Volume 17, Issue 6, June 2005
 LETTERS


Experimental study of the production of vortex rings using a variable diameter orifice
View Description Hide DescriptionA method for producing a vortex ring with an apparatus that utilizes a time varying aperture that closes during ring production is shown to have the capability for the production of rings of low nondimensional energy similar to a Hill’s spherical vortex. This represents a difference of more than 50%, in terms of the nondimensional energy, of those which have been produced with fixed diameter generators. Experiments show that modulation of the rate at which the vortex ringgenerator produces kinetic energy, circulation, and impulse is the critical factor in determining what type of ring is formed.

Visualization of nonlinear effects in reflecting internal wave beams
View Description Hide DescriptionRecent theoretical and numerical investigations predict that localized nonlinear effects in the overlapping region of an incoming and reflected internal wave beam can radiate higherharmonic beams. We present the first set of experimental visualizations, obtained using the digital Schlieren method, that confirm the existence of radiated higherharmonic beams. For arrangements in which the angle of propagation of the second harmonic exceeds the slope angle, radiated beams are visualized. When the propagation angle of the second harmonic deceeds the slope angle no radiated beams are detected, as the associated density gradient perturbations are too weak for the experimental method. The case of a critical slope is also reported.
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 ARTICLES

 Interfacial Flows

Experimental and theoretical investigation on the sloshing of a twoliquid system with free surface
View Description Hide DescriptionIn this paper a theoretical and experimental investigation is performed on the sloshing of a twoliquid system with both separation and free surface. The experimental configuration consists of an oscillating tank filled with two layers of immiscible liquids. The mathematical model is obtained by applying the Lagrangian variational approach to the potential formulation of the fluid motion, and a dynamical system which describes the dynamics of motion is derived. In order to account for the damping of the motion, generalized dissipative forces are considered. For this purpose, the logarithmic decrement coefficients are estimated by means of a wavelet analysis performed on the experimental free oscillations of the fluid system. Numerical integration of the mathematical model gives results which are in a fair agreement with the experimental results.

Thermal effects on film development during spin coating
View Description Hide DescriptionThis study is an extended version of a previous analysis of thermal effects on the freesurface liquid filmflow on a flat, heated rotating disk [A. Kitamura, “Thermal effects of liquid filmflow during spin coating,” Phys. Fluids13, 2788 (2001)]. The assumption of a constant disk temperature is extended to a nonuniform disk temperature, and the restriction on small radial distance used in the previous analysis is removed. The evolution equation for the transient film thickness is obtained and solved by the method of characteristics [B. S. Dandapat, P. Daripa, and P. C. Ray, “Asymptotic study of film thinning process on a spinning annular disk,” J. Appl. Phys.94, 4144 (2003)]. The effect of both thermocapillary forces and variable viscosity on the flow is revealed. A physical explanation is provided to justify the results.

Dynamic contact angle of spreading droplets: Experiments and simulations
View Description Hide DescriptionThis paper presents results of an experimental investigation of a single drop impact onto a dry, partially wettable substrate and its numerical simulation. Particularly, the drop spreading diameter and the dynamic contact angle are measured at different time instants after impact. Two surfaces, wax (low wettability) and glass (high wettability), are used to study the effect of surface wettability (static contact angle) on the impact dynamics. It is shown that existing empirical models for the dynamic contact angle (e.g., Hoffman–Voinov–Tanner law) do not predict well the change of the dynamic contact angle, especially at high capillary numbers. In addition to the experimental investigations, the drop impact was studied numerically, focusing primarily on the contact angle treatment. The singularity in the neighborhood of the moving contact line is removed from the computational domain and replaced by a local force with some dependence on the instantaneous advancing/receding contactline velocity. The predicted time dependence of the drop spreading diameter and of the dynamic contact angle agrees well with the experimental data for both the advancing and receding phases of the impact process.

Critical speed for capillarygravity surface flows in the dispersive shallow water limit
View Description Hide DescriptionThe stability of perfectfluid capillarygravity surface flows past a cylindrical obstacle is studied in the shallow water limit, using the twodimensional compressible Euler equations, with leadingorder dispersive corrections. Stationary solutions with different contact angles are obtained by Newton branch following, based on Fourier pseudospectral methods, using mapped Chebychev polynomials. Stable and unstable branches are found to meet, through a saddlenode bifurcation, at a critical speed beyond which no stationary solution exists. For large obstacles, the stable branch is compared with the stationary solutions of the compressible Euler equation without dispersion. Boundary layers are investigated. In this regime, the unstable dynamics are shown to lead to a finitetime dewetting singularity.

An impulsive bathtub vortex
View Description Hide DescriptionThe impulsive freesurface flow due to a point sink at the bottom of a container with rotating inviscid fluid is investigated analytically by a smalltime expansion. Before the sink is turned on impulsively, there is a steady rigidbody rotation with large Rossby number (small angular velocity). The theory is developed to first order in the inverse Rossby number. The evolution of the freesurface vorticity is investigated. The azimuthal surface velocity is a cubic function of time. It is shown that the influence of the rotation of the earth on an impulsive bathtub vortex is negligible.

Marangoni convection and heat transfer in thin liquid films on heated walls with topography: Experiments and numerical study
View Description Hide DescriptionThermocapillaryinduced motion in thin liquid films on a heated horizontal wall with parallel grooves on its upper surface is studied experimentally and numerically. The results of velocity and temperature measurements are reported. A numerical model for a liquid film on a structured wall is developed. The full incompressible Navier–Stokes equations and the energy equation are integrated by a finite difference algorithm, whereas the mobile gasliquid interface is tracked by the volumeoffluid method. The numerical model is verified by comparison with the experimental data showing a good agreement. The model is used to study flow patterns and film rupture caused by the thermocapillary forces. Heat transfer in the liquid is also investigated. In particular, it is found that the thermocapillary convection enhances heat transfer in liquid, though the effect depends on the shape of the wall surface.

Analysis of the drop weight method
View Description Hide DescriptionThe drop weight method is an accurate yet simple technique for determining surface tension. It relies on dripping a liquid of density at a low flow rate from a capillary of radius into air and measuring the combined volumes of the primary and satellite drops that are formed. The method’s origin can be traced to Tate, who postulated that the volume of the drop that falls from the capillary should be given by , where is the gravitational acceleration. Since Tate’s law is only an approximation and the actual drop volume , in practice the surface tension of the liquidair interface is determined from the experimental master curve due to Harkins and Brown (HB). The master curve is a plot of the fraction of the ideal drop volume, , as a function of the dimensionless tube radius, . Thus, once the actual drop volume , and hence , is known, is readily calculated upon determining the value of from the master curve and that . Although HB proposed their master curve more than 80 years ago, a sound theoretical foundation for the drop weight method has heretofore been lacking. This weakness is remedied here by determining the dynamics of formation of many drops and their satellites in sequence by solving numerically the recently popularized onedimensional (1–d) slenderjet equations. Computed solutions of the 1d equations are shown to be in excellent agreement with HB’s master curve when is low. Moreover, a new theory of the drop weight method is developed using the computations and dimensional analysis. The latter reveals that there must exist a functional relationship between the parameter , where is the dimensionless drop volume, and the gravitational Bond number , the Ohnesorge number , where is the viscosity, and the Weber number . When , the computed results show that depends solely on . In this limit, a new correlation is deduced which has a simple functional form, , and is more convenient to use than that of HB. The computed results are also used to show how the original drop weight method can be extended to situations where We is finite and resulting drop volumes are not independent of Oh.
 Viscous and NonNewtonian Flows

Lubrication flow between a cavity and a flexible wall
View Description Hide DescriptionLubrication flows near deformable solid boundaries occur in a diverse range of settings including coating and printing processes, biological systems, and suspensions. In order to examine the effect of surfacetopography on the elastohydrodynamic interactions that arise in these flows, the flow between a rigid cavity and a flexible wall is studied. Reynolds equation for the fluid is coupled to a model for the wall which is backed by a series of springs and/or held by a uniform tension force. The resulting nonlinear ordinary differential equations are then solved numerically to obtain pressure profiles and wall positions. When the wall modulus or tension is large relative to viscous forces, the wall hardly deforms and both a pressure mountain and valley are observed due to the gap change produced by the cavity topography. When the wall modulus and tension are small relative to viscous forces, the wall easily deforms and assumes a shape similar to that of the cavity. The pressure profiles are also dramatically altered and in some cases show only a valley without a mountain. Cavity shape is found to have a significant influence on both the pressure profiles and the wall deformation. The results suggest that surfacetopography may significantly modify the elastohydrodynamic interactions that arise in lubrication flows near deformable solid boundaries.

The dynamic mechanism for turbulent drag reduction using rigid fibers based on Lagrangian conditional statistics
View Description Hide DescriptionThe orientation moments and stresses for a suspension of rigid fibers are calculated along Lagrangian pathlines in a dragreducedturbulent channel flow. The turbulent flow fields are calculated using the methods developed by Paschkewitz et al. [“Numerical simulation of turbulentdrag reduction using rigid fibers,” J. Fluid Mech.518, 281 (2004)]. These authors investigated turbulentdrag reduction using rigid fibers with the Eulerian frame of reference direct numerical simulations, and demonstrated a correlation of drag reduction with fluctuations or “bursts” of fiber stress in intervortex extensional flow regions. These bursts are defined as events where certain components of the fiber stress exceed a specified level. Using probability density functions (PDFs) of the fiber stress contribution to the dissipation of turbulent kinetic energy weighted by the probability of occurrence of a particular stress level, we demonstrate that less than 0.001% of the dissipation is created by stress fluctuations of magnitude and . We therefore define a “small” burst or fluctuation as one of this magnitude or less and a “large” burst to be one having a magnitude greater than these thresholds. Using conditional statistics in the Lagrangian frame, the detailed dynamics responsible for the generation of these stress bursts are quantitatively characterized. As a precursor to the burst, the fibers move into extensional flow regions and align with the wallnormal or spanwise directional axis in a process that takes approximately two local strain units, where the strain rate is defined along the direction of the fiber orientation. After the stress increases past a chosen large value and the stress burst begins, the fibers continue to align and generate increasing stress until the burst is roughly half complete; at this time, the extensional character of the surrounding flow is reduced to a level such that the fibers begin to realign in the flow direction and the fiber stresses decrease. These stress bursts also have a total duration of approximately two local strain units. By considering the flowkinematics in regions near the particle, as well as the time autocorrelation of fiber stress and the second flow invariant , we demonstrate that the bursts end because the surrounding vortices are strongly weakened or destroyed by the gradients in fiber stresses. Both frequently observed small stress fluctuations and rare large stress fluctuations exhibit this behavior, with the primary difference being the strength of the nearby vortices and the resulting extensional flow region that the fiber resides in during the burst. However, the more commonly observed small stress fluctuations appear to make the largest contribution to fiber dissipation of turbulent kinetic energy and thus are responsible for the majority of the drag reduction effect. The largest contribution to fiber dissipation of turbulent kinetic energy is made by small fluctuations in the spanwise shear stress component , which result from fibers primarily confined to the plane weakly rotating towards the axis.

Imaging and quantifying mixing in a model droplet micromixer
View Description Hide DescriptionRapid mixing is essential in a variety of microfluidic applications but is often difficult to achieve at low Reynolds numbers. Inspired by a recently developed microdevice that mixes reagents in droplets, which simply flow along a periodic serpentine channel [H. Song, J. D. Tice, and R. F. Ismagilov, “A microfluidic system for controlling reaction networks in time,” Angew. Chem. Int. Ed.42, 767 (2003)], we investigate a model “droplet mixer.” The model consists of a spherical droplet immersed in a periodic sequence of distinct external flows, which are superpositions of uniform and shear flows. We label the fluid inside the droplet with two colors and visualizemixing with a method we call “backtrace imaging,” which allows us to render cross sections of the droplet at arbitrary times during the mixing cycle. To analyze our results, we present a novel scalar measure of mixing that permits us to locate sets of parameters that optimize mixing over a small number of flow cycles.
 Particulate, Multiphase, and Granular Flows

Investigation of fluidic assembly of nanowires using a droplet inside microchannels
View Description Hide DescriptionNanowires are common building blocks for the bottomup assembly of electronic and photonic devices. A significant challenge is to introduce a single nanowire into an oriented assembly in order to express its unique anisotropic properties or to fabricate a nanodevice. In this work we focused on the development of a micrometer length scale approach, based on a fluidic method for alignment and assembling of nanowires. The alignment is achieved by manipulating a droplet composed of a dilute nanowiresuspension by creating thermocapillary motion inside a microchannel. Our purpose is to explore the nanowires’ alignment mechanism in the middle region between the droplet’s front and rear menisci, and their interaction with the free surface and the contact lines. Experimental results show that nanowires which are found in the middle region of the droplet are generally aligned with the flow direction. Nanowires which reach the front meniscus move together with the displacing fluid which undergoes a “rolling” type motion, and are finally adsorbed to the surface of the microchannel. The adsorbed nanowires were found in most cases to align with the droplet’s flow direction. However, in certain cases nanowires may become reoriented by the passage of the rearcontact line.

Capture of particles of dust by convective flow
View Description Hide DescriptionInteraction of particles of dust with vortex convective flows is under theoretical consideration. It is assumed that the volume fraction of solid phase is small, variations of density due to nonuniform distribution of particles and those caused by temperature nonisothermality of medium are comparable. Equations for the description of thermal buoyancy convection of a dusty medium are developed in the framework of the generalized Boussinesq approximation taking into account finite velocity of particle sedimentation. The capture of a cloud of dust particles by a vortex convective flow is considered, general criterion for the formation of such a cloud is obtained. The peculiarities of a steady state in the form of a dust cloud and backward influence of the solid phase on the carrier flow are studied in detail for a vertical layer heated from the sidewalls. It is shown that in the case, when this backward influence is essential, a hysteresis behavior is possible. The stability analysis of the steady state is performed. It turns out that there is a narrow range of governing parameters, in which such a steady state is stable.

Energy nonequipartition, rheology, and microstructure in sheared bidisperse granular mixtures
View Description Hide DescriptionEventdriven simulations of smooth inelastic hard disks are used to probe the transport properties and the microstructure of bidisperse granular mixtures. A generic feature of such mixtures is that the two species have different levels of fluctuation kinetic energy in contrast with their elastic counterpart. The microscopic mechanism for this energy nonequipartition is shown to be directly tied to the asymmetric nature of collisional probabilities between the heavier and lighter species, compared to their purely elastic counterpart. The degree of collisional asymmetry increases with both increasing inelasticity and mass disparity, thereby increasing the energy ratio in the same limit. A phenomenological constitutive model, that incorporates energy nonequipartition, captures the nonmonotonic behavior of the transport coefficients, in agreement with the simulation results, whereas the standard constitutive model with equipartition assumption predicts monotonic variations. The sheared granular mixture readily forms clusters, having striped patterns along the extensional axis of the flow. The microstructural flow features are extracted by measuring the clustersize distributions, the paircorrelation function and the collisionangle distribution. While the inelastic dissipation is responsible for the onset of clustering, we have found that the mass disparity between the two species enhances the degree of clustering significantly in the sense that the size of the largest cluster increases with increasing mass disparity. At the microscopic level, the particle motion becomes more and more streamlined (i.e., ordered along the streamwise direction which is also a signature of enhanced shortrange correlations) with increasing dissipation and mass disparity, which is responsible for the enhanced first normal stress difference in the same limit.
 Laminar Flows

Fullspectrum nonlinear response of a sinusoidally modulated rotating disk electrode
View Description Hide DescriptionThe nonlinear behavior of a sinusoidally modulated rotating disk electrode (RDE) is analyzed to identify the flow and concentration field interactions that lead to resonance and nullification conditions in the electrochemical response of the system. Steady periodic solutions to the von Kármán selfsimilar form of the Navier–Stokes equations and convective diffusionequation are examined using a combined Fourier transform/perturbation approach that yields a set of frequency dispersionequations solved using the finite element method. Experimental measurements using a RDE with sinusoidal angular velocity waveform are compared to theory for modulation amplitudes and electrolyte Schmidt number of 1680. The modulation frequency is varied from the quasisteady to highfrequency limits to obtain the phase and amplitude of the mean, fundamental, and higherharmonic limiting current. We show that distinct features (resonances and nullifications) in the nonlinear mean and higher harmonic components of limiting current result from constructive and destructive (in phase and 180° out of phase, respectively) interactions between cross terms of the oscillating components in the axial flow and concentration fields near the disk surface.

Weakinertial flow between two rough surfaces
View Description Hide Description“Oseen–Poiseuille” equations are developed from an asymptotic formulation of the threedimensional Navier–Stokes equations in order to study the influence of weak inertia on flows between rough surfaces. The impact of the first correction on macroscopic flow due to inertia has been determined by solving these equations numerically. From the numerical convergence of the asymptotic expansion to the threedimensional Navier–Stokes flows, it is shown that, at the macroscopic scale, the quadratic correction to the Reynolds equation in the weakinertial regime vanishes generalizing a similar result in porous media.

Steady axisymmetric flow in an open cylindrical container with a partially rotating bottom wall
View Description Hide DescriptionThe steady motion of a viscousfluid in a cylindrical container with a partially rotating bottom wall and a free surface is investigated by means of axisymmetric Navier–Stokes simulations. The flow above the spinning disk at the center of the bottom wall is dominated by an Ekman boundary layer that drives the fluid radially outward. In contrast, an inwardflow ensues along the outer, stationary part of the bottom wall, where the radially increasing pressure distribution set up by the rotating fluid motion near the free surface is not balanced by a corresponding centrifugal force. As a result, flow separation occurs at an intermediate radial location close to the outer edge of the rotating disk. Thus a flow configuration results that is dominated by a meridional vortex above the spinning disk, and a counterrotating vortex above the stationary part of the bottom wall. Simulations are conducted for various aspect ratios and Reynolds numbers, in order to evaluate the resulting changes in the vortex breakdown configurations. As the ratio of container radius to disk radius increases above a value of about 2.3, the influence of the lateral container wall on the features of the central flow in the neighborhood of the spinning disk becomes insignificant. By means of a simplified model problem, it is demonstrated that this rapid loss of influence is due to the exponential decay of the azimuthal surface velocity beyond the edge of the disk. This exponential decay is confirmed by the numerical data, and it reflects the fact that as the lateral wall moves outward, the stationary part of the end wall becomes the main sink for the azimuthal momentum of the fluid.
 Instability and Transition

Simulations of crescent water wave patterns on finite depth
View Description Hide DescriptionA numerical study of the instabilities of Stokes waves on finite depth has been carried out using an efficient fully nonlinear method [D. Clamond and J. Grue, “A fast method for fully nonlinear waterwave computations,” J. Fluid Mech.447, 337 (2001)]. First, attention is given to fivewave instabilities with , being the wavenumber and the depth. Both instabilities leading to breaking and instabilities leading to recurrence are studied, yielding considerably different patterns than on infinite depth. Higherorder instabilities are exemplified, for the first time, by simulations of six and sevenwave instabilities. Simulations of interactions between four and fivewave instabilities show that a classical modulational instability can destabilize a threedimensional perturbation causing crescent waves to appear, in accordance with the hypothesis of [M.Y. Su and A. W. Green, “Coupled two and threedimensional instabilities of surfacegravity waves,” Phys. Fluids27, 2595 (1984)]. Also, a recurrent fivewave instability can boost the energy in a fourwave instability.

Simulations of oscillatory convection in mixtures in moderate aspect ratio containers
View Description Hide DescriptionSimulations of mixtures with negative separation ratios in twodimensional containers with realistic boundary conditions and moderately large aspect ratio are described. The system exhibits a large variety of states with complex time dependence including intermittent wave localization and chaotic “repeated transients.” Steady but localized states are also found. Particular attention is paid to the transitions that occur for , where is the Rayleigh number and its critical value for the primary instability, in order to clarify the gradual transition from a small number of active degrees of freedom to many active degrees of freedom .