Volume 21, Issue 3, March 2009

The response of steady state vortexflows in an enclosed circular cylinder driven by the harmonic modulation of the rotating end wall is investigated experimentally and numerically. Three dynamic regimes have been identified, with a continuous variation in forcing frequency between them. For very low forcing frequency, the synchronous flow approaches quasistatic adjustment, and for very large forcing frequencies the oscillations in the synchronous flow are localized in the boundary layers on the various cylinder walls. These localized wall oscillations drive the synchronous flow in the cylinder interior to the underlying axisymmetric steady basic state. The third regime occurs for forcing frequencies in the range of the most dangerous axisymmetric Hopf eigenfrequencies, with the 1:1 resonances leading to greatly enhanced oscillation amplitudes localized in the axis region where the flow manifests vortex breakdown recirculation zones.
 LETTERS


Maximization of granular outflow by oblique exits and by obstacles
View Description Hide DescriptionWe investigate experimentally the intermittent discharge of a granular medium out of an exit at the bottom of a vertically shaken box. Changing the orientation of the bottom shows that there exists an angle (around 20°–25° with respect to the horizontal) at which the mean discharge rate increases up to a factor 1.9, as compared to the rate with horizontal bottom. Furthermore, adjusting the diameter and the distance of a cylindrical obstacle above the exit on the (horizontal) bottom, allows to optimize the mean rate of discharge up to 3.5 times the rate without obstacle.

Prolonged residence times for particles settling through stratified miscible fluids in the Stokes regime
View Description Hide DescriptionThe behavior of settling particles in stratified fluid is important in a variety of applications, from environmental to medical. We document a phenomenon in which a sphere, when crossing density transitions, slows down substantially in comparison to its settling speed in the bottom denser layer, due to entrainment of buoyant fluid. We present results from an experimental study of the effects of the fluid interface on flight times as well as a theoretical model derived from first principles in the low Reynolds number regimes for stratified miscible fluids. Our work provides a new predictive tool and gives insight into the role of strong stratification in particle settling.
 Top

 ARTICLES

 Interfacial Flows

Early postimpact time dynamics of viscous drops onto a solid dry surface
View Description Hide DescriptionThe spreading dynamics of liquid drops normally impacting a solid dry surface at high Reynolds and Weber numbers is experimentally and numerically studied at early postimpact times starting from after impact. The focus is on the emergence and growing of the axisymmetric liquid lamella underneath the drop, that is, on the time evolution of its thickness, radius, and velocity, as a function of impact velocity and liquid viscosity. The Navier–Stokes equations for twophase flows are solved numerically by an artificial compressibility method. A “shockcapturing” method is used for the tracking of the gasliquid interface, neglecting surface tension effects. Experimental and numerical results are interpreted using a simple scaling analysis that reveals the characteristic lengths and velocities of the spreading dynamics. In particular, a finite characteristic time of appearance for the lamella is found, which is of the order of . Rescaling of the data works satisfactorily in the considered range of parameters. Thus, the lamella ejection is limited by viscosity.

Threedimensional thin film flow over and around an obstacle on an inclined plane
View Description Hide DescriptionSteady Stokes flow driven by gravity down an inclined plane over and around an attached obstacle is considered. The effects of the obstacle are examined for various flow configurations and results produced for flow over hemispherical obstacles. Comparison is made with previously published papers that assume that the obstacle is small and/or the free surface deflection and disturbance velocity are small. Values for the unit normal and curvature of the free surface are found using both finite difference approximations and Hermitian radial basis function interpolations, with the resulting solutions compared. Free surface profiles for thin film flows over hemispherical obstacles that approach the film surface are produced and the effects of near point singularities considered. All free surface profiles indicate an upstream peak, followed by a trough downstream of the obstacle with the peak decaying in a “horseshoe” shaped surface deformation. Flow profiles are governed by the plane inclination, the Bond number, and the obstacle geometry. An extension of this approach provides a new class of solutions where a thin film flows around a cylindrical obstacle. Notably, the static contact line angle between the free surface and the obstacle is introduced as an extra flow parameter and its effect investigated for a given set of flow parameters and fixed boundary conditions.Solutions are obtained where steady flow profiles can be found both over and around a cylindrical obstacle raising the awareness of possible multiple solutions.

Role of geometry and fluid properties in droplet and thread formation processes in planar flow focusing
View Description Hide DescriptionDropletformation processes in microfluidic flow focusing devices have been examined previously and some of the key physical mechanisms for dropletformation revealed. However, the underlying physical behavior is still too poorly understood to utilize it for generating droplets of precise size. In this work, we formulate scaling arguments to define dimensionless variables which capture all the parameters that control the droplet breakup process, including the flow rates and the viscosities of the two immiscible fluids, the interfacial tension between the fluids and the numerous dimensions in the flow focusing device. To test these arguments, we perform flow focusing experiments and systematically vary the dimensional parameters. Through these experiments, we confirm the validity of the scaling arguments and find a power law relationship between the normalized droplet size and the capillary number. We demonstrate that dropletformation can be separated into an upstream process for primary dropletformation and a downstream process for thread formation. These results are key to the ability to tune the flow focusing process for specific applications that require monodisperse micron and submicron droplets and particles.

Charge separation in the conical meniscus of an electrospray of a very polar liquid: Its effect on the minimum flow rate
View Description Hide DescriptionAn analysis is presented of the flow and the distribution of charge in the meniscus of an electrospray of a very polar liquid which is fed with a low flow rate. The shape of the meniscus is taken to be a Taylor cone. The characteristic value of the flow rate at which the liquid ceases to be quasineutral in a certain relaxation region of the conical meniscus under the action of the applied field is estimated, and the current/flow rate characteristic of the electrospray is numerically computed in these conditions. A state of complete charge separation, in which the ions that are pushed by the electric field away from the tip of the meniscus cease to reach the jet of the electrospray, is found for a finite value of the flow rate, and no stationary solution exists below this flow rate. For very polar liquids of small viscosity, this minimum flow rate is of the order of the experimental minimum for the conejet regime when the flux of the electric field in the jet is taken into account. The flow induced in the meniscus by the Coulomb force in the bulk of the liquid and the electric shear stress at its surface is computed and its effect on the distribution of charge and the minimum flow rate is analyzed. Estimates of the flow and the electric current in the jet are worked out for a range of flow rates above the minimum.

Marangonidriven spreading along liquidliquid interfaces
View Description Hide DescriptionMarangonidriven spreading at gasliquid interfaces has been studied extensively over the past years but so far the spreading kinetics along the interface between immiscible liquids has not been investigated systematically. In this study, the spreading kinetics of aqueous solutions of sodium dodecyl sulfate and dodecyl trimethyl ammonium bromide along the interface between thick layers of water and decane has been investigated by means of two different optical visualization techniques (dye tracer and laser shadowgraphy). The spreading kinetics follows a power law where the radius as function of time scales as indicating large similarities with Marangonidriven spreading at airliquid interfaces. The existing scaling law for spreading at airliquid interfaces is based on the balance between an interfacial tension gradient and the viscous stress in the fluid layers beneath the interface. When the viscous dissipation in the two boundary layers below and above the interface is factored into the scaling law, quantitative agreement with experimental data is obtained. Marangonidriven spreading along an interface is a fast transport mechanism. The velocity of the leading edge lies within the range of group velocities of capillary waves.

Mechanism study of deformation and mass transfer for binary droplet collisions with particle method
View Description Hide DescriptionThe binary collision of two identical liquid droplets is simulated using the moving particle semiimplicit method. We focus on various coalescence and separation mechanisms and the accompanying mass transfer characteristics. A modified surfacetension model is implemented in threedimensional numerical simulations to study the large deformation processes. Both headon collision and eccentric impact are investigated, and a mechanism map is established to qualitatively distinguish different regimes of impact.Mass transfer properties are obtained by tracking the movement of particles, which are useful for identifying the mixing rate of the droplets after coalescence or separation as well as the source of the newly formed satellite droplets. A mixing map (in terms of impact speed and impact number) is also established to provide guidelines of pursuing higher efficiency of mixing two liquids using collision. The results qualitatively agree with previous experiments and the versatile numerical protocol may also find applications in studying the free surface flows and interface deformation.

Influence of electric field on saturated film boiling
View Description Hide DescriptionElectrohydrodynamic (EHD) forces enhance heat and mass transfer in fluid flows. In twophase flows under the influence of an electric field, an EHD force acts on the interface resulting in an enhanced interfacial motion. We perform a numerical study of the effect of the application of an electric field on bubble density and heat transfer characteristics during saturated film boiling. Application of electric field results in shorter bubble separation distances, faster growth of the instability, and higher bubble release frequency. Increasing the electricfield intensity shows an increase in the space averaged Nusselt number, thus indicating the role of electric field in the enhancement of heat transfer. We perform full nonlinear simulations of saturated film boiling coupled with electrohydrodynamics using a coupled level set and volume of fluid algorithm. Simulations have been performed for water and refrigerant R123a at near critical pressures.

Capillary oscillations of a constrained liquid drop
View Description Hide DescriptionAn inviscid spherical liquid drop held by surface tension exhibits linear oscillations of a characteristic frequency and mode shape (Rayleigh oscillations). If the drop is pinned on a circle of contact the mode shapes change and the frequencies are shifted. The linear problem of inviscid, axisymmetric, volumepreserving oscillations of a liquid drop constrained by pinning along a latitude is solved here. The formulation gives rise to an integrodifferential boundary value problem, similar to that for Rayleigh oscillations, and for oscillations of a drop in contact with a spherical bowl [M. Strani and F. Sabetta, J. Fluid Mech.141, 233 (1984)], only more constrained. A spectral method delivers a truncated solution to the eigenvalue problem. A numerical routine has been used to generate the eigenfrequencies/eigenmodes as a function of the location of the pinned circle of constraint. The effect of pinning the drop is to introduce a new lowfrequency eigenmode. The centerofmass motion, important in application, is partitioned among all the eigenmodes but the lowfrequency mode is its principal carrier.

Dynamics of liquid jets and threads under the action of radial electric fields: Microthread formation and touchdown singularities
View Description Hide DescriptionWe study theoretically the axisymmetric nonlinear dynamics of viscous conducting liquid jets or threads under the action of a radial electric field. The field is generated by a potential difference between the jet surface and a concentrically placed electrode of given radius. We develop a long wave nonlinear model that is used to predict the dynamics of the system and, in particular, to address the effect of the radial electric field on jet breakup. Two canonical regimes are identified that depend on the size of the gap between the outer electrode and the unperturbed jet surface. For relatively large gap sizes, long waves are stabilized for sufficiently strong electric fields but remain unstable as in the nonelectrified case for electric field strengths below a critical value. For relatively small gaps, an electric field of any strength enhances the instability of long waves as compared to the nonelectrified case. We carry out numerical simulations based on our nonlinear models to describe the nonlinear evolution and terminal states in these two regimes. We find that jet pinching does not occur irrespective of the parameters. We identify regimes where capillary instability leads to the formation of stable quasistatic microthreads (connected to large main drops) whose radius decreases with the strength of the electric field. The generic ultimate singular event described by our models is the attraction of the jet surface toward the enclosing electrode and its contact with the electrode in finite time. A selfsimilar closed form solution is found that describes this event with the interface near touchdown having locally a cusp geometry. The theory is compared to the timedependent simulations with excellent agreement.
 Viscous and NonNewtonian Flows

Oxygen and carbon dioxide transport in timedependent blood flow past fiber rectangular arrays
View Description Hide DescriptionThe influence of timedependent flows on oxygen and carbon dioxide transport for blood flow past fiber arrays arranged in inline and staggered configurations was computationally investigated as a model for an artificial lung. Both a pulsatile flow, which mimics the flow leaving the right heart and passing through a compliance chamber before entering the artificial lung, and a right ventricular flow, which mimics flow leaving the right heart and directly entering the artificial lung, were considered in addition to a steady flow. The pulsatile flow was modeled as a sinusoidal perturbation superimposed on a steady flow while the right ventricular flow was modeled to accurately depict the period of flow acceleration (increasing flow) and deceleration (decreasing flow) during systole followed by zero flow during diastole. It was observed that the pulsatile flow yielded similar gas transport as compared to the steady flow, while the right ventricular flow resulted in smaller gas transport, with the decrease increasing with Re. The pressure drop across the fiber array (a measure of the resistance), work (an indicator of the work required of the right heart), and shear stress (a measure of potential blood cell activation and damage) are lowest for steady flow, followed by pulsatile flow, and then right ventricular flow. The pressure drop, work, shear stress, and Sherwood numbers (a measure of the gas transport efficiency) decrease with increasing porosity and are smaller for as compared to (AR is the distance between fibers in the flow direction/distance between fibers in direction perpendicular to flow), although for small porosities the Sherwood numbers are of similar magnitude. In general, for any fiber array geometry, high pressure drop, work, and shear stresses correlate with high Sherwood numbers, and low pressure drop, work, and shear stresses correlate with low Sherwood numbers creating a need for a compromise between pressure drop/work/shear stresses and gas transport.

Swimming speeds of filaments in nonlinearly viscoelastic fluids
View Description Hide DescriptionMany microorganisms swim through gels and nonNewtonian fluids in their natural environments. In this paper, we focus on microorganisms which use flagella for propulsion. We address how swimming velocities are affected in nonlinearly viscoelastic fluids by examining the problem of an infinitely long cylinder with arbitrary beating motion in the OldroydB fluid. We solve for the swimming velocity in the limit in which deflections of the cylinder from its straight configuration are small relative to the radius of the cylinder and the wavelength of the deflections; furthermore, the radius of the cylinder is small compared to the wavelength of deflections. We find that swimming velocities are diminished by nonlinear viscoelastic effects. We apply these results to examine what types of swimming motions can produce net translation in a nonlinear fluid, comparing to the Newtonian case, for which Purcell’s “scallop” theorem describes how timereversibility constrains which swimming motions are effective. We find that a leading order violation of the scallop theorem occurs for reciprocal motions in which the backward and forward strokes occur at different rates.

An efficient direct simulation Monte Carlo method for low Mach number noncontinuum gas flows based on the Bhatnagar–Gross–Krook model
View Description Hide DescriptionThe direct simulation Monte Carlo (DSMC) method is the preferred approach for simulating rarefied gas flows in complex geometries. However, the standard DSMC method becomes inefficient in the limit when the thermal velocity fluctuations which scale with the speed of sound are much larger than characteristic ensembleaveraged flow speed . In this paper, we propose a modified DSMC algorithm which simulates the linearized Bhatnagar–Gross–Krook (BGK) approximation to the Boltzmann equation. The particles in this method are uniformly distributed in space and have a reststate, equilibrium Maxwellian velocity distribution. The deviations from the equilibrium state are captured by weightings, indicating that each particle represents a noninteger expectation for finding gas molecules with the simulation particle’s velocity in the spatial cell where it is found. The BGK approximation makes the implementation of such a method simple and efficient because the BGKcollision rule can be implemented by adjusting particle weightings without the need to create new particles during the simulation. The method can be incorporated into existing DSMC simulation programs with minimal changes. We apply the new algorithm to two test problems—pressuredriven flow through a nanochannel and flow around an isolated sphere. Results obtained with the new method are in excellent agreement with previous theories and simulations.

On the flow of associative polymers past a sphere: Evaluation of negative wake criteria
View Description Hide DescriptionA study on falling spheres descending in associative polymers with spherecontainer ratios of 0.05–0.15 for various polymer concentrations and Weissenberg numbers is presented. The fluid exhibits constant viscosity over a wide range of small to moderate shear rates, and shear thinning for large shear rates. The simple shear rheology and linear viscoelasticity of these polymers are modeled with the BMP equation of state [F. Bautista, J. M. de Santos, J. E. Puig, and O. Manero, J. NonNewtonian Fluid Mech. 80, 93 (1999); O. Manero, F. Bautista, J. F. A. Soltero, and J. E. Puig, J. NonNewtonian Fluid Mech. 106, 1 (2002)], which enables the prediction of the extensional viscosity as a function of the strain rate. The particle image velocimetry technique allows the measurement of the velocity field in the rear of the sphere. The container wall affects the formation of the negative wake at a critical Weissenberg number, which closely corresponds to the region around the peak of extension thickening of the Trouton ratio in the solution. A characteristic strain rate is estimated from the distance of the sphere surface to the stagnant point where the velocity changes direction. Using these data, various criteria for the appearance of the negative wake are discussed. Conclusions reached by Dou and PhanThien [Rheol. Acta43, 203 (2004)] on the physical mechanisms for negative wake generation, are in agreement with the results exposed in this work.

Volume viscosity in fluids with multiple dissipative processes
View Description Hide DescriptionThe variational principle of Hamilton is applied to derive the volume viscosity coefficients of a reacting fluid with multiple dissipative processes. The procedure, as in the case of a single dissipative process, yields two dissipative terms in the Navier–Stokes equation: The first is the traditional volume viscosity term, proportional to the dilatational component of the velocity; the second term is proportional to the material time derivative of the pressure gradient. Each dissipative process is assumed to be independent of the others. In a fluid comprising a single constituent with multiple relaxation processes, the relaxation times of the multiple processes are additive in the respective volume viscosity terms. If the fluid comprises several relaxing constituents (each with a single relaxation process), the relaxation times are again additive but weighted by the mole fractions of the fluid constituents. A generalized equation of state is derived, for which two special cases are considered: The case of “lowentropy production,” where entropy variation is neglected, and that of “high entropy production,” where the progress variables of the internal molecular processes are neglected. Applications include acoustical wave propagation, Stokes flow around a sphere, and the structure and thickness of a normal shock. Finally, it is shown that the analysis presented here resolves several misconceptions concerning the volume viscosity of fluids.

Effective viscosity of a dilute suspension of membranebound inclusions
View Description Hide DescriptionWhen particulate suspensions are sheared, perturbations in the shear flows around the rigid particles increase the local energy dissipation so that the viscosity of the suspension is effectively higher than that of the solvent. For bulk (threedimensional) fluids, understanding this viscosity enhancement is a classic problem in hydrodynamics that originated over a century ago with Einstein’s study of a dilute suspension of spherical particles [A. Einstein, Ann. Phys.19, 289 (1906)]. In this paper, we investigate the analogous problem of the effective viscosity of a suspension of disks embedded in a twodimensional membrane or interface. Unlike the hydrodynamics of bulk fluids, lowReynolds number membranehydrodynamics is characterized by an inherent length scale generated by the coupling of the membrane to the bulk fluids that surround it. As a result, we find that the size of the particles in the suspension relative to this hydrodynamic length scale has a dramatic effect on the effective viscosity of the suspension. Our study also helps elucidate the mathematical tools needed to solve the mixed boundary value problems that generically arise when considering the motion of rigid inclusions in fluid membranes.
 Particulate, Multiphase, and Granular Flows

Effects of particle properties on segregationband drift in particleladen rimming flow
View Description Hide DescriptionWe experimentally study rimming flow of a particleladen fluid. We begin to investigate the details of the spatiotemporal segregationband dynamics that were first documented by us elsewhere [E. Guyez and P. J. Thomas, Phys. Rev. Lett.100, 074501 (2008)]. There exist eight relevant nondimensional parameters that must be expected to affect the drift dynamics of segregation bands in particleladen rimming flow. Here we summarize results from experiments investigating the effects of three of these parameters that involve the particle size and the particle density. It is shown that two of the parameters are crucial to the initiation of the band drift and that bands become stationary whenever either one of the two parameters adopts values below an associated critical threshold. Based on the physical relevance of the two parameters it is concluded that the initiation of band drift is strongly affected by a competition between capillary forces and gravitational forces. The third nondimensional parameter studied here characterizes the bulk particle concentration and it is found that it controls the banddrift speed in the parameter regime where band drift exists.

Forces on a finitesized particle located close to a wall in a linear shear flow
View Description Hide DescriptionTo understand and better model the hydrodynamic force acting on a finitesized particle moving in a wallbounded linear shear flow, here we consider the two limiting cases of (a) a rigid stationary spherical particle in a linear wallbounded shear flow and (b) a rigid spherical particle in rectilinear motion parallel to a wall in a quiescent ambient flow. In the present computations, the particle Reynolds number ranges from 2 to 250 at separation distances to the wall from nearly sitting on the wall to far away from the wall. First we characterize the structure of the wake for a stationary particle in a linear shear flow and compare with those for a particle moving parallel to a wall in a quiescent ambient [see L. Zeng, S. Balachandar, and P. Fischer, J. Fluid Mech.536, 1 (2005)]. For both these cases we present drag and lift results and obtain composite drag and lift correlations that are valid for a wide range of Re and distance from the wall. These correlations have been developed to be consistent with all available low Reynolds numbertheories and approach the appropriate uniform flow results at large distance from the wall. Particular attention is paid to the case of particle in contact with the wall and the computational results are compared with those from experiments.

Hydrodynamic diffusion and mass transfer across a sheared suspension of neutrally buoyant spheres
View Description Hide DescriptionWe present experimental, theoretical, and numerical simulation studies of the transport of fluidphase tracer molecules from one wall to the opposite wall bounding a sheared suspension of neutrally buoyant solid particles. The experiments use a standard electrochemical method in which the mass transfer rate is determined from the current resulting from a dilute concentration of ions undergoing redox reactions at the walls in a solution of excess nonreacting ions that screen the electric field in the suspension. The simulations use a latticeBoltzmann method to determine the fluid velocity and solid particle motion and a Brownian tracer algorithm to determine the chemical tracer mass transfer. The mass transport across the bulk of the suspension is driven by hydrodynamicdiffusion, an apparent diffusive motion of tracers caused by the chaotic fluid velocity disturbances induced by suspended particles. As a result the dimensionless rate of mass transfer (or Sherwood number) is a nearly linear function of the dimensionless shear rate (Peclet number) at moderate values of the Peclet number. At higher Peclet numbers, the Sherwood number grows more slowly due to the mass transport resistance caused by a moleculardiffusion boundary layer near the solid walls. Fluid inertia enhances the rate of mass transfer in suspensions with particle Reynolds numbers in the range of 0.5–7.