Volume 17, Issue 3, March 2005
 ARTICLES

 Interfacial Flows

Spraying modes in coaxial jet electrospray with outer driving liquid
View Description Hide DescriptionCoaxial jet electrospray is a technique to generate microencapsules, which uses electric forces to create a coaxial microjet from two immiscible liquids. Compound droplets with narrow size distribution are produced after the jet breaks up. In this paper, the spraying modes are investigated experimentally with proper flow rates of the inner and outer liquids. Ethanol/glycerol/tween mixture (outer liquid) and cooking oil (inner liquid) are fed into the gap between outer and inner capillaries and the inner capillary, respectively. The spraying modes presented in our experiments are “dripping mode,” “dripping mode in spindle,” “conejet mode,” “pulse mode in cone,” and “multijets mode” sequentially, as the applied voltage increases. The region of stable conejet mode extends with decrease of the outer liquidflow rate and increase of the inner one. It is found that the spray phenomena are mainly determined by properties of the outer liquid, which is viscous and electric conductive enough. A rudimentary physical model is developed, in which both the viscosity and liquid interface tension are taken into account.

The stability of an encapsulated cylindrical liquid bridge subject to offcentering
View Description Hide DescriptionLiquid bridges are usually encapsulated with another liquid in floatzone crystal growth processes to ensure the containment of volatile components. This paper is concerned with the stability of an inviscid liquid bridge that is offcentered with respect to its encapsulant. Perturbation theory is used to study the stability of such a bridge subject to inertial disturbances. It is concluded that while the offcentered nature does not change the neutral point it does affect the rate of growth and decay of the disturbances causing the unstable regions to become less unstable and stable regions to become less stable. Limiting conditions are considered in order to provide a better understanding of the physics of offcentering.

Breakup and capture of two sedimenting drops in a vertical temperature gradient
View Description Hide DescriptionA curvatureless threedimensional boundaryintegral algorithm has been developed for tangential Marangoni stresses and used to study breakup and capture of two deformable drops for arbitrary droptomedium viscosity and thermal conductivity ratios in parallel and antiparallel arrangements of gravity and an applied vertical temperature gradient. When the driving forces are opposed, the previously observed inhibition of breakup by a weak thermocapillary effect for drops with equal viscosity and thermal conductivity ratios is shown to be almost exclusively the result of changing interfacial tension, with Marangoni stresses having virtually no influence. Alignment of gravity and the temperature gradient in the same direction enhances breakup more than opposing driving forces reduce it, with the limitation that the drops are moving toward a region of zero interfacial tension. The thermal conductivity ratio has negligible impact on these interactions. For bubbles, the effect of a temperature gradient on gravitational results is much less pronounced than for drops. Under certain conditions, parallel orientation of the driving forces weakly inhibits the capture interaction for bubbles, while antiparallel orientation enhances the phenomenon.

Electrically induced pattern formation in thin leaky dielectric films
View Description Hide DescriptionThe stability of the interface between two thin leaky dielectric liquid layers bounded between two flat electrodes is considered. A coupled system of evolution equations is derived for the interfacial location and charge density using lubrication theory. This system is parametrized by the dielectric constants of the two fluids in addition to ratios of their conductivities, viscosities, and thicknesses. A linear stability analysis is conducted and the behavior of the system in the nonlinear regime is also examined. The system is destabilized by electrical stresses that are resisted by capillarity and modified by viscous dissipation. Our results suggest that decreasing the thickness ratio is destabilizing, giving rise to periodic structures of decreasing wavelength. Decreasing the viscosity ratio was also found to lead to the formation of sharpedged structures whose vertical extent is virtually equal to the gap width between the electrodes. Similar structures were also determined upon increasing the ratio of the dielectric constants and electric conductivities.

Enhanced damping of capillary bridge oscillations using velocity feedback
View Description Hide DescriptionIn reduced gravity, the stability of cylindrical liquid bridges and other systems having free surfaces is affected by ambient vibrations of the spacecraft. Such vibrations are expected to excite capillary modes. The lowestorder unstable mode of a liquid bridge is particularly susceptible to vibration as the length of the bridge approaches the stability limit. This mode is known as the (2,0) mode and is an axisymmetric varicose mode of one wavelength in the axial direction. In this work, an optical system is used to detect the (2,0)mode amplitude. The derivative of the error signal produced by this detector is used to produce the appropriate voltages on a pair of annular disk electrodes which are concentric with the bridge. A modecoupled Maxwell stress profile is thus generated in proportion to the modal velocity. Depending on the sign of the gain, the damping of the capillary oscillation can be either increased or decreased. This effect has been demonstrated in Plateautank experiments. Increasing the damping of the capillary modes on free liquid surfaces in space could be beneficial for containerless processing and other technologies.

Viscous versus inviscid instability of twophase mixing layers with continuous velocity profile
View Description Hide DescriptionWe consider the temporal instability of parallel twophase mixing layers. The viscous case is examined using a composite errorfunction velocity profile. The inviscid case is considered for the brokenline velocity profile, where the thickness of the boundary layer in each fluid next to the interface is chosen to match the viscous errorfunction profile at the interface and far away from it. Viscosity modifies the inviscid stability properties quantitatively, but we can also discern an additional unstable mode exclusively related to viscous shear. In the absence of interfacial tension, this mode dominates at large wavenumbers when the Reynolds number is sufficiently high. The various viscous modes cannot generally be attributed to either one of the phases due to mode mixing or exchange. For parameters resembling those of atomization experiments and applications, the most unstable wavelength and growthrate in the viscous case can exceed the inviscid values significantly. The viscous stability analysis also provides better agreement with recent experimental results for air and water than inviscid stability calculations.

Onshell description of stationary flames
View Description Hide DescriptionThe problem of nonperturbative description of stationary flames with arbitrary gas expansion is considered. On the basis of the Thomson circulation theorem an implicit integral of the flow equations is constructed. With the help of this integral, a simple explicit expression for the vortex mode of the burnt gas flow near the flame front is obtained. Furthermore, a dispersion relation for the potential mode at the flame front is written down, thus reducing the initial system of bulk equations and jump conditions for the flow variables to a set of integrodifferential equations for the flame front position and the flow velocity at the front. The developed approach is applied to the case of thin flames. Finally, an asymptotic expansion of the derived equations is carried out in the case where is the gas expansion coefficient, and a single equation for the front position is obtained in the second postSivashinsky approximation. It is demonstrated, in particular, how the wellknown problem of correct normalization of the front velocity is resolved in our approach. It is verified also that in the first postSivashinsky approximation, the equation reduces to the Sivashinsky–Clavin equation corrected according to Cambray and Joulin. Analytical solutions of the derived equations are found, and compared with the results of numerical simulations.
 Viscous and NonNewtonian Flows

Aging and free surface flow of a thixotropic fluid
View Description Hide DescriptionFree surface flows of thixotropic fluids such as paints, selfcompacting concrete, or natural mudflows are of noticeable practical interest. Here we study the basic characteristics of the uniform flow of a layer of thixotropic fluid under gravity. A theoretical approach relying on a simple thixotropy constitutive equation shows that after some time at rest over a small slope angle the fluid layer should start to flow rather abruptly beyond a new, larger, critical slope angle. The theory also predicts that the critical time at which the layer velocity should significantly increase is proportional to the duration of the preliminary rest and tends to infinity when the new slope approaches the critical slope. Experiments carried out with different suspensions show that the qualitative trends of the flows are in very good agreement with the theoretical predictions, except that the critical time for flow start appears to be proportional to a power 0.6 of the time of rest whereas the theory predicts a linear dependence. We show that this indicates a restructuration process at rest differing from the restructuration process under flow.

Collapse and merging of cavity regions in a granular material due to viscous flow
View Description Hide DescriptionExperimental studies are made on the collapse of macroscopic cavity regions placed in an otherwise homogeneous granular material due to viscousflow. First, an initially circular twodimensional hole is exposed to a uniform flow at infinity and the process of boundary shape deformation due to locally enhanced stress components is clarified. Above a certain critical velocity, particles on the upstreamside boundary lose contact with neighboring ones, and are carried to the other side of the boundary. At the same time the fluidized region develops towards upstream direction in the granular material. A simple scaling law on the reduction of the void area is obtained. Second, attention is paid to the interaction of two circular cavities of equal radii, whose centertocenter distance and angle of attack are varied. In some configuration of cavities, lowering of the critical velocity of collapse is recognized. Successive processes on the collapse of upstreamside and downstreamside cavities are classified depending on the magnitude of velocity and the configuration of cavities. A possible scenario on the growth of global scale patterns in granular material such as water vein formation and landslides is suggested.

Distributed forcing of flow over a circular cylinder
View Description Hide DescriptionIn the present study, we apply a distributed (i.e., spatially varying) forcing to flow over a circular cylinder for drag reduction. The distributed forcing is realized by a blowing and suction from the slots located at upper and lower surfaces of the cylinder. The forcing profile from each slot is sinusoidal in the spanwise direction but is steady in time. We consider two different phase differences between the upper and lower blowing/suction profiles: zero (inphase forcing) and (outofphase forcing). The Reynolds numbers considered are from 40 to 3900 covering various regimes of flow over a circular cylinder. For all the Reynolds numbers larger than 47, the present inphase distributed forcing attenuates or annihilates the Kármán vortex shedding and thus significantly reduces the mean drag and the drag and lift fluctuations. The optimal wavelength and amplitude of the inphase forcing for maximum drag reduction are also obtained for the Reynolds number of 100. It is shown that the inphase forcing produces the phase mismatch along the spanwise direction in the vortex shedding, weakens the strength of vortical structures in the wake, and thus reduces the drag. Unlike the inphase forcing, the outofphase distributed forcing does not reduce the drag at low Reynolds numbers, but it reduces the mean drag and the drag and lift fluctuations at a high Reynolds number of 3900 by affecting the evolution of the separating shear layer, although the amount of drag reduction is smaller than that by the inphase forcing.

The effect of added polymers on the formation of drops ejected from a nozzle
View Description Hide DescriptionThe presence of small amounts of polymer in Newtonian solvents can have a significant impact on the flow behavior of these fluids in extensiondominated flows. This study investigates the effect of elasticity on the ondemand drop formation through the use of low viscosityelastic liquids. A highspeed camera is employed to observe drops ejected from a nozzle. The drops are created using a piezoelectric sleeve that contracts around a nozzle forcing fluid out. We observe that the satellite drops commonly produced with Newtonian fluids of identical shear viscosity in the same geometry can be suppressed when polymers with sufficient molecular weight are added. For our nozzle, the minimum required molecular weight is 300k PEO at a concentration of 25 ppm. However, the resultant increased elasticity in the solution requires a greater pulse strength to eject the drop. In addition, the fluids containing polymers have a longer thread, a longer time to separation, and a lower velocity than the Newtonian fluids with similar shear viscosity.

Ferrofluid pipe flow in an oscillating magnetic field
View Description Hide DescriptionFerrofluid pipe flow in an oscillating magnetic field along the pipe axis is studied theoretically in a wide range of the flow rate. The fielddependent part of viscosity (it can be positive or negative) reveals significant dependence on the flowvorticity, i.e., ferrofluids exhibit nonNewtonian behavior. This is manifested in an alteration of the velocity profile—it ceases to be parabolic—and deviation of the flow rate from the value prescribed by Poiseuille’s formula. The presented model based on the conventional ferrohydrodynamic equations and an assumption of the ferrofluid structure fits well experimental data recently obtained by Schumacher, Sellien, Konke, Cader, and Finlayson [“Experiment and simulation of laminar and turbulent ferrofluid pipe flow in an oscillating magnetic field,” Phys. Rev. E67, 026308 (2003)].
 Laminar Flows

Chaotic mixing in thermocapillarydriven microdroplets
View Description Hide DescriptionLiquid microdroplets represent a convenient system for studies of mixing by chaotic advection in discrete microscopic volumes. The mixing properties of the flows in microdroplets are governed by their symmetries, which give rise to invariant surfaces serving as barriers to transport. Thorough mixing via chaotic advection requires destruction of all such invariant surfaces. To illustrate this idea, we demonstrate that quick and thorough mixing inside a spherical microdroplet suspended in a layer of substrate fluid can be obtained by moving the droplet along a twodimensional path using temperatureinduced surface tension gradients. The use of flow invariants also provides a convenient way to analyze the mixing properties of flows in many other experimental implementations.

Sharp scalar and tensor bounds on the hydrodynamic friction and mobility of arbitrarily shaped bodies in Stokes flow
View Description Hide DescriptionWe prove rigorous inequalities for the hydrodynamic translational friction and mobility matrices and of an arbitrarily shaped rigid particle in terms of the electrostaticcapacitance of a conducting particle of identical shape. Specifically, we derive the scalar and matrix inequalities and , where all quantities are normalized by the corresponding values for a sphere, and the mobility matrix is evaluated in the centerofmobility reference frame. These bounds are obtained using a variational approach with the energy dissipation functional expressed in terms of the induced force distribution on the surface of the particle. To relate the hydrodynamic problem to the solution of the corresponding electrostatic problem, the trial force field is expressed in terms of the charge distribution on the equipotential particle surface. This procedure yields the first rigorous bounds on hydrodynamicfriction that apply to bodies with translationrotation coupling. We demonstrate that the error of the Hubbard–Douglas approximation , corresponding to our scalar bound, is quadratic in the deviation of the trial inducedforce field from the exact form—which explains why this relation is highly accurate for many particle shapes. Our numerical results confirm that the Hubbard–Douglas approximation is accurate for a variety of objects, including helices with translational–rotational coupling. In addition, we establish a rigorous, sharp bound on the effective (scalar) Brownian diffusion coefficient of an arbitrarily shaped particle.
 Instability and Transition

Richtmyer–Meshkov instability of arbitrary shapes
View Description Hide DescriptionWe consider the effect of a shock passing through an arbitrarily shaped interface between two fluids. The evolution of the interface into a new shape, written formally as , is found by applying the linear, classical Richtmyer–Meshkov instability result to each mode in the Fourier expansion of the original interface. We provide several examples where the new shape can be found analytically. For any interface we define an associated dual interface and show that . Representing a shock by a new mathematical operator we find how , and transform under the effect of a shock. Kinksingularities are found in when and where has a discontinuous change in its first derivative. These are the locations where jetting occurs. We briefly discuss the effects of nonlinearity, compressibility, viscosity, etc., all of which suppress kinksingularities, and present hydrocode simulations of shock tube and highexplosivedriven experiments to highlight the influence of compressibility, nonlinearity, and material strength.

Instability of isolated compressible entropystratified vortices
View Description Hide DescriptionThe normal mode linear analysis is applied to investigate the stability of a circular isolated compressible vortex with the emphasis on studying the effect of entropy stratification of the basic flow on stability properties. We study a family of vortices that have zero total circulation, with the swirl velocity being presented by a Taylortype distribution in the radial direction. The stratification of entropy is modeled by a Gaussian twoparametric profile whose parameters control the maximal deviation from the homentropic distribution and the extent of the entropic zone. Results presented concern the effect of these parameters and the vortex intensity on instability characteristics. In particular, in the case of homentropic flow,vortices considered are found to be stable only for sufficiently high intensities and cease to be stable as the intensity weakens. As an example of the situation where unstable normal modes can be excited, the scattering of sound waves by the vortexflow is considered. By simulating this process numerically, we show that the scattered field becomes unstable in the course of time and acquires a typical periodical pattern in polar angle. The characteristics of this instability (viz., the growth rate and the azimuthal phase frequency) coincide with those of the linear analysis very closely. This fact shows conclusively that the instability is caused by the induction of unstable normal modes in the vortex by means of sound irradiation.

Characteristic modes and evolution processes of shearlayer vortices in an elevated transverse jet
View Description Hide DescriptionCharacteristics and evolution processes of the traveling coherent flow structure in the shear layer of an elevated round jet in crossflow are studied experimentally in an openloop wind tunnel. Streak pictures of the smoke flow patterns illuminated by the laserlight sheet in the median and horizontal planes are recorded with a high speed digital camera. Time histories of the instantaneous velocity of the vortical flows in the shear layer are digitized by a hotwire anemometer through a highspeed data acquisition system. By analyzing the streak pictures of the smoke flow visualization, five characteristic flow structures, mixinglayer type vortices, backwardrolling vortices, forwardrolling vortices, swinginduced mushroom vortices, and jettype vortices, are identified in the shear layer evolving from the upwind edge of the jet exit. The behaviors and mechanisms of the vortical flow structure in the bent shear layer are prominently distinct in different flow regimes. The frequency characteristics, Strouhal number, powerspectrum density functions, autocorrelation coefficient, as well as the time and length scales of the coherent structure and the Lagrangian integral scales are obtained by processing the measured instantaneous velocity data. The Strouhal number is found to decay exponentially with the increase of the jettocrossflow momentum flux ratio. The autocorrelation coefficients provide the information for calculating the statistical time scales of the coherent structure and the integral time scales of turbulencefluctuations. The corresponding length scales of the vortical structure and the integral length scales of turbulence in the shear layer are therefore obtained and discussed.

Precession driven dynamos
View Description Hide DescriptionThe dynamo effect is demonstrated numerically in precession driven flow in a spherical container. Both laminar as well as unstable flows act as dynamos. At low Ekman numbers in unstable flows, the dynamo mechanism relies predominantly on the components of the flow excited by instabilities. All calculations are performed in a frame of reference attached to the boundaries. In this frame, the rotation axis of the fluid executes a periodic motion with a period equal to the rotation period of the boundaries, whereas the magnetic dipole moment undergoes slower variations interrupted by reversals.

Experiments on the latetime development of singlemode Richtmyer–Meshkov instability
View Description Hide DescriptionThe latetime development of Richtmyer–Meshkov instability is studied in shock tubeexperiments. This investigation makes use of the experimental apparatus and visualization methods utilized in the earlier study of Collins and Jacobs [J. Fluid Mech.464, 113 (2002)] but employs stronger shocks and initial perturbations with shorter wavelengths to obtain much latertime (in the dimensionless sense) images of the singlemode instability. These modifications produce a very detailed look at the evolution of the latetime singlemode instability, revealing the transition and development of turbulence in the vortex cores that eventually results in the disintegration of the laminar vortex structures into small scale features. Amplitude measurements taken from these images are shown to be effectively collapsed when plotted in dimensionless variables defined using the wave number and the initial growth rate. The amplitude measurements are compared with several latetime nonlinear models and solutions. The best agreement is obtained with the model of Sadot et al. [Phys. Rev. Lett.80, 1654 (1998)] which can be slightly improved by modifying the expression for the latetime asymptotic growth rate.

Linear stability analysis of axisymmetric perturbations in imperfectly conducting liquid jets
View Description Hide DescriptionA discussion is presented on the role of limited conductivity and permittivity on the behavior of electrified jets. Under certain conditions, significant departures with respect to the perfectconductor limit are to be expected. In addition, an exploration is undertaken concerning the validity of onedimensional average models in the description of charged jets. To that end, a temporal linear modal stability analysis is carried out of poorconductor viscousliquid jets flowing relatively to a steady radial electric field. Only axisymmetric perturbations, leading to highest quality aerosols, are considered. A grounded coaxial electrode is located at variable distance. Most available studies in the literature are restricted to the perfectconductor limit, while the present contribution is an extension to moderate and low electrical conductivity and permittivityjets, in an effort to describe a situation increasingly prevalent in the sector of smallscale freesurface flows. The influence of the electrode distance , a parameter defined as the ratio of the electric relaxation time scale to the capillary time scale, and the relative permittivity on the growth rate has been explored yielding results on the stability spectrum. In addition, arbitrary viscosity and electrification parameters are contemplated. In a wide variety of situations, the perfectconductor limit provides a good approximation; however, the influence of and on the growth rate and most unstable wavelength cannot be neglected in the general case. An interfacial boundary layer in the axial velocity profile occurs in the lowviscosity limit, but this boundary layer tends to disappear when or are large enough. The use of a onedimensional (1D) averaged model as an alternative to the 3D approach provides a helpful shortcut and a complementary insight on the nature of the jet’s perturbative behavior. Lowestorder 1D approximations (average model), of widespread application in the literature of electrified jets, are shown to be inaccurate in lowviscosity imperfectconductor jets.