Volume 20, Issue 4, April 2008

The effect of solublesurfactants on the unsteady motion and deformation of a bubble rising in an otherwise quiescent liquid contained in an axisymmetric tube is computationally studied by using a finitedifference/fronttracking method. The unsteady incompressible flow equations are solved fully coupled with the evolution equations of bulk and interfacial surfactant concentrations. The surface tension is related to the interfacial surfactant concentration by a nonlinear equation of state. The nearly spherical, ellipsoidal, and dimpled ellipsoidalcap regimes of bubble motion are examined. It is found that the surfactant generally reduces the terminal velocity of the bubble but this reduction is most pronounced in the nearly spherical regime in which the bubble behaves similar to a solid sphere and its terminal velocity approaches that of an equivalent solid sphere. Effects of the elasticity number and the bulk and interfacial Peclet numbers are examined in the spherical and ellipsoidal regimes. It is found that the surface flow and interfacial surfactant concentration profiles exhibit the formation of a stagnant cap at the trailing end of the bubble in the ellipsoidal regime at low elasticity and high interfacial Peclet numbers. Bubble deformation is first reduced due to rigidifying effect of the surfactant but is then amplified when the elasticity number exceeds a critical value due to overall reduction in the surface tension.
 ARTICLES

 Interfacial Flows

Dependency of the apparent contact angle on nonisothermal conditions
View Description Hide DescriptionThe dynamic behavior of liquids in partly filled containers is influenced to a large extend by the angle between the gasliquid phase boundary and the solid container wall at the contact line. This contact angle in turn is influenced by nonisothermal conditions. In the case of a cold liquid meniscus spreading over a hot solid wall, the contact angle apparently becomes significantly larger. In this paper we want to establish a quantitative equation for this enlargement, both from experimental and numerical data. Our findings can be used to build a subgrid model for computations, where the resolution is not sufficient to resolve the boundary layers. This might be the case for large containers which are exposed to low accelerations and where the contact angle boundary condition determines the position of the free surface. These types of computation are performed, for example, to solve propellant management problems in launcher and satellite tanks. In this application, the knowledge of the position of the free surface is very important for the withdrawal of liquid and the calculation of heat and mass transfer.

Conical tips inside conejet electrosprays
View Description Hide DescriptionIn coaxial jet electrosprays inside liquid baths, a conductive liquid forms a conejet electrospray within a bath containing a dielectric liquid. An additional dielectric liquid is injected inside the Taylor cone, forming a liquidmeniscus. The motion of the conductive liquid that flows toward the vertex cone deforms the inner dielectricmeniscus until a liquid jet is issued from its tip. Both the conductive and inner dielectric liquid jets flow coaxially and, further downstream, they will eventually be broken up by capillary instabilities. Coaxial jet electrosprays inside liquid baths is a useful technique to generate fine simple or double emulsions. However, in certain circumstances, we have observed that the dielectricmenisci present extremely sharp tips that can be stabilized and made completely steady without mass emission. In this paper, we will first explore the parametrical range of liquid properties, mainly viscosities and surface tensions, under which these sharp tips take place. In addition, a simplified analytical model of the very complex electrohydrodynamical flow is presented for a more complete approach to the phenomena.

Effect of an electric field on film flow down a corrugated wall at zero Reynolds number
View Description Hide DescriptionThe effect of an electric field on a liquid layer flowing down an inclined, corrugated wall at zero Reynolds number is investigated. The layer is taken to be either a perfect conductor or a perfect dielectric. The region above the layer is assumed to be a perfect dielectric. Steady flow down a wall with smallamplitude sinusoidal corrugations is considered, and it is shown how the electric field can be used to control the amplitude of the freesurface deflection and the phase shift between the free surface and the wall profile. Steady flow over walls with large amplitude sinusoidal corrugations or othershaped indentations is studied by using the boundaryelement method. Results for flow into a wide rectangular trench are compared to previous model predictions based on the lubrication approximation. For a perfectconductor film, the results confirm that the height of the capillary ridge, which appears above a downward step, monotonically decreases as the electric field strength increases. Solutions for a perfectdielectric film with relative permittivity larger than unity are similar to those for a perfectconductor film, although the height of the capillary ridge nonmonotonically varies with the electric field strength. The behavior of the solutions for a perfectdielectric film with relative permittivity less than unity is qualitatively different. The height of the capillary ridge monotonically increases as the electric field strength increases. Flows into narrow trenches and over narrow mounds are also computed.
 Viscous and NonNewtonian Flows

Coalescence of two viscoelastic droplets connected by a string
View Description Hide DescriptionA new type of coalescence of viscoelasticdroplets is described and analyzed. A viscoelasticdrop of polyisobutene (PIB) containing high molar mass PIB (HPIB), suspended in polydimethylsiloxane, is separated into two droplets connected by a string, which is called “beadstringbead” (BSB), through repeated elongation and relaxation in a fourroll mill rheometer. Upon flow cessation, the two drops are pulled to approach each other by the string and eventually coalesce. This process exhibits interesting features: The string remains stable like a rod without capillary breakup; the string length and the merging force decay exponentially with time; the string diameter may not change significantly through the approach. The string in BSB is more stable than that of the capillary thinning process of beads on string in jet or fixedend rheometer. These phenomena are modeled based on the force balance of the viscoelasticity of HPIB systems, viscous drag, and the Laplace force. The model prediction agrees with the experimental observation reasonably well, revealing the viscoelastic nature of the coalescence of two droplets. The characteristic time of drop approach is comparable to that of the pulling stress decay. The condition of keeping a constant diameter in BSB approach creates a status where both drop approach and stress decay exponentially with the same characteristic time, which is comparable to the material relaxation time. This case allows the model to analyze the relationships between critical string diameter and the material parameter as well as the process parameters and to discuss the microscopic images of the BSB process.

A mesoscopic rheological model of immiscible blends with the interface covered with a surface active agent
View Description Hide DescriptionThe Maffettone–Minale mesoscopic rheological model of immiscible blends is extended to blends that include a surface active agent. Its nonuniform distribution on the interface (assumed to be the surface of droplets) and the associated with it nonuniform distribution of the surface tension and the Marangoni stress are incorporated into the model. Instead of using one ellipsoid (one conformation tensor in the mathematical formulation) to characterize the shape of the droplet, we use a one parameter family of ellipsoids (a “necklace” of ellipsoids) to play this role. Both rheology and morphology (including large deviations form the ellipsoidal shape of droplets) are calculated and compared to predictions of microscopic models (i.e., models based on microhydrodynamics) and with results of experimental observations.

Asymptotic regimes of ridge and rift formation in a thin viscous sheet model
View Description Hide DescriptionWe numerically and theoretically investigate the evolution of the ridges and rifts produced by the convergent and divergent motions of two substrates over which an initially uniform layer of a Newtonian liquid rests. We put particular emphasis on the various asymptotic selfsimilar and quasiselfsimilar regimes that occur in these processes. During the growth of a ridge, two selfsimilar stages occur; the first takes place in the initial linear phase, and the second is obtained for a large time. Initially, the width and the height of the ridge increase as . For a very large time, the width grows as , while the height increases as . On the other hand, in the process of formation of a rift, there are three selfsimilar asymptotics. The initial linear phase is similar to that for ridges. The second stage corresponds to the separation of the current in two parts, leaving a dry region in between. Last, for a very large , each of the two parts in which the current has separated approaches the selfsimilar viscous dam break solution.
 Particulate, Multiphase, and Granular Flows

Formation of a submillimeter bubble from an orifice using pulsed acoustic pressure waves in gas phase
View Description Hide DescriptionThe mechanism of a bubble production method using pulsed acoustic pressurewaves in gas phase is investigated using highspeed photography. The acoustic characteristics of the present bubblegenerator are also investigated. We found out the optimal acoustic waveform for producing only one bubble per one action; the bubble detachment radius is accurately controlled by first applying the positive onsetassistance acoustic pressurewave and then the negative detachmentassistance acoustic pressurewave with an accurately controlled time lapse. From an orifice with a radius of submerged in water, bubbles with radii in the range of with an extremely small standard deviation of less than are obtained. The shrinking and pinchoff motions of a capillary bridge connecting the bubble and orifice at the time of bubble detachment are observed in detail. The balancing force on a growing bubble, which is based on a spherical bubble formation model is also estimated. As a result, we reveal that when the gas pressure is decreased due to a negative pressurewave, the capillary bridge submerges into the orifice and an upward added mass force is applied on the bubble, both of which cause the detachment of the bubble from the orifice.

Spin coating of a colloidal suspension
View Description Hide DescriptionA discreteelement approach is employed to model the transport, collision, adhesion, and deposition of small colloidal particles in a spin coating process. The computations are used to predict particle distribution and wall adhesion during the nonevaporative phase of spin coating of a thin film, which is important for controlling the abrasiveness, opacity, conductivity, and other properties of the film, as well as for using the deposited particles for growing new materials (e.g., nanotubes). The computations examine the particle distribution and the effect of particle adhesive force on particle deposition during spin coating. Particles are observed to preferentially collect within the film ridge just behind the moving contact line. An increase in the particle adhesive force is observed to lead to enhanced deposition of particles within an inner radius of the film and increase in the aggregate size.

Response of an emulsion of leaky dielectric drops immersed in a simple shear flow: Drops more conductive than the suspending fluid
View Description Hide DescriptionDirect numerical simulation is used to examine the rheological properties of an emulsion of leaky dielectric fluids when an electric field is applied to the system. The emulsion consisting of neutrally buoyant drops is immersed in a simple shear flow where an electric potential difference is applied between the plates. It is assumed that drops are more conductive than the suspending fluid and that the electrical conductivity ratio between the drops and the suspending fluid, , is larger than the dielectric permittivity ratio, . If a single leaky dielectric drop is immersed in an electric field, this combination of properties leads to a viscous fluid motion from the equator to the poles. The response of an emulsion depends on the competition between the electrical forces and the fluid shear. This relation is quantified by the Mason number, . The significance of drop deformability is measured through the electric capillary number, . The microstructure and properties of an emulsion depend mainly on Mn, , and . An emulsion immersed in an electric field exhibits three different regimes for increasing Mn. When the electrical forces are substantially larger than the fluid shear, , the drops aggregate in structures oriented parallel to the electric field that dictate the response of the system. At intermediate shear rates,, the competition between the electrical forces and the fluid shear results in a continuous rearrangement of the aggregated structures. When the shear rate is increased further, , the aggregated structures are broken up, and the effect of the electrical interaction weakens. The application of an electric field leads electrorheological emulsions to exhibit an increase in their effective viscosity for the range of properties examined here, . However, this variation is strongly nonlinear and depends on the microstructure of the emulsion. The deformation and aggregation of the drops caused by the electric field also modifies the elastic properties of the systems. When the strength of the electrical forces is larger than that of the viscous forces, , the electrorheological emulsions exhibit negative values for the first normal stress difference. The electric field causes the drops to deform into a prolate shape in the direction parallel to the electric field. The prolate deformation leads to stronger interfacial stresses in the direction normal to the applied shear, which results in negative values of the first normal stress difference. The dependency of the microstructure and rheological properties on the drop deformability, the electrical conductivity ratio, and the drop volumetric fraction is also discussed. Results for emulsions with a drop volumetric fraction of up to 0.56 are presented.

Response of an emulsion of leaky dielectric drops immersed in a simple shear flow: Drops less conductive than the suspending fluid
View Description Hide DescriptionDirect numerical simulations of the effects of an electric field on an emulsion of drops are presented. A simple shear flow configuration is adopted where the electric field is applied perpendicular to the sliding plates. Both the drops and the suspending fluid are assumed to behave as leaky dielectric fluids. Here, drops less conductive than the suspending fluid with an electrical conductivity ratio smaller than the dielectric permittivity ratio are considered. This combination of electrical properties leads to a viscous fluid motion from the poles to the equator. The response of an emulsion is governed by the competition between the electrical forces, the fluid shear, and the capillary forces. The Mason number and the electric capillary number are used to describe the response of the systems. As previously observed in experiments at low shear rates,, the drops aggregate in chains that tilt under a shear. The competition between the electrical forces and the fluid shear results in shorter chains at intermediate shear rates,. As the fluid shear becomes stronger than the electrical attraction, , the chains of drops break up. The rheological properties mainly depend on the emulsion microstructure. The effective viscosity exhibits a strong shearthinning response because the chains of drops, which appear at low shear rates, increase the resistance of the system to shear. As the chains shorten and break up, the effective viscosity decreases. The elastic properties of the emulsion are also affected by the presence of the electric field. Normal stress differences arise as a consequence of the deformation of the drops and the surface tension acting on the interface between the fluids. The shape of the drops is determined by the deformation caused by the viscous forces and the deformation due to the electric stresses. At low shear rates, the electric effects are predominant and the application of an electric field leads the drops to deform into oblate shapes. The oblate deformation results in higher stresses in the direction parallel to the shearing motion than perpendicular to it, which results in a significant increase in the first normal stress difference. As the shear rate is increased, the oblate deformation is supplemented by the deformation due to the fluid shear. The deformation caused by the electric field is also responsible for the negative magnitude of the second normal stress difference in threedimensional emulsions.

Velocity fluctuations and hydrodynamic diffusion in finiteReynoldsnumber sedimenting suspensions
View Description Hide DescriptionThe velocity variance and the hydrodynamic diffusivity for a finiteReynoldsnumber settling suspension are determined from latticeBoltzmann simulations of many particles in cubic cells with periodic boundary conditions. The velocity variance is found to grow logarithmically with the size of the computational domain in contrast to the algebraic growth found in comparable Stokesflow simulations. The growth rate and size of the velocity variance are found to be smaller than the theoretical prediction for a random suspension owing to a deficit in particle pair probability distribution in the wake of a test particle that screens the velocity disturbance felt by other particles. The particle velocity variance is smaller than the fluid velocity variance because a particle does not follow fluid motions on length scales comparable to or smaller than its own size. The hydrodynamic diffusivity of particles is proportional to the product of the rootmeansquare velocity and the size of the computational domain.

Shearinduced particle migration in binary colloidal suspensions
View Description Hide DescriptionWe present experimental investigations of the spatial and temporal evolution of particle migration in pressure driven flows of Brownian particle suspensions. Binary suspensions of 1.4 and diameter colloidal particles are pumped through a rectangularcrosssection capillary tube. Shear rate gradients caused by the resulting parabolic velocity profile drive the particles away from the walls toward the center of the channel, where the shear rate is lowest. The flows are directly imaged using highspeed laser scanning confocal microscopy. Size segregation of the particles is observed. Depending on the conditions, either the large or the small particles enrich the center. We measure the development of the size segregation by tracking the evolution of the crossstream concentrations of the particles.
 Laminar Flows

Plane model of fluid interface rupture in an electric field
View Description Hide DescriptionModeling of an airfluid interface in an electric field is presented. Specifically, equilibrium of the interface under the dominant forces—electric stress, surface tension, and pressure—is investigated. Since interface shape and equilibrium are related, the shape of an electrified interface is also addressed. To determine the electric stress, an analytical expression for the electric field in the vicinity of the interface is determined. The operating point of the interface is shown to exist in a threedimensional parameter space that is divided by a critical surface into equilibrium, quasiequilibrium, and nonequilibrium subdomains. The three parameters are applied voltage, electrode separation, and pressure difference. Interface size, counterelectrode size, and fluid properties are also considered. The subdomain in which the operating point resides defines the important characteristics of the interface. The operating point moves within, and transfers between, equilibrium subdomains, and points on the critical surface represent “rupture points” of the interface. The final shape of the interface is solved iteratively using this equilibrium model. Interfaces emitting an electrospray can have a range of apex angles, and it is shown that the magnitude of this angle impacts equilibrium. It is revealed that the excess pressure difference term is critical in determining the interface shape (specifically the cone generatrix) and that minimization of the potential energy of all forces can be used to predict the magnitude of the apex angle and pressure immediately after interface rupture. The equilibrium model is important from an operational and optimization perspective, as it is useful to predict the conditions required to break equilibrium and transfer to a quasiequilibrium state (i.e., an electrospray), and the conditions necessary to maintain quasiequilibrium once it is formed.

Implications of hydrophobic interactions and consequent apparent slip phenomenon on the entrance region transport of liquids through microchannels
View Description Hide DescriptionThe implications of entrance region transport in hydrophobic microchannels are theoretically and experimentally investigated in this work. Detailed analytical solutions are derived, depicting the dependences of the liquid phase velocity profiles, entrance lengths, and friction factor variations on the relative thickness of a nanobubbledispersed layer formed in the vicinity of the microchannel wall as a consequence of localized hydrophobic interactions. It is revealed that even for a layer of nanobubbles formed with a typical thickness in the tune of three orders of magnitude lower than the characteristic microchannel dimensions, the entrance length can be enhanced to the limit of about 1.5 times than that for the cases devoid of any hydrophobic interactions. The pressure drop characteristics in the entrance region, as obtained for such cases, can turn out to be of significant consequence with regard to the design of typical pressuredriven microflow systems involving hydrophobic substrates. Closedform expressions for the effective friction factor are also derived so that more accurate and scientific guidelines can be provided for design of hydrophobic microchannels, rather than trivially overruling the consequences of entrance region transport that is commonly exercised on a routine basis.

On slip velocity boundary conditions for electroosmotic flow near sharp corners
View Description Hide DescriptionThe Helmholtz–Smoluchowski (HS) slip velocity boundary condition is often used in computational models of microchannel flows because it allows the motion of the electric double layer(EDL) to be approximated without resolving the charge density profiles close to the walls while dramatically reducing the computational effort required to solve the flowmodel. The approximation works well for straight channel flows but breaks down in areas of high wall curvature such as sharp corners, where large nonphysical velocities are generated. Many microfluidic applications such as the onchip focusing and separation of biomolecules rely on the interaction of electroosmosis and electrophoresis in complex channel geometries. In order for these effects to be properly treated using the slip velocity boundary condition, the errors introduced into the solution at corners must be understood. In this article, a complete model for the ion concentrations, electric field, and fluid flow in complex microchannel geometries is presented and is used to compute a pure electroosmotic flow in a twodimensional microchannel cross slot. The full model solution near the corner at the edge of the EDL is compared to the approximate solution computed by using the HS boundary condition, and it is shown that the accuracy of the approximate solution may be greatly increased by “patching” the full solution as a boundary condition for the approximate solution at the edge of the double layer region. Finally, an empirically derived modified slip velocity boundary condition for electroosmotic flow is proposed. It is shown to improve the accuracy of the flow solution at sharp corners by about 60% when compared to the original boundary condition while also delivering a modest improvement in computational performance because of the elimination of a singularity in the velocity field.
 Instability and Transition

Convective instability of an Ohmic liquid layer in an unsteady thermal field
View Description Hide DescriptionThe linear stability problem for an incompressible viscous poorly conducting Ohmic liquid between two rigid horizontal boundaries with timeperiodic temperature distribution and under steady transverse electric field is considered. The free charge in the liquid is due only to the nonuniform electroconduction. The electrohydrodynamic (EHD) approximation is used. Floquet theory is applied for finding various instability thresholds in linear approximation. The influence of timedependent temperature field modulation on the liquid layer behavior is studied with and without an additional steady component. The boundaries of instability and the characteristics of critical disturbances are determined. Depending on the frequency and amplitude of modulation, the temperature gradient can destabilize the equilibrium of the liquid or stabilize otherwise unstable base state. In addition to synchronous and subharmonic responses to external modulation, the instability can be associated with quasiperiodic disturbances. The critical electric Rayleigh number is given as a function of frequency and heating level. The limit of low frequency modulation is studied by an asymptotic method.

Threedimensional effects in shockcylinder interactions
View Description Hide DescriptionThe shockcylinder interaction is important for understanding the mixing of fluids by impulsively accelerated material interfaces. The flowfield development is highly sensitive to the initial conditions, which can be difficult to fully characterize in experimental facilities. In this work, simulations based on experimental measurements are used to model the initial flowfield at the Los Alamos shocktube facility. While nominally twodimensional, our simulations show the initial column has significant axial variation. These numerical solutions are then used as the initial conditions for threedimensional simulations of the shockcylinder interaction. A genuinely threedimensional flowfield develops as a consequence of the initial, axial variation of the cylinder.

Thermoacoustic instability in a Rijke tube: Nonnormality and nonlinearity
View Description Hide DescriptionThe role of nonnormality and nonlinearity in thermoacoustic interaction in a Rijke tube is investigated in this paper. The heat release rate of the heating element is modeled by a modified form of King’s law. This fluctuating heat release from the heating element is treated as a compact source in the onedimensional linear model of the acoustic field. The temporal evolution of the acoustic perturbations is studied using the Galerkin technique. It is shown that any thermoacoustic system is nonnormal. Nonnormality can cause algebraic growth of oscillations for a short time even though the eigenvectors of the system could be decaying exponentially with time. This feature of nonnormality combined with the effect of nonlinearity causes the occurrence of triggering, i.e., the thermoacoustic oscillations decay for some initial conditions whereas they grow for some other initial conditions. If a system is nonnormal, then there can be large amplification of oscillations even if the excited frequency is far from the natural frequency of the system. The dependence of transient growth on time lag and heater positions is studied. Such amplifications (pseudoresonance) can be studied using pseudospectra, as eigenvalues alone are not sufficient to predict the behavior of the system. The geometry of pseudospectra can be used to obtain upper and lower bounds on the growth factor, which provide both necessary and sufficient conditions for the stability of a thermoacoustic system.

Threedimensional structure of confined swirling jets at moderately large Reynolds numbers
View Description Hide DescriptionWe have performed a series of threedimensional (3D) numerical simulations of the incompressible flow discharging from a rotating pipe into a coaxial static cylindrical container through a sudden expansion. We have considered several values of the Reynolds number based on the pipe flow rate between 50 and 300, and an expansion diameter ratio of , and have analyzed the emerging 3D flow structures in the swirling jet exiting from the rotating pipe as the swirl parameter is increased. The results are compared to axisymmetric numerical simulations of the same problem. Threedimensional, nonlinear instabilities are found in the swirling jet when above a critical value of , which depends on , that obviously do not appear in the axisymmetric simulations. These nonlinear instabilities are initially triggered by the linear instabilities inside the rotating pipe, which are already present in the pipe from a much lower value of , and are transformed in the jet. As increases further, there exists another critical value above which the swirling jet undergoes vortex breakdown, producing a flow in the jet which is basically axisymmetric. This critical value of the swirl parameter for breakdown is significantly larger than that found in the axisymmetric simulations. Thus, one of the main results of the present work is that 3D instabilities delay the formation of vortex breakdown in the jet, in relation to the same axisymmetric flow, but once the vortex breakdown phenomenon occurs, the 3D instabilities coming from the rotating pipe appear to be suppressed in the jet, and the swirling flow becomes basically axisymmetric again. Finally, the axisymmetric simulations show that the jet becomes unstable to axisymmetric perturbations, when , above another critical value of . However, these axisymmetric instabilities do not appear in the 3D simulations because the flow becomes unstable to asymmetric perturbations at much lower values of .

Reynolds number dependence of plane jet development in the transitional regime
View Description Hide DescriptionThe transitional characteristics of plane turbulent jets have been investigated in the present study. Hot wire measurements have been performed for a jet issuing from a rectangular nozzle of aspect ratio 20, for Reynolds number varying in the range . In this range, the characteristics of flow development are found to be Reynolds number dependent, in contrast to the fully developed turbulent jets which show features independent of initial conditions such as inlet Re. For low Re jets, the jet spread is significantly influenced by the low frequency oscillations caused by shear layer instability. Large sized vortices are formed in the shear layers at the fundamental frequency of the instability, which lead to subharmonic low frequency oscillations due to vortex pairing and merger, at larger axial distances. Consequently, the far field flow structure of a low Re jet is dominated by large size vortices which give rise to a higher level of flow intermittency, larger entrainment of ambient fluid, and faster jet decay, as compared to high Re jets. Also, in the absence of finer scales and broader spectrum of eddies, the mean flow field achieves selfsimilar structure much ahead of the fluctuating components and fully developed turbulent flow characteristics are not observed, even in the far field. In high Re jets, on the other hand, the vortex breakup processes also simultaneously occur along with vortex pairing and merger. Therefore, energy transfers to a broad spectrum of scales and finer scales are observed even in the near field of the jet. Although the achievement of selfsimilarity for the mean field is slightly faster than that for the fluctuating components, turbulence also attains a fully developed state at about a nondimensional axial distance of 20. The associated probability density functions of the fluctuating components evolve into Gaussian profiles, implying isotropic turbulence. Due to the dominance of finer scales, the overall entrainment level is less and decay is slower for a high Re jet.