Volume 18, Issue 4, April 2006
 LETTERS


Laserinduced fluorescence measurements of buoyancy driven mixing in tilted tubes
View Description Hide DescriptionThe front of a light fluid rising into a miscible heavier fluid inside a long tilted tube has been analyzed by laserinduced fluorescence. Both the local concentration field and the front velocity have been studied in the inertial flow regime as a function of the tilt angle for a constant density contrast . We demonstrate experimentally that the velocity is directly related to the local density contrast by the relation . This relation reflects a local instantaneous equilibrium between inertia and buoyancy; it is valid in the transient relaxation phase as well as in the quasistationary regime reached thereafter.

Mechanism of drag reduction by dimples on a sphere
View Description Hide DescriptionIn this Letter we present a detailed mechanism of drag reduction by dimples on a sphere such as golfball dimples by measuring the streamwise velocity above the dimpled surface. Dimples cause local flow separation and trigger the shear layer instability along the separating shear layer, resulting in the generation of large turbulence intensity. With this increased turbulence, the flow reattaches to the sphere surface with a high momentum near the wall and overcomes a strong adverse pressure gradient formed in the rear sphere surface. As a result, dimples delay the main separation and reduce drag significantly. The present study suggests that generation of a separation bubble, i.e., a closedloop streamline consisting of separation and reattachment, on a body surface is an important flowcontrol strategy for drag reduction on a bluff body such as the sphere and cylinder.

Reverse rotation of a cylinder near a wall
View Description Hide DescriptionA heavy cylinder is free to move inside a rotating horizontal drum filled with a very viscous fluid. Over a range of speeds, the cylinder sits at a set of fixed points adjacent to the rising drum wall with a lubrication region between the cylinder and the surface of the drum. The cylinder is observed to rotate slowly either with or, counterintuitively, against the direction of the drum. An explanation for this surprising effect is provided. The sense of rotation is controlled by a chain of vapor bubbles formed in the expansion area of the lubrication region. These regulate the bulk flow which determines the resulting torque on the cylinder.
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 ARTICLES

 Interfacial Flows

Equations of state in a lattice Boltzmann model
View Description Hide DescriptionIn this paper we consider the incorporation of various equations of state into the singlecomponent multiphase lattice Boltzmann model. Several cubic equations of state, including the van der Waals, RedlichKwong, and PengRobinson, as well as a noncubic equation of state (CarnahanStarling), are incorporated into the lattice Boltzmann model. The details of phase separation in these nonideal singlecomponent systems are presented by comparing the numerical simulation results in terms of density ratios, spurious currents, and temperature ranges. A comparison with a real fluid system, i.e., the properties of saturated water and steam, is also presented.

Nonlinear stability of a charged electrified viscous liquid sheet under the action of a horizontal electric field
View Description Hide DescriptionIn a recent paper [D. T. Papageorgiou and P. G. Petropoulos, J. Eng. Math.50, 223 (2004)] we considered the linear stability of a twodimensional incompressible leaky dielectricviscousliquid sheet surrounded by a hydrodynamically passive conducting medium, when an electric field is applied parallel to the initially flat bounding fluid interfaces. It was established that for orderone Reynolds numbers and when the dielectricpermittivity ratio, , and the electric conductivity ratio, , satisfy , the flow is linearly stable in the absence of an electric field. When a field is present a band of unstable long waves emerges whose size increases as the field increases (the band remains finite, that is short waves are stable, for large fields). In the present study we consider the nonlinear dynamics in the vicinity of the zero electric field bifurcation. The scalings determined from the linear stability calculations are used to derive canonical strongly nonlinear evolution equations for the leading order shape of the sheet and the corresponding horizontal velocity. Numerical simulations indicate that for a wide class of initial conditions, a quasisteady state is reached in the long time when the layer organizes into a number of lobes connected by slowly draining threads whose height vanishes asymptotically in time. The number of lobes and their volumes depend on initial conditions. Using this insight, we construct an ordinary differential equation which describes the shape of the sheet in the limit .

Macroscopic analysis of gasjet wiping: Numerical simulation and experimental approach
View Description Hide DescriptionCoating techniques are frequently used in industrial processes such as paper manufacturing, wire sleeving, and in the iron and steel industry. Depending on the application considered, the thickness of the resulting substrate is controlled by mechanical (scraper), electromagnetic (if the entrained fluid is appropriated), or hydrodynamic (gasjet wiping) operations. This paper deals with the latter process, referred to as gasjet wiping, in which a turbulent slot jet is used to wipe the coating film dragged by a moving substrate. This mechanism relies on the gasjet–liquid film interaction taking place on the moving surface. The aim of this study is to compare the results obtained by a lubrication one–dimensional model, numerical volume of fluid–large eddy simulation (VOFLES) modeling and an experimental approach. The investigation emphasizes the effect of the controlling wiping parameters, i.e., the pressure gradient and shear stress distributions induced by the jet, on the shape of the liquid film. Those profiles obtained experimentally and numerically for a jet impinging on a dry fixed surface are compared. The effect of the substrate motion and the presence of the dragged liquid film on these actuators are analyzed through numerical simulations. Good agreement is found between the film thickness profile in the wiping zone obtained from the VOFLES simulations and with the analytical model, provided that a good model for the wiping actuators is used. The effect of the gasjet nozzle to substrate standoff distance on the final coating thickness is analyzed; the experimental and predicted values are compared for a wide set of conditions. Finally, the occurrence of the splashing phenomenon, which is characterized by the ejection of droplets from the runback filmflow at jet impingement, thus limiting the wiping process, is investigated through experiments and numerical simulations.

Asymptotic theory for a moving droplet driven by a wettability gradient
View Description Hide DescriptionAn asymptotic theory is developed for a moving drop driven by a wettability gradient. We distinguish the mesoscale where an exact solution is known for the properly simplified problem. This solution is matched at both the advancing and the receding side to respective solutions of the problem on the microscale. On the microscale the velocity of movement is used as the small parameter of an asymptotic expansion. Matching gives the droplet shape, velocity of movement as a function of the imposed wettability gradient, and droplet volume.

Oscillatory convective structures and solutal jets originated from discrete distributions of droplets in organic alloys with a miscibility gap
View Description Hide DescriptionThe pattern formation process driven by droplets out of thermodynamic equilibrium, uniformly distributed on the bottom of a container filled with a partially miscible organic liquid, is investigated for different values of by means of a multiprocessor solution of the NavierStokes equations. The considered system is intended to model the typical phenomena occurring during the thermal processing of liquidliquid systems exhibiting a miscibility gap (the socalled “immiscible alloys”). These alloys undergo sedimentation of the separated heavier phase to the bottom of the container under normal gravity conditions. Droplets in nonequilibrium conditions are responsible for the occurrence of still poorly known fluiddynamic instabilities. In the present analysis we provide a clear and quite exhaustive picture of the different stages of evolution of fluid motion inside the container. The distribution of solute is found to depend on the complex multicellular structure of the convective field and on associated “pluming phenomena.” Significant adjustments in the pattern take place as time passes. The structure of the velocity field and the number of rising solutal plumes exhibit sensitivity to the number of droplets and to the possible presence of surfaceMarangoni effects. New classes of possible instability mechanisms (pulsating, traveling, erratic) are identified and described. The investigation provides “local” details as well as general rules and trends about the macroscopic evolution (i.e., “ensemble behaviors”) of the system.

Numerical simulation of multiple bubbles growing in a Newtonian liquid filament undergoing stretching
View Description Hide DescriptionOur recent finite elementbased study of the deformation of a single bubble in a Newtonian or viscoelastic filament undergoing stretching is extended here to the case of multiple bubbles simultaneously growing in the stretched medium. The filament, having initially the shape of a cylinder with uniform radius, is confined between two disks and is continuously stretched by pulling the upper disk along the filament axis with a constant velocity; the lower disk is assumed stationary. All bubbles are taken to lie along the axis of symmetry of the filament and undergo deformation and/or growth with the medium being stretched. The governing equations are solved by a finite element/Galerkin method coupled with an implicit Euler method for the time integration, using an adaptive time step. The problem of the multiple bubbleliquid interfaces is addressed by a robust meshgeneration scheme that solves a set of elliptic differential equations for the locations of the nodal points. The resulting numerical scheme is accurate and extremely stable, independently of the number of bubbles assumed in the filament. It has allowed us to address multiple bubble growth in the filament, and investigate how their interaction affects the response of the system to the applied deformation. Numerical results are presented quantifying the dependence of bubble dynamics on bubbleliquid surface tension, filament aspect ratio (especially as this is decreased to very low values), relative bubble size, and bubblebubble separation. The tensile force on the upper plate is also calculated and reported as a function of time and number of bubbles present in the film. Overall, our numerical calculations demonstrate the dominant role of the pressure field in the elongating filament: for the case of Newtonian fluids considered here, the force needed to maintain the flow comes solely from the pressure field, while for geometries with small aspect ratio, pressure attains large negative values near the centerline, in accord with the predictions of simple arguments from lubricationtheory to the extent this applies to this problem.

Steady threedimensional thermocapillary flows and dryout inside a Vshaped wedge
View Description Hide DescriptionWe consider a liquid meniscus inside a wedge of included angle that wets the solid walls with a contact angle . Under an imposed axial temperature gradient, the Marangoni stress moves fluid toward colder regions whereas the capillary pressure gradient drives a reverse flow, leading to a steady state. The fluxes driven by these two mechanisms are found by numerical integration of the parallel flow equations. Perturbation theory is applied to derive an expression for the capillary pressure, which is typically dominated by the transverse curvature of the circular arc inside the cross section perpendicular to the flow axis, and corrected by a higher order axial curvature resulting from the axial variation of the interface. Lubricationtheory is then used to derive a thin film equation for the shape of the interface. Solutions are determined by two primary parameters: , a geometric parameter giving the relative importance of the two curvatures and , a modified Marangoni number. Numerical solutions indicate that for sufficiently large , the Marangoni stress creates a virtual dry region. The value of at dryout is found to depend linearly on . A simplified analytical model is developed which agrees very well with the numerical solution for large values of . It is found that dryout occurs more easily for larger wedge and/or contact angles except for a special case of . In that case the axial curvature dominates and the dependence of the dryout condition on and is nonmonotonic, but only weakly so.

A numerical study of electrostatic interactions between two charged conducting droplets
View Description Hide DescriptionWith numerous applications prevailing in science and engineering, the dynamics of charged droplets coupled together via electrostatics is of great interest to many researchers. In this study, the liquiddroplets are assumed to be inviscid, incompressible, and electrically conductive. The boundary integral method is used to solve the three Laplace equations governing the dynamics of the two axisymmetric, inviscid charged droplets (two for hydrodynamic problems with interior domains and one for the electrostatic problem with exterior domains). Time integration of the associated dynamical system is achieved by using the fourth order RungeKutta method. The present results suggest that the electrostatic interaction has only a localized effect on the motions of the two coupled droplets. Global analysis based upon Legendre modes may not be a meaningful approach in the current study. Consequently, some interesting aspects of the dynamics of two electrostatically coupled charged droplets are illustrated via various examples.

Flow deformation of polymer blend droplets and the role of block copolymer compatibilizers
View Description Hide DescriptionWe use a multiscale Brownian dynamics simulation approach to study the influence of block copolymer compatibilizers upon the dynamics of a nanoscale polymerdroplet in a matrix of another polymer. The present study focuses on the influence of the physical characteristics of the copolymer, viz. its coverage and chain lengths upon the dropletdeformation characteristics and the rheological properties of the polymer blend system. At a fixed chain length, the copolymer coverage is found to affect the dropletdeformation in a nonmonotonic manner as a function of the capillary number, while it increases monotonically with an increase in the copolymer chain length at a fixed coverage. We identify the interplay between interfacial tension reduction, bending modulus enhancement, and Marangoni stresses as responsible for the preceding characteristics. We also study the rheological effects arising from the presence of block copolymers. Our results suggest increased shear thinning with either increasing the copolymer coverage or the copolymer chain lengths. Moreover, the normal stresses of the mixture are dominated by the inherent normal stresses of the matrix and the droplet phases. The rheological results are rationalized by invoking the interplay between deformation characteristics and the dynamical effects of block copolymers at polymer blend interfaces.

A note on the RayleighTaylor instability with phase change
View Description Hide DescriptionA commonly used approximation in the study of the effect of phase change on RayleighTaylor instability is recalled and reexamined. This approximation incorporates the use of a constant phenomenological coefficient to simplify the model. However, the phenomenological coefficient depends on the wavenumber, and therefore its use as a constant input must be questioned. We employ the solution of the full model to find the variation of this coefficient with the wavenumber and show that this dependence can be strong.

Suppressing falling film instabilities by Marangoni forces
View Description Hide DescriptionThe linear stability of a thin liquid layer falling down an inclined wall heated by a downstream linearly increasing temperature distribution is investigated. It is shown that hydrodynamic and Marangoni instabilities yield two types of transverse instabilities: long surface waves and convective rolls, and longitudinal convective rolls, much like in the case of a uniformly heated wall. However, in contrast to the problem of a uniformly heated wall, where the thermocapillary forces have a destabilizing influence on all instability modes, here they can either destabilize or stabilize the flow. For liquids with sufficiently large Prandtl numbers, increasing the temperature gradient first destabilizes the flow and then stabilizes it. On the other hand, for small Prandtl numbers, increasing the temperature gradient leads to a monotonic stabilization of all instability modes.
 Viscous and NonNewtonian Flows

Drop formation and breakup of low viscosity elastic fluids: Effects of molecular weight and concentration
View Description Hide DescriptionThe dynamics of drop formation and pinchoff have been investigated for a series of low viscosityelastic fluids possessing similar shear viscosities, but differing substantially in elastic properties. On initial approach to the pinch region, the viscoelastic fluids all exhibit the same global necking behavior that is observed for a Newtonian fluid of equivalent shear viscosity. For these low viscosity dilute polymer solutions, inertial and capillary forces form the dominant balance in this potential flow regime, with the viscous force being negligible. The approach to the pinch point, which corresponds to the point of rupture for a Newtonian fluid, is extremely rapid in such solutions, with the sudden increase in curvature producing very large extension rates at this location. In this region the polymer molecules are significantly extended, causing a localized increase in the elastic stresses, which grow to balance the capillary pressure. This prevents the necked fluid from breaking off, as would occur in the equivalent Newtonian fluid. Alternatively, a cylindrical filament forms in which elastic stresses and capillary pressure balance, and the radius decreases exponentially with time. A dimensional finitely extensible nonlinear elastic dumbbell theory incorporating inertial, capillary, and elastic stresses is able to capture the basic features of the experimental observations. Before the critical “pinch time” , an inertialcapillary balance leads to the expected power scaling of the minimum radius with time: . However, the diverging deformation rate results in large molecular deformations and rapid crossover to an elastocapillary balance for times . In this region, the filament radius decreases exponentially with time where is the characteristic time constant of the polymer molecules. Measurements of the relaxation times of polyethylene oxide solutions of varying concentrations and molecular weights obtained from high speed imaging of the rate of change of filament radius are significantly higher than the relaxation times estimated from RouseZimm theory, even though the solutions are within the dilute concentration region as determined using intrinsic viscosity measurements. The effective relaxation times exhibit the expected scaling with molecular weight but with an additional dependence on the concentration of the polymer in solution. This is consistent with the expectation that the polymer molecules are in fact highly extended during the approach to the pinch region (i.e., prior to the elastocapillary filament thinning regime) and subsequently as the filament is formed they are further extended by filament stretching at a constant rate until full extension of the polymer coil is achieved. In this highly extended state, intermolecular interactions become significant, producing relaxation times far above theoretical predictions for dilute polymer solutions under equilibrium conditions.

A new resistance function for two rigid spheres in a uniform compressible lowReynoldsnumber flow
View Description Hide DescriptionThe pressure moment of a rigid particle is defined as the trace of the first moment of the surface stress acting on the particle. We calculate the pressure moments of two unequal rigid spheres immersed in a uniform compressible linear flow, using twin multipole expansions and lubrication theory. Following the practice established in previous studies on twobody hydrodynamic interactions at low Reynolds number, the results are expressed in terms of a new (stresslet) resistance function.

Electricfieldinduced force on a charged spherical colloid embedded in an electrolytesaturated Brinkman medium
View Description Hide DescriptionWhen an electric field is applied to an electrolytesaturated polymer gel immobilizing charged colloidal particles, the force that must be exerted by the hydrogel on each particle reflects a delicate balance of electrical and hydrodynamic stresses. This article adopts a simple boundarylayer analysis to derive a convenient formula for the force in terms of the particle, electrolyte and gel characteristics. Comparisons with numerically exact solutions of the full set of electrokinetic transport equations are presented. These reveal that a fortuitous cancellation of errors leads to reasonably accurate predictions of the force over a much wider range of the parameter space than should be expected. It is noteworthy that, in gels with low permeability, an adverse pressure gradient yields a net force that exceeds the bare electrical force. The analytical theory also provides a convenient formula for the incremental pore mobility, which is a convenient measure of the electroosmotic pumping capacity of dilute random arrays of charged inclusions.
 Particulate, Multiphase, and Granular Flows

The unsteady drag force on a cylinder immersed in a dilute granular flow
View Description Hide DescriptionThis paper presents results from hardparticle discrete element simulations of a twodimensional dilute stream of particles accelerating past an immersed fixed cylinder. Simulation measurements of the drag force are expressed in terms of a dimensionless drag coefficient, , where is the particle density, is the upstream solid fraction, is the upstream instantaneous velocity, and and are the cylinder and particle diameters, respectively. Measurements indicate that the cylinder’s unsteady drag coefficient does not vary significantly from its steady (nonaccelerating) drag coefficient for both frictionless and frictional particles implying that the added mass for the flow is negligible. However, the drag coefficient is larger than its nominal value during an initial transient stage, during which a shock wave develops in front of the cylinder. Once the shock has developed, the drag coefficient remains constant despite the stream’s acceleration. The duration of the shock development transient stage is a function of the number of particle/cylinder collisions.

Instabilities in vertically vibrated granular beds at the single particle scale
View Description Hide DescriptionThe dynamics of granular motion have been studied in a vertically vibrated bed using positron emission particle tracking, which allows the motion of a single tracer particle to be followed in a noninvasive way. The particle movement is closely correlated with the oscillation of the bottom plate. Two types of granular motion have been observed in beds with heap formation: convection and fluctuation. The effects of system parameters, including vibration amplitude, frequency, and bed weight, have been studied. The particle cycle frequency was found to correlate well with the dimensionless acceleration. Cycle frequency appears to be inversely proportional to bed mass. The particle dispersion was determined by following the tracer particle trajectory. The system is highly anisotropic, as the horizontal dispersion is stronger than the vertical dispersion.

Local and global dynamics of shallow gasfluidized beds
View Description Hide DescriptionShallow fluidized beds (SFB) are studied in this paper as a paradigm for deep fluidized beds. We take advantage of their much simpler behavior to analyze their dynamics with a onedimensional EulerianLagrangian model. After a validation with available experiments, we examine how particlescale collisional and hydrodynamic phenomena combine to produce the overall dynamics of SFB. Characteristic times governing the particle's motion are evidenced and the coupling with the fluid is also examined. As a perspective to this work we finally discuss the implications of the SFB dynamics for deep beds. We show that despite the severe simplifications, the model is able to reproduce the temporal dynamics of deep beds and that the internal structuring of SFB is similar in nature to bubbling beds.