lamellae vanish in a finite time, because individual droplets evaporate at their border in contact with the dry environment. Experiments demonstrate that the lifetime of an individual droplet embedded in a lamellae is much larger than expected from the usual $d^2$ law describing the fate of a single drop evaporating in a quiescent environment. By analogy with the way mixing times are understood from the convection-diffusion equation for passive scalars, we show that the lifetime of a spray lamellae stretched at a constant rate $\gamma$ is $t_v=\frac{1}{\gamma }ln\left(\frac{1+\varphi }{\varphi }\right)$, where $\varphi$ is a parameter that incorporates the thermodynamic and diffusional properties of the vapor in the diluting phase. The case of time-dependent stretching rates is examined too. A dense spray behaves almost as a (nonconserved) passive scalar.

Evaporation of a liquid on a hot solid can be exploited to induce both levitation and self-propulsion if the vapor flow under the liquid is made directional. Here we propose to force an anisotropic flow of vapor with a herringbone pattern at a solid surface. Liquids are found to self-propel on such hot materials, drawn by the underlying flow of vapor. The simplicity of the geometry allows us to predict the direction of vapor flow rectification, to quantitatively model the propulsion, and to discuss how the pattern must be designed to optimize both driving force and drop velocity.

The adhesion and detachment of a capsule on a solid boundary surface is studied via a combination of scaling theory and numerical simulation and the behavior is compared and contrasted with a vesicle. It is shown that the dominant physical property for both capsules and vesicles is the area dilation modulus $K_s$ of the membrane. The nonzero shear modulus $G_s$ for capsules increases the resistance to deformation and thus decreases slightly the equilibrium contact radius for an adhered capsule compared to an adhered vesicle. The detachment process in this study is due to an external axisymmetric flow. Unlike a rigid body that must be pulled away without change of shape, capsules (and vesicles) almost always detach dominantly by peeling in which the contact radius decreases but the minimum separation distance does not change until the final moments of detachment. Compared to a vesicle with the same $K_s$, a capsule maintains a more compact shape and is harder to elongate under a given external flow. Hence, the detachment process is slower for capsules compared to vesicles with the same $K_s$.

With the aim to predict energy transport by internal waves incident upon observed density staircases in the ocean, the theory for internal wave tunneling through a single finite-size mixed region is extended to predict the transmission of internal waves through a piecewise-constant density profile with an arbitrary number of steps. An analytic solution is found if all steps have equal vertical extent. From this solution, bounds are established for the relative incident horizontal wave number and frequency of waves that have negligible transmission, a result that is independent of the number of steps. If the step sizes vary randomly about a mean vertical extent, the transmission can be computed numerically for several random realizations. Provided the horizontal wavelength is much longer than the total vertical extent of the staircase, the standard deviation of the computed transmission coefficients is small and the mean agrees well with the analytic prediction for equally spaced steps. The results are applied to thermohaline staircases observed in the ocean in order to put lower bounds on the wavelengths of long inertia gravity waves that significantly transmit to great depth.