Volume 17, Issue 7, July 2005
 LETTERS


Tensorial time scale in turbulent gradient transport of Reynolds stresses
View Description Hide DescriptionOn the notion that the Reynolds stresses are transported with different time scales depending on the transport direction, the thirdorder velocity correlations are represented by a new turbulent gradient transport model with a tensorial Lagrangian time scale. In order to verify the proposed model, DNS data are first obtained in a turbulent channel flow at and 300, and tensorial Lagrangian time scales are computed. The present model predictions are compared with direct numerical simulation data and those predicted by the thirdorder turbulent transport model of Hanjalic and Launder that uses a scalar time scale. The result demonstrates that the Reynolds stresses are indeed transported with different time scales depending on the transport direction.

Control of the natural frequency of the (2,0) mode of liquid bridges using active electrostatic fields
View Description Hide DescriptionWe demonstrate that active stiffening of a capillary bridge through the application of an appropriate Maxwell stress projection can be used to raise the frequency of bridge oscillations. The stress projection is proportional to the mode amplitude. This approach may be useful for suppressing the response of capillary systems to ambient vibrations. The amount of stiffening has a simple dependence on a dimensionless gain and it also depends on the slenderness of the bridge. Using a Plateau tank, we demonstrate a simple method for inferring the dimensionless gain.

On the coalescence speed of bubbles
View Description Hide DescriptionTwo bubbles brought into contact will coalesce rapidly. We study experimentally the initial coalescence motions in the neck region connecting the two bubbles, using an ultrahighspeed video camera. The air bubbles are grown in ethyl alcohol, at the ends of two opposing vertical needles, until they come into contact. The growth rate of the neck is measured and modeled with a capillary/inertial balance. The instantaneous profile in the neck forms a circular arc, with the ratio of the two principal radii of curvatures increasing linearly in time.

Iterated stretching and multiple beadsonastring phenomena in dilute solutions of highly extensible flexible polymers
View Description Hide DescriptionThe dynamics of elastocapillary thinning in high molecular weight polymer solutions are reexamined using highspeed digital video microscopy. At long times, the evolution of the viscoelastic thread deviates from selfsimilar exponential decay and the competition of elastic, capillary, and inertial forces leads to the formation of a periodic array of beads connected by axially uniform ligaments. This configuration is itself unstable and successive instabilities propagate from the necks connecting the beads and the ligaments. This iterated process results in the development of multiple generations of beads in agreement with the predictions of Chang, Demekin, and Kalaidin [“Iterated stretching of viscoelastic jets,” Phys. Fluids11, 1717 (1999)] although experiments yield a different recursion relation between successive generations. At long times, finite molecular extensibility truncates the iterated instability and axial translation of the bead arrays along the interconnecting threads leads to a progressive coalescence before the rupture of the filament.
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 ARTICLES

 Interfacial Flows

Selfsimilar flow and contact line geometry at the rear of cornered drops
View Description Hide DescriptionPartially wettingdrops sliding down an inclined plane develop a “corner singularity” at the rear, consisting of two dynamic contact lines that intersect. We analyze the threedimensional flow in the vicinity of this singularity by exploring similarity solutions of the lubricationequations. These predict a selfsimilar structure of the velocity field, in which the fluid velocity does not depend on the distance to the corner tip; this is verified experimentally by particle image velocimetry. The paper then addresses the smallscale structure of the corner, at which the singularity is regularized by a nonzero radius of curvature of the contact line. Deriving the lubricationequation up to the lowest order in , we show that contact line curvature postpones the destabilization of receding contact lines to liquid deposition, and that increases dramatically close to the “pearling” instability. The general scenario is thus that sliding drops avoid a forced wetting transition by forming a corner of two inclined contact lines, which is regularized by a rounded section of rapidly decreasing size.

Threedimensional solutions for coating flow on a rotating horizontal cylinder: Theory and experiment
View Description Hide DescriptionWe present threedimensional numerical simulations of the flow of a thin liquid coating on a rotating horizontal right circular cylinder. The liquid motion is described using a lubrication model. The model evolution equation is discretized and solved numerically using an alternatingdirection implicit algorithm. The cylinder rotates about its axis, carrying liquid around its circumference, resulting in the formation of a relatively thick coating where the cylinder surface moves upward. For coatings which are initially nearly uniform along the cylinder axis, this results in a ridge of liquid aligned with the cylinder axis. Over time, this ridge may break up into one of several possible configurations, including drops near the underside, and rings enveloping the cylinder. Simulations show that on larger cylinders, under certain circumstances this ridge may develop undulations which grow to form long fingers. These fingers drain down the cylinder. The simulation results are compared with a simple laboratory experiment, which exhibits similar fingering.

Generalized linear stability of noninertial coating flows over topographical features
View Description Hide DescriptionThe transient evolution of perturbations to steady lubrication flow over a topographically patterned surface is investigated via a nonmodal linear stability analysis of the nonnormal disturbance operator. In contrast to the capillary ridges that form near moving contact lines, the stationary capillary ridges near trenches or elevations have only stable eigenvalues. Minimal transient amplification of perturbations occurs, regardless of the magnitude or steepness of the topographical features. The absence of transient amplification and the stability of the ridge are explained on physical grounds. By comparison to unstable ridge formation on smooth, flat, and homogeneous surfaces, the lack of closed, recirculating streamlines beneath the capillary ridge is linked to the linear stability.

Adjustment of sand ripples under changing water waves
View Description Hide DescriptionThe results of an experimental investigation on the adjustment of vortex sand ripples under shoaling waves to changing of wave conditions are presented in this paper. A large wave tank was used to generate shoaling waves.Waves with small (S), moderate (M), and large (L) intensities (as specified by the wave paddle excursion) were used to model three basic cases of cyclic variation of wave forcing, namely, MLM, LML, and LSL. Depending on the forcing transitions (LM, ML, or LS), three main ripple adjustment processes were identified: (i) ripple splitting, (ii) ripple regrowth, and (iii) ripple flattening. Quantitative data on the time evolution of ripple characteristics were collected using the structured light technique. The results of the observations were explained by extending a simplified physical model proposed earlier for ripples under constant wave forcing to the case of changing wave forcing.

Deformation of a capsule in simple shear flow: Effect of membrane prestress
View Description Hide DescriptionWe investigate the effect of an initial isotropic prestress on the mechanics of a spherical capsule freely suspended in a simple shear flow. The capsule consists of a drop of liquid enclosed by a very thin hyperelastic membrane. A sufficient prestress may prevent the buckling instability that arises at low shear rates when the membrane bending resistance is neglected. A small perturbation analysis of the capsule deformation is first obtained for small prestress and weak viscous stresses. When the viscous shear stress and the preinflation are large, a numerical model of the capsule deformation shows how prestress decreases the capsule deformation for a given shear rate and how it significantly increases the elastic tension in the membrane at a given deformation level. These results may help the experimentalists to detect the presence of an osmotic pressure difference in a semipermeable capsule.

Asymmetryinduced particle drift in a rotating flow
View Description Hide DescriptionWe report on an intriguing phenomenon taking place in a liquid rotating around a fixed horizontal axis. Under suitable conditions, bubbles and particles are observed to drift along the axis of rotation maintaining a constant distance from it and a constant angle of elevation above the horizontal. Absence of foreaft symmetry of the bubble or particle shape is a prerequisite for this phenomenon. For bubbles, this requires a volume sufficiently large for surfacetension effects to be small and large deformations to be possible. Particle imagevelocimetry and flow visualization suggest that the wake does not play a role. The dependence on bubble radius, particle shape, liquid viscosity, and speed of rotation is investigated.

Stretch flow of thin layers of Newtonian liquids: Fingering patterns and lifting forces
View Description Hide DescriptionWe study the stretch flow of a thin layer of Newtonian liquid constrained between two circular plates. The evolution of the interface of the originally circular bubble is studied when lifting one of the plates at a constant velocity and the observed pattern is related to the measured lifting force. By comparing experimental results to numerical simulations using a Darcy’s law model we can account for the fully nonlinear evolution of the observed fingering pattern. One observes an initial destabilization of the interface by growth of air fingers due to a Saffman–Taylorlike instability and then a coarsening of the pattern toward a circular interface until complete debonding of the two plates occurs. Numerical simulations reveal that when relating the observed patterns to the lifting force not only the number of fingers but also the amplitude of the fingering growth has to be taken into account. This is consistent with the experimental observations.
 Viscous and NonNewtonian Flows

A simple paradigm for active and nonlinear microrheology
View Description Hide DescriptionIn microrheology, elastic and viscous moduli are obtained from measurements of the fluctuating thermal motion of embedded colloidal probes. In such experiments, the probe motion is passive and reflects the nearequilibrium (linear response) properties of the surrounding medium. By actively pulling the probe through the material, further information about material properties can be obtained, analogous to largeamplitude measurements in (macro) rheology. We consider a simple model of such systems: a colloidal probe pulled through a suspension of neutrally buoyant bath colloids. We choose a system with hardsphere interactions but neglect hydrodynamic interactions, which is simple enough to permit analytic solutions, but nontrivial enough to raise issues important for the interpretation of experiments in active and nonlinear microrheology. We calculate the microstructural deformation for arbitrary probe size and pulling rate (expressed as a dimensionless Péclet number ). From this, we determine the average retarding effect on the probe due to the microstructure, as well as fluctuations about this average. The high limit is singular, giving a finite Brownian contribution even in the limit of negligible diffusion. Significantly, different results are obtained for probes driven at constant velocity and constant force. Furthermore, we demonstrate that a probe pulled with an optical tweezer (roughly a harmonic well) can behave as fixedforce, fixedvelocity, or as a mixture of those modes, depending on the strength of the trap and on the pulling speed. More generally, we discuss how these results relate to previous work on the rheology of colloidalsuspensions. Not surprisingly, the present theory (which ignores hydrodynamic interactions) gives shear thinning but no shear thickening; we expect that the incorporation of hydrodynamics would result in shear thickening as well. The effective micro and macroviscosities, when appropriately scaled, are in semiquantitative agreement. This seems remarkable, given the rather significant difference in the two methods of measurement. However, for more complicated or unknown materials, where such scaling relations may not be known in advance, the comparison between micro and macro may not be so favorable, which raises important questions about the relation between micro and macrorheology. Finally, by analogy with previous work on macrorheology, we propose methods to scale up the present (dilute) theory to account for more concentrated suspensions, and suggest new active microrheological experiments to probe different aspects of suspension behavior.

Small amplitude oscillations of a thin beam immersed in a viscous fluid near a solid surface
View Description Hide DescriptionThe hydrodynamic loading on a solid body moving in a viscous fluid can be strongly affected by its proximity to a surface. In this article, we calculate the hydrodynamic load on an infinitely long rigid beam of zero thickness that is undergoing small amplitude oscillations. The presence of a solid surface an arbitrary distance from the beam is rigorously accounted for using a boundary integral formulation.
 Particulate, Multiphase, and Granular Flows

Clustering and collisions of heavy particles in random smooth flows
View Description Hide DescriptionFinitesize impurities suspended in incompressible flows distribute inhomogeneously, leading to a drastic enhancement of collisions. A description of the dynamics in the full positionvelocity phase space is essential to understand the underlying mechanisms, especially for polydisperse suspensions. These issues are studied here for particles much heavier than the fluid by means of a Lagrangian approach. It is shown that inertia enhances collision rates through two effects: correlation among particle positions induced by the carrier flow and uncorrelation between velocities due to their finite size. A phenomenological model yields an estimate of collision rates for particle pairs with different sizes. This approach is supported by numerical simulations in random flows.

Particle concentration evolution and sedimentationinduced instabilities in a stably stratified environment
View Description Hide DescriptionWe present the results of a combined experimental and theoretical investigation of sedimentation in a stratified fluid. The theory of sedimentation in a homogeneous ambient is extended to include the influence of a spatially varying particle settling speed. The results of an experimental investigation of latex particles settling in a stably stratified salt water solution are reported. Density variations in the suspending fluid reduce the particle settling speed which increases particle concentrations, thus enhancing the effects of hindered settling. A criterion is developed for the convective instability of an initially uniform suspension settling in a stably stratified ambient. If, as depth increases, the magnitude of the ambient density gradient decreases sufficiently rapidly, an initially uniform particle concentration will give rise to a statically unstable density profile. Experimental observations provide qualitative verification of this new stability criterion.

The flow of a very concentrated slurry in a parallelplate device: Influence of gravity
View Description Hide DescriptionWe investigate, both experimentally and theoretically, the flow and structure of a slurry when sheared between two horizontal plates. The slurry, otherwise called “wet granular material,” is made of nonBrownian particles immersed in a viscous fluid. The particles are heavier than the immersion fluid, in contrast to the socalled “suspensions,” corresponding to densitymatched fluid and particles. Consequently, gravity influences the structure and flow profiles of the sheared material. Experiments are carried out in a plane Couette device with a model slurry composed of quasimonodispersed spherical polymethylmetacrylate particles in oil, at high average solid concentration (about 59%). Optical observation reveals a typical twophase configuration, with a fluidized layer in contact with the upper plate and on top of an amorphous solid phase. We provide data on velocity profiles, wall slip, average shear stress, and average normal stress, versus the angular velocity of the upper plate. To interpret the data, we propose a model for the ideal case of infinite horizontal flat plates (plane Couette flow). The model, of meanfield type, is based on local constitutive equations for the tangential and normal components of the stress tensor and on material expressions relating the material viscometric coefficients (the shear viscosity and the normal stress viscosity) with the local concentration () and the local shear rate. One, two, and threephase configurations are predicted, with nonlinear flow and concentration profiles. We conclude that model equations correctly describe the experimental data, provided that appropriate forms are chosen for the divergences of and near the packing concentration (), namely, a singularity.

Velocity profiles, stresses, and Bagnold scaling of sheared granular system in zero gravity
View Description Hide DescriptionWe report the results of threedimensional eventdriven simulations of sheared granular system in a Couette geometry. The simulations use realistic boundary conditions that may be expected in physical experiments. For a range of boundary properties we report velocity and density profiles, as well as forces on the boundaries. In particular, we find that the results for the velocity profiles throughout the shearing cell depend strongly on the interaction of the system particles with the physical boundaries. Even frictional boundaries can allow for significant slippage of the particles, therefore reducing the shear in the system. Next, we present stress distributions both for controlled volume and for controlled stress configurations. We discuss the dependence of solid volume fraction on shear rate under the constantpressure condition, and Bagnold scaling in volumecontrolled simulations. In addition, we study the influence of oscillatory driving of one of the domain boundaries on the systemproperties.

Mixing in a drop moving through a serpentine channel: A computational study
View Description Hide DescriptionThe chaotic mixing in a drop moving through a winding channel is studied computationally in a twodimensional setting. The molecular mixing is ignored and only the mixing due to the chaotic advection is considered. Passive tracer particles are used to visualize the mixing patterns and mixing is quantified by two distinct methods. It is found that both the quantification methods are consistent with visual observations as well as with each other. The effects of various nondimensional parameters on the quality of mixing are studied and it is found that the capillary number, the ratio of the drop phase fluid viscosity to that of the ambient fluid and the relative size of the drop compared to the average channel width are the most critical parameters influencing the mixing. The mixing is found to be weakly dependent on Reynolds number.
 Laminar Flows

Universal wake structures of Kármán vortex streets in twodimensional flows
View Description Hide DescriptionFormation of Kármán vortex streets by circular rods is studied in a flowing soap film channel. The twodimensional fluid flow in the film allows stable vortex streets to be generated and studied over a broad range of Reynolds numbers,. There have been many studies of the Kármán vortex streets. The current study focuses on the universal feature of the nearwake structures generated by circular cylinders in a uniform twodimensional (2D) flow. The new approach in the investigation is to decompose the street into a series of vortex couples, which defines the outer envelope of the wake, and to study the expansion rate of these couples. Our experiment shows that motion of vortex couples are powered by bursts of transverse jets, and the jet intensity determines the asymptotic width and the expansion rate of the wake. A simple model is constructed that not only allows us to extract the kinetic energy encapsulated in the vortices but also explains the universal appearance of the 2D laminar wakes generated by large and small rods. To the best of our knowledge, this is the first time that the efficiency of energy injected into vortices is estimated and this efficiency turns out to be .

Analytical solutions for a spherical particle near a wall in axisymmetrical polynomial creeping flows
View Description Hide DescriptionAnalytical solutions are presented for axisymmetric creeping flows around a spherical particle near a wall, when the unperturbed flow velocity varies as polynomial with the coordinates. The perturbed fluid velocity and pressure are calculated directly with the method of bipolar coordinates. They are obtained with a precision, even for small gaps of the order of sphere radius. For a constant unperturbed flow, the problem is equivalent to that of a sphere moving perpendicularly to a wall in a fluid at rest. An alternative indirect solution for the fluid velocity and pressure is obtained in this case from the solution of Brenner [Chem. Eng. Sci.16, 242 (1961)] and Maude [Brit. J. Appl. Phys.12, 293 (1961)] for the stream function. For a small gap between the sphere and the wall, the values of the pressure are compatible with the ones from the lubrication approximation but are systematically larger; this may be important for applications. Calculations are also performed when the unperturbed flow velocity is a polynomial of degree 2 and 3 in the coordinates, viz., for different types of stagnation point flows. Various flow structures are obtained, depending on the particle to wall distance. When the sphere approaches the wall, there is an increasing number of nested toroidal eddies, providing a link between the case of a single toroidal eddy when the sphere is far from the wall and the infinite set of Moffatt eddies in the gap between bodies in contact. The flow structure is analogous to that for two equal spheres in a uniform flow field, cf. Davis, O’Neill, Dorrepaal, and Ranger [J. Fluid Mech.77, 625 (1976)]. Precise results for the force on the sphere are provided.