Volume 19, Issue 7, July 2007

Simple exact solutions for horizontal flows, with a stagnant funnel consisting of a lighter fluid and narrowing toward the ground, exhibit the main features of real tornadoes. The twofluid model is quite robust, as it can be adapted to any vorticity distribution, any vertical stratification, and the effects of compressibility.
 LETTERS


Inertial waves in rotating grid turbulence
View Description Hide DescriptionUsing liquid helium,liquid nitrogen, and water as test fluids, we attempt to generate homogeneous turbulence in a steadily rotating system. We create turbulence by pulling a grid in rotating channels with both square and round cross sections, and observe largescale inertial waves in the flow. These inertial waves quickly sense the boundaries, and resonate at frequencies characteristic of the container. We describe some of their properties and argue that the resultant inhomogeneity is a feature of any real system.

Smallscale aspects of flows in proximity of the turbulent/nonturbulent interface
View Description Hide DescriptionThe work reported below is the first of its kind to study the properties of turbulent flow without strong mean shear in a Newtonian fluid in proximity of the turbulent/nonturbulent interface, with emphasis on the smallscale aspects. The main tools used are a threedimensional particle tracking system allowing one to measure and follow in a Lagrangian manner the field of velocity derivatives and direct numerical simulations. The comparison of flow properties in the turbulent (A), intermediate (B), and nonturbulent (C) regions in the proximity of the interface allows for direct observation of the key physical processes underlying the entrainment phenomenon. The differences between smallscale strain and enstrophy are striking and point to the definite scenario of turbulent entrainment via the viscous forces originating in strain.

Particle image velocimetry measurements of the velocity profile in forced flow
View Description Hide DescriptionMeasurements of the velocity profile of forced flow within a square crosssection channel are reported. The particle imagevelocimetry(PIV) technique is used for these measurements with micron scale solid deuterium particles as tracers of the velocity field. Steady state velocity profiles at different bath temperatures clearly show the existence of a turbulent boundary layer in forced flow. These observed velocity profiles are in reasonable agreement with correlations based on measurements in classical fluids.
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 ARTICLES

 Interfacial Flows

Breakup behavior of a conducting drop suspended in a viscous fluid subject to an electric field
View Description Hide DescriptionThe slow axisymmetric deformation of a conducting drop surrounded by a viscous insulating fluid subject to a uniform electric field is considered. Numerical computations, based on a boundary integral formulation, are used to follow the behavior of relatively inviscid and viscousdrops right up to breakup. The type of breakup seen depends on the ratio of the viscosity of the drop to that of the surrounding fluid, and on the electric field strength. The different types of breakup possible are small droplets being emitted from the ends of the drop with a charge greater than the Rayleigh limit, the formation of what appear to be pointed ends with the subsequent ejection of thin jetlike structures, or the formation of thin jetlike structures without the pointed ends. The different types of breakup are examined and the different regions of the parameter space where each occurs is determined. Also, by considering different stress balances, local equilibrium solutions that allow for a conical drop end are derived. Several sets of solutions, corresponding to different stress balances, are found. However, none of the conical end solutions appear to be compatible with the pointed ends seen in the numerical results.

Formation of dynamic particle accumulation structures in oscillatory thermocapillary flow in liquid bridges
View Description Hide DescriptionWe report on the behavior of small particles of dilute concentration in timedependent (oscillatory) thermocapillary flow in cylindrical liquid bridges. The particles accumulate in a dynamic string for certain aspect ratios of the liquid bridge and at, typically, two times the critical Marangoni number for the onset of time dependence. This was observed for particles with a density larger and smaller than that of the fluid and for the isodense case. If looked at in a snapshot, this string would be wound times around the thermocapillary vortex as a deformed spiral. If one looked at the full dynamics, it would be seen that the spiral string is rotating around its ringshaped axis. The phenomenon is called a dynamical particle accumulation structure (dynamical PAS). The mode is the mode number of the oscillatory flow field with wavetrains of the hydrothermal wave (HTW) traveling in the azimuthal direction. We visualize and describe the different modes in detail. We give direct experimental evidence for the gathering of liquid with particles during the cold phases of the HTW and the injection of liquid with particles into the return flow in azimuthally traveling “cold spots.” We varied the particle diameter at constant density and the ratio of the particle density to fluid density at constant particle diameter to measure the time of the formation of PAS and discuss and explain the experimental results in comparison with possible mechanisms underlying the formation process. We describe the results of an experiment under microgravity to exclude gravity as a PASforming mechanism. We conclude by describing a possible mechanism that could account for the observed particle accumulation in certain regions of the flow. This mechanism involves the observed gathering and injection of liquid during the cold phases of the HTW and the particle enrichment of the injected fluid due to particle migration in sheared flow.PAS occurs at a resonance between the azimuthally traveling wave and the “turnover time” of the PASstring in the thermocapillary vortex.

Stability of twophase vertical flow in homogeneous porous media
View Description Hide DescriptionImmiscible twophase flow in porous media, which results from the downward injection of a heavier fluid or upward injection of a lighter fluid, is characterized by two shocks, one at each end of a rarefaction wave. The specific details of the saturation profile, such as the shock speeds and the shock saturations, are determined by the fractional flow function for given values of the mobility ratio and the gravity number. We employ a normal mode, matched asymptotic expansion analysis to obtain analytical expressions governing the stability behavior of such flows.Instability occurs at both ends of the onedimensional base saturation profile with unique characteristics such that the maximum growth rate decreases both when the mobility ratio is increased at the front end and decreased at the back end. This unusual behavior is explained in terms of vorticity eigenfunctions related to nonmonotonic mobility profiles. Analysis of stability behavior as a function of fractional flow profile shows that although the fundamental mechanisms are qualitatively similar, they are associated with different parameter values in view of particular mobility profiles. Growth rates and wavenumbers predicted by the linear stabilityanalysis are observed at the onset of the nonlinear displacement process, which is followed by the fully developed nonlinear flow with largescale unstable structures.

Reflections on cavitation nuclei in water
View Description Hide DescriptionThe origin of cavitation bubbles, cavitation nuclei, has been a subject of debate since the early years of cavitation research. This paper presents an analysis of a representative selection of experimental investigations of cavitation inception and the tensile strength of water. At atmospheric pressure, the possibility of stabilization of free gas bubbles by a skin has been documented, but only within a range of bubble sizes that makes them responsible for tensile strengths up to about , and values reaching almost have been measured. However, cavitation nuclei can also be harbored on the surface of particles and bounding walls. Such nuclei can be related to the full range of tensile strengths measured, when differences of experimental conditions are taken into consideration. The absence or presence of contamination on surfaces, as well as the structure of the surfaces, are central to explaining why the tensile strength of water varies so dramatically between the experiments reported. A model for calculation of the critical pressure of skincovered free gas bubbles as well as that of interfacial gaseous nuclei covered by a skin is presented. This model is able to bridge the apparently conflicting results of the many scientists, who have been working in the field over the years.

Threedimensional oscillatory longwave Marangoni convection in a binary liquid layer with the Soret effect: Bifurcation analysis
View Description Hide DescriptionThreedimensional longwave oscillatory Marangoni convection in a thin layer of binary mixture with a nondeformable interface is investigated in the presence of the Soret effect. Both thermocapillary and solutocapillary effects are taken into account. A set of amplitude equations is obtained and studied analytically near the critical value of the Marangoni number. It is shown that alternating rolls (either rhombic or square) are selected and they bifurcate supercritically. Subcritical bifurcation takes place only for alternating rolls on a square lattice in a narrow range of parameters.

Nonlinear oscillations and collapse of elongated bubbles subject to weak viscous effects: Effect of internal overpressure
View Description Hide DescriptionThe details of nonlinear oscillations and collapse of elongated bubbles, subject to large internal overpressure, are studied by a boundary integral method. Weak viscous effects on the liquid side are accounted for by integrating the equations of motion across the boundary layer that is formed adjacent to the interface. For relatively large bubbles with initial radius on the order of millimeters, and , and an almost spherical initial shape, , RayleighTaylor instability prevails and the bubble breaks up as a result of growth of higher modes and the development of regions of very small radius of curvature; , , , and denote the surface tension, density, viscosity, and dimensional static pressure in the host liquid while is the ratio between the length of the minor semiaxis of the bubble, taken as an axisymmetric ellipsoid, and its equivalent radius . For finite initial elongations, , the bubble collapses either via two jets that counterpropagate along the axis of symmetry and eventually coalesce at the equatorial plane, or in the form of a sink flow approaching the center of the bubble along the equatorial plane. This pattern persists for the above range of initial elongations examined and large internal overpressure amplitudes, , irrespective of Oh. It is largely due to the phase in the growth of the second Legendre mode during the afterbounce of the oscillating bubble, during which it acquires large enough positive accelerations for collapse to take place. For smaller bubbles with initial radius on the order of micrometers, and , and small initial elongations, , viscosity counteracts growth and subsequent jet motion, thus giving rise to a critical value of below which the bubble eventually returns to its equilibrium spherical shape, whereas above it collapse via jet impact or sink flow is obtained. For moderate elongations, , and large overpressures, , jet propagation and impact along the axis of symmetry prevails irrespective of Oh. For very large elongations, , and above a certain threshold value of Oh the counterpropagating jets pinch the contracting bubble sidewalls in an offcentered fashion.

On the breakup of fluid films of finite and infinite extent
View Description Hide DescriptionWe study the dewetting process of thin fluid films that partially wet a solid surface. Using a longwave (lubrication) approximation, we formulate a nonlinear partial differential equation governing the evolution of the film thickness, . This equation includes the effects of capillarity, gravity, and an additional conjoining/disjoining pressure term to account for intermolecular forces. We perform standard linear stability analysis of an infinite flat film, and identify the corresponding stable, unstable, and metastable regions. Within this framework, we analyze the evolution of a semiinfinite film of length in one direction. The numerical simulations show that for long and thin films, the dewetting fronts of the film generate a pearling process involving successive formation of ridges at the film ends and consecutive pinchoff behind these ridges. On the other hand, for shorter and thicker films, the evolution ends up by forming a single drop. The time evolution as well as the final dropspattern show a competition between the dewetting mechanisms caused by nucleation and by free surface instability. We find that precise computations, requiring quadrupole precision of computer arithmetic, are often needed to avoid spurious results.

Surface tension dominated impact
View Description Hide DescriptionWe study the impact of a line mass onto a liquidgas interface. At early times we find a similarity solution for the interfacial deformation and show how the resulting surface tension force slows the fall of the mass. We compute the motion beyond early times using a boundary integral method, and find conditions on the weight and impact speed of the mass that determine whether it sinks or is trapped by the interface. We find that for given impact speed there is a critical weight above which the mass sinks, and we investigate the asymptotic behavior of this critical weight in the limits of small and large impact speeds. Below this critical weight, the mass is trapped by the interface and subsequently floats. We also compare our theoretical results with some simple tabletop experiments. Finally, we discuss the implications of our work for the vertical jumps of waterwalking arthropods.

A computational study of axial dispersion in segmented gasliquid flow
View Description Hide DescriptionAxial dispersion of a tracer in a twodimensional gasliquidflow is studied computationally using a finitevolume/fronttracking method. The effects of Peclet number, capillary number, and segment size are examined. At low Peclet numbers, the axial dispersion is mainly controlled by the convection through the liquid films between the bubbles and channel walls. In this regime, the computational results are found to be in a very good agreement with the existing model due to Pedersen and Horvath [Ind. Eng. Chem. Fundam.20, 181 (1981)]. At high Peclet numbers, the axial dispersion is mainly controlled by the molecular diffusion, with some convective enhancement. In this regime, a new model is proposed and found to agree well with the computational results. These Peclet number regimes are shown to persist for different slug lengths. The axial dispersion is found to depend weakly on the capillary number in the diffusioncontrolled regime. Finally, computational simulations are performed for the cases of six bubbles to mimic bubble trains, and results are compared with the theoretical models.

The initial coalescence of miscible drops
View Description Hide DescriptionWhen two drops of different miscible liquids are brought into contact, their coalescence speed is governed by the liquid having the weaker surface tension. Marangoni waves propagate along the drop with the stronger surface tension. We present surface profiles and propagation speeds of these waves, from experiments with a pendent water dropcoalescing with a flat ethanolsurface or with a sessile drop of ethanol. We find in the former case that the capillaryMarangoni waves along the water drop show selfsimilar character when measured in terms of arc length along the original surface. The coalescence of two liquids of different viscosities is also studied. For large viscosity difference, mobility is confined to the lower viscosity fluid and a sharp corner forms where the two liquids meet along the free surface. The coalescence speed of a water drop with a much more viscousliquid is nearly independent of the strength of the viscosity difference.

Reappraisal of criticality for twolayer flows and its role in the generation of internal solitary waves
View Description Hide DescriptionA geometric view of criticality for twolayer flows is presented. Uniform flows are classified by diagrams in the momentummassflux space for fixed Bernoulli energy, and cuspoidal curves on these diagrams correspond to critical uniform flows. Restriction of these surfaces to critical flow leads to new subsurfaces in energymassflux space. While the connection between criticality and the generation of solitary waves is well known, we find that the nonlinear properties of these bifurcating solitary waves are also determined by the properties of the criticality surfaces. To be specific, the case of two layers with a rigid lid is considered, and application of the theory to other multilayer flows is sketched.

Cavitation in an orifice flow
View Description Hide DescriptionThe purpose of this study is to identify the potential locations for cavitation induced by total stress on the flow of a liquid through an orifice of an atomizer. A numerical simulation of twophase incompressible flow is conducted in an axisymmetric geometry of the orifice for Reynolds numbers between 100 and 2000. The orifice has a rounded upstream corner and a sharp downstream corner with lengthtodiameter ratio between 0.1 and 5. The total stress including viscous stress and pressure has been calculated in the flow field and, from there, the maximum principal stress is found. The totalstress criterion for cavitation is applied to find the regions where cavitation is likely to occur and compared with those of the traditional pressure criterion. Results show that the viscous stress has significant effects on cavitation. The effect of geometry and occurrence of hydraulic flip in the orifice on the total stress are studied. The NavierStokes equations are solved numerically using a finitevolume method and a boundaryfitted orthogonal grid that comes from the streamlines and potential lines of an axisymmetric equipotential flow in the same geometry. A levelset formulation is used to track the interface and model the surface tension.

Emission of drops from the tip of an electrified jet of an inviscid liquid of infinite electrical conductivity
View Description Hide DescriptionA numerical description is presented of the emission of drops from the tip of a long axisymmetric jet, which may develop when an inviscid liquid of infinite electrical conductivity is injected into a dielectric medium under the action of a strong electric field. The applied field is intensified by the presence of the equipotential surface of the jet, leading to a strong electric stress normal to the surface that accelerates a stretch of liquid and cuts it from the jet. The jet consists of a long region of stationary flow followed by a long oscillatory region where the drops develop and detach. The process of drop generation comprises different stages and is dominated by the electric stress and the inertia of the liquid, with little effect of its surface tension. Orderofmagnitude estimates are used to determine conditions under which these results can be applied to jets of liquids of finite electrical conductivity.
 Viscous and NonNewtonian Flows

Miscible viscous fingering with linear adsorption on the porous matrix
View Description Hide DescriptionViscous fingering between miscible fluids of different viscosities can affect the dispersion of finite samples in porous media. In some applications, as typically in chromatographic separations or pollutant dispersion in underground aquifers, adsorption onto the porous matrix of solutes (the concentration of which rules the viscosity of the solution) can affect the fingering dynamics. Here, we investigate theoretically the influence of such an adsorption on the stability and nonlinear properties of viscous samples displaced in a twodimensional system by a less viscous and miscible carrying fluid. The model is based on Darcy’s law for the evolution of the fluid velocity coupled to a diffusionconvection equation for the concentration of a solute in the mobile phase inside the porous medium. The adsorptiondesorption dynamics of the solute onto the stationary phase is assumed to be at equilibrium, to follow a linear isotherm and is characterized by a retention parameter equal to the adsorptiondesorption equilibrium constant multiplied by the phase ratio . In practice, retention on the porous matrix renormalizes the logmobility ratio by a factor . Correspondingly, a linear stability analysis and nonlinear simulations of the model show that an increase of leads to a stabilization of viscous fingering with fingers appearing on a dimensional time scale multiplied by and with a dimensional wavelength multiplied by .

Transient flow caused by a sudden impulse or twist applied to a sphere immersed in a viscous incompressible fluid
View Description Hide DescriptionAn analytic solution of the linearized NavierStokes equations is presented for the flow of an incompressible viscous fluid about a sphere suddenly set in motion by an impulse or twist. Mixed slipstick boundary conditions are assumed to hold at the surface of the sphere. The transient flow pattern depends on the mass and moment of inertia of the sphere. The impulsive translational motion of the sphere generates an expanding vortex ring. Conservation of total momentum is discussed and the momentum of the fluid is expressed as a sum of a potential and a viscous contribution. Conservation of total angular momentum is verified in the case of rotational motion.

Dynamics of viscoelastic fluid filaments in microfluidic devices
View Description Hide DescriptionThe effects of fluid elasticity and channel dimension on polymericdroplet formation in the presence of a flowing continuous Newtonian phase are investigated systematically by using different molecular weight (MW) poly(ethylene oxide) (PEO) solutions and varying microchannel dimensions with constant orifice width to depth ratio and , , , and . The flow rate is varied so that the mean shear rate is practically identical for all cases considered. Relevant times scales include inertiacapillary Rayleigh time , viscocapillary Tomotika time , and the polymerrelaxation time, where is the fluid density of the dispersed phase, is the interfacial tension, is the zero shear viscosity of the dispersed polymer phase, and is the maximum filament radius. Dimensionless numbers include the elasticity number , elastocapillary number , and Deborah number, , where is the kinematic shear viscosity of the fluids. Experiments show that higher MW Boger fluids possessing longer relaxation times and larger extensional viscosities exhibit longer thread lengths and longer pinchoff times . The polymer filament dynamics are controlled primarily by an elastocapillary mechanism with increasing elasticity effect at smaller length scales (larger and ). However, with weaker elastic effects (i.e., larger and lower MW), pinchoff is initiated by inertiacapillary mechanisms, followed by an elastocapillary regime. A high degree of correlation exists between the dimensionless pinchoff times and the elasticity numbers. We also observe that higher elasticity number yields smaller effective . Based on the estimates of polymer scission probabilities predicted by Brownian dynamics simulations for uniaxial extensional flows,polymer chain scission is likely to occur for ultrasmall orifices and high MW fluids, yielding smaller . Finally, the inhibition of beadonastring formation is observed only for flows with large Deborah number .

Behavior near critical for a conducting drop in an electric field
View Description Hide DescriptionThe time scales for the behavior of a conducting drop undergoing slow deformation in a uniform electric field are examined. Below the critical electric field strength, equilibrium shapes are possible and approached exponentially. At the critical value, the convergence behaves as . Above (but still near) critical, there is a period of slow elongation near the critical equilibrium shape. The time scale, , for this period of slow growth behaves like , where is the electric field strength and is the critical electric field strength. In addition, the dependence of the time scales on the viscosities of the drop and of the surrounding fluid is examined. The theoretical results compare favorably with boundary integral numerical computations. Asymptotically, the time scales depend linearly on the viscosity ratio for both large and small values. These linear relations remain accurate well outside the expected regions of validity, and the entire range of viscosity ratios can be reasonably approximated by a piecewise linear combination of the two. Thus, an equation is obtained that determines the duration of the period of slow deformation and predicts the time to breakup.