We consider the long-wave Marangoni convection in a horizontally oriented heated thin liquid layer with weak heat flux from the free surface. Two modes of instability are possible in this problem: monotonic and oscillatory. The system of nonlinear evolution equations is modified for the case when the surface tension is a nonlinear function of temperature determined by a set of coefficients in the Taylor series expansion about the surface temperature. The coefficients which are used are taken mainly from experimental data. In the case of monotonic instability, we investigate the influence of nonlinear temperature dependence of the surface tension on the stability of roll, square, and hexagonal patterns. In the case of oscillatory instability, we consider the stability of single traveling waves and traveling rectangles.

We study viscous fingering formation in an immiscible three-layer radial Hele-Shaw flow, where two coupled interfaces spread radially. More specifically, we examine how the initial distance $d$ between the inner and outer interfaces (i.e., annulus' thickness) influences the shape of the emerging fingering patterns. Our theoretical analysis extends the perturbation theory beyond linear stability to a second-order mode-coupling theory. An important feature of our second-order perturbative approach is the fact that through the coupling of the appropriate Fourier modes, one is able to extract key analytical information about the morphology of the interface at the onset of nonlinearities. Under the circumstances where the inner interface is unstable and the outer one is stable, our theoretical results indicate that as $d$ decreases, the coupling between the interfaces becomes stronger and the nearly matched final shapes exhibit the formation of wide fingers with bifurcated tips. However, if $d$ is reduced further, we observe an unexpected change in the morphology of the patterns, where the conventional finger-splitting morphologies are replaced by polygonal-like structures with narrow fingers. The opposite scenario where the inner interface is stable and the outer one is unstable reveals two interesting situations: the onset of formation of drops due to the rupture of the intermediate layer of fluid, and an overall stabilization of the outer interface by the presence of the fluid annulus.

Accurately predicting profiles of mean-mass fraction $\overline{Y}$ and mean-molar/volume fraction $\overline{\mathrm{\Phi }}$ (equivalent to renormalized mean density $\overline{\rho })$ of turbulent mixing induced by Rayleigh-Taylor instability at different density ratio $R$ is of fundamental importance for many natural phenomena and engineering applications but has not been quantified yet. Based on asymptotic analysis, we give an $R$-invariant hybrid profiles $C\left(X\right)\equiv F\overline{Y}+(1-F)\overline{\mathrm{\Phi }}$ with $A\equiv (R-1)/(R+1)$ and $F\equiv (1+A)(1-X)/2$ and derive the analytical expression of $C\left(X\right)$, where $X\in [0,1]$ is a mixing-width-rescaled dimensionless coordinate. The invariance are validated by both experiments and simulations. Consequently, combining the $R$ invariance of $C\left(X\right)$ and the relation of $\overline{\mathrm{\Phi }}=\overline{Y}/[\overline{Y}+(1-\overline{Y})R]$ derived from definitions, the mean profiles of density, mass fraction, and molar/volume fraction at different density ratios are accurately predicted.

Particle resuspension due to shoaling internal solitary waves is studied through laboratory experiments and direct numerical simulations. Experiments examine particles placed along the sloping bottom to observe when and where bed load transport and particle resuspension occur. Through a comparison of velocimetry measurements between the experiments and the simulations, the accuracy of the numerical results is established. A suite of simulations is conducted to investigate the dependence of the location of incipient particle resuspension on the bottom slope and incident wave parameters. While a rapid increase in the Shields parameter in the lee of the shoaling wave corresponds to the observed location of resuspension, we find that particle transport away from the bottom is better assessed by the resuspension criterion $w_L/w_s>1$, where $w_s$ is the particle settling velocity and $w_L=w-su$ is the vertical Lagrangian velocity in which $\vec{u}=(u,w)$ is the Eulerian velocity and $s$ is the slope, which measures the rise of fluid from the bottom following a streamline.

In this study we present numerical and scaling analyses of deformation and freezing processes of ferrofluid droplets under magnetic field effects. A multiphase flow model coupled with an enthalpy-based lattice Boltzmann model is developed to directly simulate the deformation and subsequent freezing processes of a ferrofluid droplet with considerations of both volume expansion and magnetization effects. Meanwhile, analytical models and scaling analyses are derived to reveal how the morphology of the ferrofluid droplet responds to the magnetic field and how the morphology evolution affects the freezing time. We find that the magnetic force induced by magnetic field gradient is much larger than that induced by the magnetization effect, leading to the flattening or elongation of a ferrofluid droplet under magnetic squeeze or lift conditions. The height of the ferrofluid droplet almost linearly decreases (increases) in the low magnetic strength regime for magnetic squeeze (lift) cases, while it follows a nonlinear scaling law under high magnetic squeeze conditions. Besides, the propagation of the freezing front well obeys the scaling law $h\sim t^{0.5}$ for high magnetic squeeze cases, but deviates much from that at the final freezing stage for both magnetic absence and lift cases.

We investigate freely expanding liquid sheets made of either simple Newtonian fluids or solutions of high molecular water-soluble polymer chains. A sheet is produced by the impact of a drop on a quartz plate covered with a thin layer of liquid nitrogen that suppresses shear viscous dissipation due to an inverse Leidenfrost effect. The sheet expands radially until reaching a maximum diameter and subsequently recedes. Experiments indicate the presence of two expansion regimes: the capillary regime, where the maximum expansion is controlled by surface tension forces and does not depend on the viscosity, and the viscous regime, where the expansion is reduced with increasing viscosity. In the viscous regime, the sheet expansion for polymeric samples is strongly enhanced as compared to that of Newtonian samples with comparable zero-shear viscosity. We show that data for Newtonian and non-Newtonian fluids collapse on a unique master curve where the maximum expansion factor is plotted against the relevant effective biaxial extensional Ohnesorge number that depends on fluid density, surface tension, and the biaxial extensional viscosity. For Newtonian fluids, this biaxial extensional viscosity is six times the shear viscosity. By contrast, for the non-Newtonian fluids, a characteristic Weissenberg number-dependent biaxial extensional viscosity is identified, which is in quantitative agreement with experimental and theoretical results reported in the literature for biaxial extensional flows of polymeric liquids.

Viscous streaming has emerged as an effective method to transport, trap, and cluster inertial particles in a fluid. Previous work has shown that this transport is well described by the Maxey-Riley equation augmented with a term representing Saffman lift. However, in its straightforward application to viscous streaming flows, the equation suffers from severe numerical stiffness due to the wide disparity between the timescales of viscous response, oscillation period, and slow mean transport, posing a severe challenge for drawing physical insight on mean particle trajectories. In this work, we develop equations that directly govern the mean transport of particles in oscillatory viscous flows. The derivation of these equations relies on a combination of three key techniques. In the first, we develop an inertial particle velocity field via a small Stokes number expansion of the particle's deviation from that of the fluid. This expansion clearly reveals the primary importance of Faxén correction and Saffman lift in effecting the trapping of particles in streaming cells. Then, we apply the generalized Lagrangian mean theory to unambiguously decompose the transport into fast and slow scales, and ultimately, develop the Lagrangian mean velocity field to govern mean transport. Finally, we carry out an expansion in small oscillation amplitude to simplify the governing equations and to clarify the hierarchy of first- and second-order influences, and particularly, the crucial role of Stokes drift in the mean transport. We demonstrate the final set of equations on the transport of both fluid and inertial particles in configurations involving one cylinder in weak oscillation and two cylinders undergoing such oscillations in sequential intervals. Notably, these equations allow numerical time steps that are $O\left({10}^3\right)$ larger than the existing approach with little sacrifice in accuracy, allowing more efficient predictions of transport.

Blood flow in sickle cell disease (SCD) can substantially differ from normal blood flow due to significant alterations in the physical properties of the red blood cells (RBCs). Chronic complications, such as inflammation of the endothelial cells lining blood vessel walls, are associated with SCD, for reasons that are unclear. Here, detailed boundary integral simulations are performed to investigate an idealized model flow in SCD, a binary suspension of flexible biconcave discoidal fluid-filled capsules and stiff curved prolate capsules that represent healthy and sickle RBCs, respectively, subjected to pressure-driven flow in a planar slit. The stiff component is dilute. The key observation is that, unlike healthy RBCs that concentrate around the center of the channel and form an RBC-depleted layer (i.e., a cell-free layer) next to the walls, sickle cells are largely drained from the bulk of the suspension and aggregate inside the cell-free layer, displaying strong margination. These cells are found to undergo a rigid-body-like rolling orbit near the walls. A binary suspension of flexible biconcave discoidal capsules and stiff straight (noncurved) prolate capsules is also considered for comparison, and the curvature of the stiff component is found to play a minor role in the behavior. Additionally, by considering a mixture of flexible and stiff biconcave discoids, we reveal that rigidity difference by itself is sufficient to induce the segregation behavior in a binary suspension. Furthermore, the additional shear stress on the walls induced by the presence of cells is computed for the various cases. Compared to the small fluctuations in wall shear stress for a suspension of healthy RBCs, large local peaks in wall shear stress are observed for the binary suspensions, due to the proximity of the marginated stiff cells to the walls. This effect is most marked for the straight prolate capsules. As endothelial cells are known to mechanotransduce physical forces such as aberrations in shear stress and convert them to physiological processes such as activation of inflammatory signals, these results may aid in understanding mechanisms for endothelial dysfunction associated with SCD.

Nonlinear small-amplitude high-frequency vibrations affect mechanical systems leading to such effects as the stabilization of an inverted pendulum on a vibrating foundation and size separation of particles of a granular material. These effects can be studied using the mathematical technique called “the method of separation of fast and slow motions.” When applied to fluid mechanics applications, the method predicts various unusual effects including jamming holes in a vibrating tank with liquid, vibrational propulsion, and multiphase flow separation. Many of these effects have been confirmed experimentally. Here we discuss the possibility of similar microfluidic effects in applications where capillary forces dominate over viscosity and inertia, such as vibrational aeration, vibration-induced superhydrophobicity (the elimination of the contact angle hysteresis), and vibrational two-phase flow separation for microparticle extraction.

In many natural granular systems, the interstitial pores are filled with a fluid. Deformation of this two-phase system is complex, is highly coupled, and depends on the initial and boundary conditions. Here we study granular compaction and fluid flow in a saturated, horizontally shaken, unconfined granular layer, where the fluid is free to flow in and out of the layer through the free upper surface during shaking (i.e., drained boundary condition). The geometry, boundary conditions, and parameters are chosen to resemble a shallow soil layer, subjected to horizontal cyclic acceleration simulating that of an earthquake. We develop a theory and conduct coupled discrete element and fluid numerical simulations. Theoretical and simulation results show that under drained conditions and above a critical acceleration, the grain layer compacts at a rate governed by the fluid flow parameters of permeability and viscosity and is independent of the shaking parameters of frequency and acceleration. A compaction front develops, swiping upward through the system. Above the front, compaction occurs and the fluid becomes pressurized. Pressure gradients drive fluid seepage upward and out of the compacting layer while supporting the granular skeleton. The rate of compaction and the interstitial fluid pressure gradient coevolve until fluid seepage forces balance solid contact forces and grain contacts disappear. As an outcome, the imposed shear waves are not transmitted and the region is liquefied. Below the compaction front (i.e., after its passage), the grains are well compacted, and shaking is transmitted upward. We conclude that the drained condition for the interstitial pore fluid is a critical ingredient for the formation of an upward-moving compaction front, which separates a granular region that exhibits a liquidlike rheology from a solidlike region.

Herein we numerically examine the relative dispersion of Lagrangian particle pairs in two-dimensional inverse energy-cascade turbulence. Behind the Richardson-Obukhov $t^3$ law of relative separation, we discover that the second-order moment of the relative velocity have a temporal scaling exponent different from the prediction based on the Kolmogorov's phenomenology. The results also indicate that time evolution of the probability distribution function of the relative velocity is self-similar. The findings are obtained by enforcing the Richardson-Obukhov law either by considering a special initial separation or by conditional sampling. In particular, we demonstrate that the conditional sampling removes the initial-separation dependence of the statistics of the separation and relative velocity. Furthermore, we demonstrate that the conditional statistics are robust with respect to the change in the parameters involved, and that the number of the removed pairs from the sampling decreases when the Reynolds number increases. We also discuss the insights gained as a result of conditional sampling.

Criteria to identify transitions between dynamic self-similar linear thinning regimes of liquid bridges are of utmost importance in order to accurately interpret results in capillary break-up rheometry. Currently available criteria encompass many experimental difficulties or rely on numerical approaches. Here, we introduce a different set of nondimensional groups, ${\text{Oh}}_{\text{L}}={\eta }_{\text{in}}/\sqrt{\gamma \rho L}$ and $a=R/L$, based on the experimentally relevant axial length scale of a liquid bridge $L$, for viscous-dominated fluids undergoing capillary break-up in air. This framework is further extended to encompass the effect of outer viscous fluids. As a result, we present a two-dimensional operating map in which the boundaries are set by fluid properties and a single geometrical parameter, related to the experimental configuration. This approach establishes guidelines to correctly interpret experimental data and identify transitions in capillary break-up experiments of liquid bridges surrounded by fluids of different viscosities.

The formation and evolution of micron-sized droplets of a Newtonian liquid generated on demand in an industrial inkjet printhead are studied experimentally and simulated numerically. The shapes and positions of droplets during droplet formation are observed using a high-speed camera and compared with their numerically obtained analogs. Both the experiments and the simulations use practical length scales for inkjet printing. The results show how fluid properties, specifically viscosity and surface tension, affect the drop formation, ligament length, and breakoff time. We identify the parameter space of fluid properties for producing single drops at a prescribed speed and show this is not simply a restriction on the Ohnesorge number, but that there is an additional restriction on the Reynolds number that is distinct from the Reynolds number limit associated with the prevention of splashing. This phase diagram provides more precise guidance on the space of fluid parameters for jetting single droplets in drop-on-demand inkjet printers.

We present a statistical analysis of the behavior of the kinetic-energy spectrum in the dissipative range of fully developed three-dimensional turbulence, with the aim of testing a recent prediction obtained from the nonperturbative renormalization group. Analyzing spectra recorded in experiments of grid turbulence, generated in the Modane wind tunnel, and spectra obtained from high-resolution direct numerical simulations (DNSs) of the forced Navier-Stokes equation, we observe that the spectra decay as a stretched exponential in the dissipative range. The theory predicts a stretching exponent $\alpha =2/3$, and the data analyses of the numerical and experimental spectra are in close agreement with this value. This result also corroborates previous DNS studies which found that the spectrum in the near-dissipative range is best modeled by a stretched exponential with $\alpha <1$.

A numerical and experimental study of the process of droplet formation in a T-junction microchannel with a shear-thinning continuous phase is presented. Simulations are performed using a three-dimensional lattice Boltzmann multicomponent model. The shear-thinning behavior is modeled using the Carreau-Yasuda model and validated with our experiments involving both Newtonian and non-Newtonian continuous phases. Using the validated numerical framework, a parametric study is performed to quantify the effect of the rheological parameters, namely, the power-law index $n$ and the Carreau number Cr, on the droplet size for two different capillary numbers. To understand the variation in the droplet size and shape with the shear-thinning parameters, the dynamics of the droplet breakup process is analyzed in terms of the upstream pressure buildup and the shear stresses acting on the interface of the two fluids. Our numerical results show that the dynamics of droplet formation shifts from the shear-dominated dripping regime to the pressure-dominated squeezing regime for continuous phase fluids having lower values of $n$ and higher values of Cr. Hence, an increase in the droplet size and a change in its shape from spherical to plug are observed for fluids showing a higher shear-thinning tendency (higher Cr and lower $n$). Finally, an analysis of the transition in the droplet flow regimes with a shear-thinning continuous phase at high capillary numbers is presented. For $\text{Ca}\ge 0.1$, an increase in the shear thinning of the continuous fluid is shown to shift the droplet flow pattern from parallel flow to droplets in the channel to eventually forming droplets at the T junction.

Turbulence modulation in particle-laden stationary homogeneous isotropic turbulence is investigated using one-dimensional turbulence (ODT), a low-dimensional stochastic flow simulation model. For this purpose, ODT is extended in two ways. First, a forcing scheme that maintains statistical stationarity is introduced. Unlike direct numerical simulation (DNS) of forced turbulence, the ODT framework accommodates forcing that is not directly coupled to the momentum equation. For given forcing the ODT energy dissipation rate is therefore the same in particle-laden cases as in the corresponding single-phase reference case. Second, previously implemented one-way-coupled particle phenomenology is extended to two-way coupling using the general ODT methodology for flow modulation through interaction with any specified energy and momentum sources and sinks. As in a DNS comparison case for ${\text{Re}}_{\lambda }=70$, turbulence modulation is diagnosed primarily on the basis of the fluid-phase kinetic-energy spectrum. Because ODT involves subprocesses with straightforward physical interpretations, the ODT mechanisms of particle-induced turbulence modulation are clearly identified and they are plausibly relevant to particle-laden Navier-Stokes turbulence. ODT results for the ratio of particle-phase and fluid-phase kinetic energies as a function of particle Stokes number and mass loading are reported for the purpose of testing these predictions in the future when these quantities are evaluated experimentally or using DNS.

A complete theory able to assess the longitudinal dispersion of a passive solute injected into an annular cavity subject to a pulsatile flow and a porous medium is provided. The Aris-Taylor method of statistical moments is combined with the Brinkman approach for porous flows and morphological dispersion, in order to get an analytical relationship for the time-dependent enhanced dispersion coefficient. The application of intrathecal drug delivery in the cerebrospinal fluid contained in the subaracnoid space is discussed in detail. The main result of the theory and its assumptions are also numerically validated through the use of a finite-volume solver. The role of several physiological features, such as the geometry, temporal frequency, and wavelength, of the pressure forcing are analyzed. It turns out that the presence of delicate strands of connective tissue, called trabeculae, that fill the cavity and link the innermost layer of meninges plays a crucial role. They in fact induce extra terms of morphological dispersion and act synergistically with pulsation to produce realistic times of drug delivery in clinically significant conditions. The results have potential for the optimization of delivery protocols of drug therapies of the central nervous system.

We consider the drainage of blood plasma across the capillary wall, focusing on the flow through the endothelial glycocalyx layer that coats the luminal surface of vascular endothelial cells. We investigate how the presence of a sub-glycocalyx space between the porous glycocalyx and the impermeable endothelial cells affects the flow, using the Darcy and Stokes equations to model the flow in the glycocalyx and sub-glycocalyx space, respectively. Using an asymptotic analysis, we exploit the disparity of length scales to reduce the problem complexity to reveal the existence of several asymptotic regions in space. We provide a detailed characterization of the flow through the glycocalyx layer in terms of the microscale system parameters, and we derive analytic macroscale results, such as for the flux through and hydraulic conductivity of the glycocalyx layer. We show that the presence of a sub-glycocalyx space results in a higher flux of blood plasma through the glycocalyx layer, and we use our theoretical predictions to suggest experiments that could be carried out to shed light on the extent of the layer.

The flow behavior of entangled polymer solutions in simple shear flow is often assumed to be uniform, both in terms of the velocity gradient and polymer concentration. However, there is growing evidence of nonuniform “banded” transient or steady state flows of entangled polymer solutions. The present work considers a distinct phenomenon, whereby transient banding is concomitant with the presence of local shear-enhanced concentration fluctuations. Experimental observations are made using combined rheological measurements with simultaneous particle tracking velocimetry and microscopy (rheomicroscopy) during startup shear in Taylor-Couette flow of entangled polymer solutions of polystyrene in a marginal solvent, dioctyl phthalate, at various entanglement numbers ($Z$). At high $Z$, the flow develops transient, nonhomogeneous “banded” velocity profiles over a wide range of imposed shear rates that are inverted from those expected for Taylor-Couette flow, which then relax to a nearly uniform shear rate at steady state. Rheomicroscopy reveals that these transient banded states are accompanied by strong shear-enhanced concentration fluctuations localized to the region of lowest local shear rate. We hypothesize that such finite-amplitude concentration fluctuations lead to increased dissipation that contributes a higher local “effective” fluid viscosity that, in turn, could produce the observed flow nonuniformity. In spite of the nonuniformity of the flow and the concentration fluctuations, the measured apparent rheology of the fluids is in qualitative agreement with both prior experimental studies and model predictions based on a Rolie-Poly model under the assumption that the fluid remains homogeneous, suggesting that this behavior is only observable through spatially resolved measurements of fluid flow and concentration.

We introduce a microfluidic design that is desirable for three-dimensional (3D) micromanipulation, achieved through a set of double-layer channels with embedded symmetries. For the six-channel configuration, we show that a zero strain rate at the center of the device is protected by these symmetries, leading to a nonperturbative manipulation flow along any direction in 3D. We visualize such a nonperturbative flow structure through a finite element simulation and confirm this symmetry-protected strain-free condition. In addition to such 3D nonperturbative manipulations, we reveal two distinct perturbative flow modes available in this six-channel device, corresponding to a total of five independent modes that agrees with the degree-of-freedom counting. This symmetry-based micromanipulation is fully compatible with conventional microscopes and can be easily extended to other channel geometries for rich biological and physical applications.