Volume 18, Issue 8, August 2006
 LETTERS


Hybrid synthetic jets as the nonzeronetmassflux synthetic jets
View Description Hide DescriptionThe “hybrid synthetic jet” combines the zeronetmassflux synthetic jet and fluidic pumping through a valveless pump. No fluid is supplied from an external source (blower or compressor). Hotwire experiments on a demonstration model confirm that the hybrid synthetic jet exhibits a higher extrusion volume flow rate than the ordinary synthetic jet. For the tested configuration, the increase is by 25% at the 1.30 ratio of the extrusion and suction volume fluxes.

Effects of vortex filaments on the velocity of tracers and heavy particles in turbulence
View Description Hide DescriptionThe Lagrangian statistics of heavy particles and of fluid tracers transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. The Lagrangian velocity structure functions are measured in a time range spanning about three decades, from a tenth of the Kolmogorov time scale, , up to a few largescale eddy turnover times. Strong evidence is obtained that fluid tracer statistics are contaminated in the time range by a bottleneck effect due to vortex filament. This effect is found to be significantly reduced for heavy particles which are expelled from vortices by inertia. These findings help in clarifying the results of a recent study by H. Xu et al. [Phys. Rev. Lett.96, 024503 (2006)], where differences between experimental and numerical results on scaling properties of fluid tracers were reported.
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 ARTICLES

 Interfacial Flows

Pertubations on a liquid curtain near breakup: Wakes and free edges
View Description Hide DescriptionWe report experiments on liquid curtains falling between two vertical wires. The flow is mainly driven by gravity, so that the Weber number (We) (ratio of momentum flux to twice the surface tension) is close to zero at the top of the curtain and increases downstream, with the possible existence of a location where We equals 1 (which turns out to be a singular point in the sheet, in terms of waves propagation). In the present paper, we focus on the curtain response to localized perturbations, i.e., formation of either surface waves or free edges behind a thin needle touching the curtain, with a special emphasis to what happens near the breakup limit. We extract and compare the shapes of two kind of “wakes” left behind the obstacle: classical triangular wake of standing sinuous waves and stationary hole involving two free edges pinned on the needle. It is found that these two wakes are very similar for high enough We, but behave very differently when We reaches 1 from above; the sinuous wake disappears, while the “hole wake” still exists, and its shape becomes rounded. Below , the hole can either stay stable, oscillate or expand and break the curtain. We provide exact analytical expressions for stationary freeedges that compare very well with experiments.

Tip streaming from a liquid drop forming from a tube in a coflowing outer fluid
View Description Hide DescriptionDynamics of formation of a drop of an incompressible Newtonian fluid of viscosity and density from the tip of a tube of radius into a coflowing immiscible, incompressible Newtonian fluid of viscosity and density that is enclosed in a concentric cylindrical tube of radius are investigated under creeping flow conditions. Transient drop shapes, and fluid velocities and pressures, are calculated numerically by solving the governing Stokes equations with the appropriate boundary and initial conditions using the Galerkin/finite element method for spatial discretization and an adaptive finite difference method for time integration. In accord with previous studies, the primary effect of increasing the ratio of the volumetric injection rate of the outer fluid to that of the inner fluid , , is shown to be a reduction in the volume of primary drops that are formed. When is small, calculations show that drop formation occurs in a slug flow regime where the primary drops that are about to be formed are elongated axially and occupy virtually the entire cross section of the outer tube. In this slug flow regime, the primary drops at breakup resemble cylinders that are terminated by hemispherical caps and their aspect ratios , where denotes their axial lengths and their maximum diameters. As increases, the dynamics are shown to transition to the dripping regime, where the primary drops are more globular, , and their radii are of the order of or smaller than . As increases, the importance of viscous stress exerted by the outer fluid relative to the surface tension or capillary pressure increases. Thus, the drop length measured from the tube exit to the drop tip at breakup increases while the primary drop volume decreases as increases. When is sufficiently large, viscous stress exerted by the outer fluid induces a recirculating flow within a formingdrop. Once exceeds a critical value , viscous stress exerted by the outer fluid becomes so large that the growing drop takes on a conical shape and a thin fluid jet with a radius that is a few orders of magnitude smaller than that of the radius of the inner tube emanates from its tip. This latter regime of drop breakup, which is henceforward referred to as tip streaming, is remarkably similar to electrohydrodynamic jetting that is seen from the tips of conical drops in electric fields and tip streaming that occurs from the pointed ends of surfactantcovered free drops subjected to linear extensional or shear flows, but takes place here in the absence of electric fields or surfactants. Scaling arguments for fixed show that for exterior viscous stress to overcome capillary pressure and cause tip streaming, , where is the capillary number, is the interfacial tension, and is the viscosity ratio. In accordance with the scaling arguments, the computed predictions show that the critical value of the flow rate ratio signaling transition from dripping to tip streaming varies inversely with for small to moderate but becomes independent of both and as viscosity ratio grows without bound.

Marangoni flows during drying of colloidal films
View Description Hide DescriptionIn this study, we consider the drying of a thin film that contains a stable dispersion of colloidal particles so that a coating of these particles is formed after the liquid is driven off by evaporation. For sufficiently thin films, we show that evaporative cooling can drive a Marangoni flow that results in surface deformation of the drying film. A thinfilm approximation is used to describe the velocity and temperature fields, and the particle transport equation with convective terms retained is used to describe the concentration field. A coupled finite difference/spectral element scheme is implemented numerically to solve the particle transport equation, where high accuracy is required to resolve sharp gradients within the film and to ensure particle conservation during drying. The model employed is capable of describing the evolution of film thickness and concentration field up to the time when maximum packing is nearly reached at some point in the domain. Three types of film structures are observed, all characterized by a final nonuniform thickness. In the first type, observed at low Peclet numbers, the maximum concentration is reached at the thinnest points in the film, which surround elevations with lower particle concentrations. This mode of instability suggests that dried coatings will have pronounced nonuniformities, resulting in the formation of craters or pinholes. In the second type, observed at high Peclet numbers, a closely packed skin of nonuniform thickness is formed, with low concentration fluid remaining beneath the elevations. In the final stages of drying one would expect capillary pressure to pull particles in the underlying fluid toward the skin, thus creating voids under a seemingly homogeneously applied coating. Finally, still at relatively large particle Peclet numbers and when the destabilizing Marangoni stresses are sufficiently strong, floating lumps of closely packed particles may form in the vicinity of film elevations.

Capillary pressure of a liquid in a layer of closepacked uniform spheres
View Description Hide DescriptionThe capillary pressure in liquid partially filling the pore space in a layer of equidimensional closepacked spheres has been calculated numerically and studied experimentally. The case of square packing when the centers of the spheres are in the same plane and lie at the corners of a square receives primary consideration for zero gravity. In the absence of gravity, the menisci shapes of a liquid that occupies some fraction of the pore space are constructed using the Surface Evolver code. The mean curvature (and, hence, the capillary pressure) of the liquid surface is calculated. The dependence of capillary pressure on the liquid volume is obtained for selected contact angles in the range . The evolution of the shape of the liquid’s free surface and the capillary pressure under quasistatic infiltration and drainage can be deduced from these results. The maximum pressure difference between liquid and gas required for a meniscus passing through a pore is calculated and compared with that for hexagonal packing and with an approximate solution given by Mason and Morrow. The lower and upper critical liquid volumes that determine the stability limits for the equilibrium of a capillary liquid in contact with spheres in a square packed array are tabulated for a set of contact angles. To assess the applicability of the obtained results to multiple layers, the possibility that the constructed menisci intersect spheres from adjacent layers has been analyzed. The effect of gravity has also been examined. For square packing the dependence of capillary pressure on liquid volume is constructed for selected contact angles and Bond numbers, , and compared with the case of zero gravity . Maximum capillary pressures are calculated for a set of contact angles and in the limiting cases of square packing and hexagonal packing. Experiments were performed for a layer of squarepacked spheres to compare with numerical predictions.

An experimental analysis of the linear vibration of axisymmetric liquid bridges
View Description Hide DescriptionThe linear vibration of axisymmetric liquid bridges of millimetric size was analyzed experimentally. The equilibrium shape and the evolution of the interface position were recorded by a highspeed video camera. To calculate the surface tension, the Theoretical Image Fitting Analysis method was adapted to process images of liquid bridges at equilibrium. An image processing method was developed to obtain both the position of the solid supports delimiting the liquid bridge and the interface deformation as a function of time. The former allows one to accurately measure the inertial force acting on the liquid bridge. The interface deformation was determined at the subpixel level, so that vibrations of very small amplitude could be analyzed. The results are compared with various theoretical approaches in the linear regime.

A threedimensional spectral boundary element algorithm for interfacial dynamics in Stokes flow
View Description Hide DescriptionIn the present study we describe a novel threedimensional spectral boundary element algorithm for interfacialdynamics in Stokes flow and/or gravity. The main attraction of this approach is that it exploits all the benefits of the spectral methods (i.e., exponential convergence and numerical stability) with the versatility of the finite element method. In addition, it is not affected by the disadvantage of the spectral methods used in volume discretization to create denser systems. Our algorithm also exploits all the benefits of the boundary element techniques, i.e., a reduction of the problem dimensionality and great parallel scalability. To achieve continuity of the interfacial geometry and its derivatives at the edges of the spectral elements during the droplet deformation, a suitable interfacial smoothing is developed based on a Hermitianlike interpolation. An adaptive mesh reconstructing procedure based on the relevant lengths of the spectral elements is also described. In addition, we consider the inertialess motion of a buoyant droplet left to rise (or sediment) near a vertical solid wall and compare our numerical results with analytical predictions. In our study we emphasize the need for computational studies for the accurate determination of droplet migration near solid walls.

Nonlinear dynamics and breakup of compound jets
View Description Hide DescriptionFiniteamplitude deformation and breakup of a compound jet, whose core and shell are both incompressible Newtonian fluids, that is surrounded by a passive gas are analyzed computationally by a temporal analysis. The means is a method of lines algorithm in which the Galerkin/finite element method with elliptic mesh generation is used for spatial discretization and an adaptive finite difference method is employed for time integration. The dynamics are initiated by subjecting the inner and the outer interfaces of a quiescent compound jet to axially periodic perturbations that are either in phase () or radians out phase (), where is the phase shift between the disturbances imposed on the two interfaces. The initial growth rates of disturbances obtained from computations are compared and demonstrated to be in excellent agreement with predictions of linear theory [Chauhan et al., J. Fluid Mech.420, 1 (2000)]. Computations reveal that recirculating flows occur commonly during the deformation and pinchoff of compound jets, and hence render inapplicable the use of slenderjet type approximations for analyzing the dynamics in such cases. Moreover, as the deformations of one or both of the interfaces of the compound jet grow, the resulting shapes at the incipience of pinchoff are asymmetric and lead to the formation of satellite drops. Calculations are carried out over a wide range of Reynolds numbers of the core fluid, ratios of the viscosity and density of the shell fluid to those of the core fluid, ratio of the surface tension of the outer interface to the interfacial tension of the inner interface, the ratio of the unperturbed radius of the outer cylindrical interface to that of the inner cylindrical interface, wavenumber, and perturbation amplitudes to determine their effects on breakup time and whether both interfaces pinch at the same instant in time to result in the formation of compound drops. Conditions are also identified for which the dynamical response of compound jets subjected to initial perturbations with differ drastically from those subjected to ones with .
 Viscous and NonNewtonian Flows

A freeboundary theory for the shape of the ideal dripping icicle
View Description Hide DescriptionThe growth of icicles is considered as a freeboundary problem. A synthesis of atmospheric heat transfer, geometrical considerations, and thinfilm fluid dynamics leads to a nonlinear ordinary differential equation for the shape of a uniformly advancing icicle, the solution to which defines a parameterfree shape which compares very favorably with that of natural icicles. Away from the tip, the solution has a powerlaw form identical to that recently found for the growth of stalactites by precipitation of calcium carbonate. This analysis thereby explains why stalactites and icicles are so similar in form despite the vastly different physics and chemistry of their formation. In addition, a curious link is noted between the shape so calculated and that found through consideration of only the thin coating water layer.

Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions
View Description Hide DescriptionWe consider the twodimensional topology of streamlines near a surface where the Navier slip boundary condition applies. Using transformations to bring the streamfunction in a simple normal form, we obtain bifurcation diagrams of streamline patterns under variation of one or two external parameters. Topologically, these are identical with the ones previously found for noslip surfaces. We use the theory to analyze the Stokes flow inside a circle, and show how it can be used to predict new bifurcation phenomena.

Viscoelastic flow in rotating curved pipes
View Description Hide DescriptionFully developed viscoelasticflows in rotating curved pipes with circular cross section are investigated theoretically and numerically employing the OldroydB fluid model. Based on Dean’s approximation, a perturbationsolution up to the secondary order is obtained. The governing equations are also solved numerically by the finite volume method. The theoretical and numerical solutions agree with each other very well. The results indicate that the rotation, as well as the curvature and elasticity, plays an important role in affecting the friction factor, the secondary flow pattern and intensity. The corotation enhances effects of curvature and elasticity on the secondary flow. For the counterrotation, there is a critical rotational number , which can make the effect of rotation counteract the effect of curvature and elasticity. Complicated flow behaviors are found at this value. For the relative creeping flow, can be estimated according to the expression . Effects of curvature and elasticity at different rotational numbers on both relative creeping flow and inertial flow are also analyzed and discussed.
 Particulate, Multiphase, and Granular Flows

Confined suspension jet and longrange hydrodynamic interactions: A destabilization scenario
View Description Hide DescriptionThe collective dynamics of a quasitwodimensional suspension jet, of nonBrownian particles, confined in a thin cell and driven by gravitational force is studied both numerically and theoretically. We present a theoretical scheme aimed to describe such a system in the Stokes regime. We focus on the dynamics of the interface between the suspension and the pure fluid. Numerical simulations solving Newton’s equations for all particles show that the jet free surface becomes unstable: the fastest growing modes at small sizes coarsen up to the largest structures reaching the jet lateral scale. In the bulk, structural waves develop and travel at slightly slower speed than the jet average fall. An analytical model, based on hydrodynamiclike equations for the suspension, is derived and predicts the development of the interfacial instability. It captures in essence the collective effects driving the interface destabilization, i.e., the longrange hydrodynamic interactions coupled with the abrupt interface, and no relation to surface tension is found.

Stokes experiment in a liquid foam
View Description Hide DescriptionThe paper reports on the quasistatic steady flow of a dry liquid foam around a fixed spherical bead, a few times larger than the typical bubble size. The force exerted on the bead is recorded with a precision and a time resolution large enough to show the succession of elastic loading of the foam, separated by sudden force drops. The foam structure is observed by direct light transmission, synchronized with the force measurement, thus allowing us to correlate the plastic events with the force variations. Scaling laws for the force signal as a function of the bubble size are detailed and interpreted with a simple elastoplastic model. The spatial distribution of the plasticity is strongly localized in the first bubble layers around the bead and the average size of the bubble rearrangements increases with the corresponding force jump amplitude.

Rheology of particle suspensions with low to moderate fluid inertia at finite particle inertia
View Description Hide DescriptionThe lattice Boltzmann method is applied to simulate the rheology of particle suspensions with lowtomoderate fluid inertia and a wide range of particle inertia. The viscous dissipation in a suspension of particles with a Maxwellian velocity distribution is shown to have a linear dependence on the Reynolds number similar to the Ergun correlation for the drag in a fixed bed of particles. Dynamic simulations of the flow of a suspension of elastic particles being sheared between two rough walls are used to determine the range of Reynolds and Stokes numbers for which a treatment of a homogeneous, sheared suspension based on kinetic theory and the simulated viscous dissipation can accurately describe the particlephase kinetic energy and effective viscosity of the suspension. The dependence of the particlephase slip velocity and the depletion of particles near the boundaries on the particle volume fraction and Stokes number is determined.

Compaction and dilation rate dependence of stresses in gasfluidized beds
View Description Hide DescriptionA particle dynamicsbased hybrid model, consisting of monodisperse spherical solid particles and volumeaveraged gas hydrodynamics, is used to study traveling planar waves (onedimensional traveling waves) of voids formed in gasfluidized beds of narrow crosssectional areas. Through ensembleaveraging in a cotraveling frame, we compute solid phase continuum variables (local volume fraction, average velocity, stress tensor, and granular temperature) across the waves, and examine the relations among them. We probe the consistency between such computationally obtained relations and constitutive models in the kinetic theory for granular materials which are widely used in the twofluid modeling approach to fluidized beds. We demonstrate that solid phase continuum variables exhibit appreciable “path dependence,” which is not captured by the commonly used kinetic theorybased models. We show that this path dependence is associated with the large rates of dilation and compaction that occur in the wave. We also examine the relations among solid phase continuum variables in beds of cohesive particles, which yield the same path dependence. Our results both for beds of cohesive and noncohesive particles suggest that pathdependent constitutive models need to be developed.

Mass and heat fluxes for a binary granular mixture at low density
View Description Hide DescriptionThe Navier–Stokes order hydrodynamicequations for a lowdensity granular mixture obtained previously from the Chapman–Enskog solution to the Boltzmann equation are considered further. The six transport coefficients associated with mass and heat flux in a binary mixture are given as functions of the mass ratio, size ratio, composition, and coefficients of restitution. Their quantitative variation across this parameter set is demonstrated using loworder Sonine polynomial approximations to solve the exact integral equations. The results are also used to quantify the violation of the Onsager reciprocal relations for a granular mixture. Finally, the stability of the homogeneous cooling state is discussed.
 Laminar Flows

Effect of slip on existence, uniqueness, and behavior of similarity solutions for steady incompressible laminar flow in porous tubes and channels
View Description Hide DescriptionThe existence and multiplicity of similarity solutions for steady, fully developed, incompressible laminar flow in uniformly porous tubes and channels with one or two permeable walls is investigated from first principles. A fourthorder ordinary differential equation obtained by simplifying the NavierStokes equations by introducing Berman’s stream function [A. S. Berman, J. Appl. Phys.24, 1232 (1953)] and Terrill’s transformation [R. M. Terrill, Aeronaut. Q.15, 299 (1964)] is probed analytically. In this work that considers only symmetric flows for symmetric ducts; the noslip boundary condition at porous walls is relaxed to account for momentum transfer within the porous walls. By employing the Saffman [P. G. Saffman, Stud. Appl. Math.50, 93 (1971)] form of the slip boundary condition, the uniqueness of similarity solutions is investigated theoretically in terms of the signs of the guesses for the missing initial conditions. Solutions were obtained for all wall Reynolds numbers for channel flows whereas no solutions existed for intermediate values for tube flows. Introducing slip did not fundamentally change the number or the character of solutions corresponding to different sections. However, the range of wall Reynolds numbers for which similarity solutions are theoretically impossible in tube flows was found to be a weak function of the slip coefficient. Slip also weakly influenced the transition wall Reynolds number corresponding to flow in the direction of a favorable axial pressure gradient to one in the direction of an adverse pressure gradient. Momentum transfer from the longitudinal axis to the walls appears to occur more efficiently in porous channels compared to porous tubes even in the presence of slip.

Gravitydriven film flow with variable physical properties
View Description Hide DescriptionThe laminar flow and heat transfer in an accelerating thin liquid film are considered with the view to examine the influence of variable density and transport properties. A new similarity transformation is proposed which exactly transforms the hydrodynamic and thermal boundary layerequations for vertically falling filmflow into a coupled set of ordinary differential equations. The resulting twopoint boundary value problem is integrated numerically with empirical data for the physical properties of water. For given inflow conditions, the temperaturedependency of the dynamic fluid viscosity makes both the hydrodynamic and thermal boundary layers thinner with increasing wall temperature. The expected thickening of the thermal boundary layer due to the increasing thermal diffusivity is therefore more than outweighed by the decreasing viscosity. The nonlinear variation of the physical properties makes these effects more pronounced at the lower inflow temperature.

Inertiainduced coherent structures in a timeperiodic viscous mixing flow
View Description Hide DescriptionInertiainduced changes in transport properties of an incompressible viscous timeperiodic flow due to fluid inertia (nonzero Reynolds numbers) are studied in terms of the topological properties of volumepreserving maps. In the noninertial Stokes limit (vanishing ), the flow relates to a socalled oneaction map. However, the corresponding invariant surfaces are topologically equivalent to spheres rather than the common case of tori. This has fundamental ramifications for the response to small departures from the noninertial limit and leads to a new type of response scenario: resonanceinduced merger of coherent structures. Thus several coexisting families of twodimensional coherent structures are formed that make up two classes: fully closed structures and leaky structures. Fully closed structures restrict motion as in a oneaction map; leaky structures have open boundaries that connect with a locally chaotic region through which exchange of material with other leaky structures occurs. For large departures from the noninertial limit the above structures vanish and the topology becomes determined by isolated periodic points and associated manifolds. This results in unrestricted chaotic motion.