Volume 21, Issue 10, October 2009

Imposing axial flow in the annulus and/or radial flow through the cylindrical walls in a Taylor–Couette system alters the stability of the flow.Theoretical methods and numerical simulations were used to determine the impact of imposed axial and radial flows, homogeneous in the axial direction, on the first transition of Taylor–Couette flow in the framework of convective and absolute instabilities. At low axial Reynolds numbers the convective instability is axisymmetric, but convective helical modes with an increasing number of helices having a helicity opposite that of the base flow dominate as the axial flow increases. The number of helices and the critical Taylor number are affected only slightly by the radial flow. The flow becomes absolutely unstable at higher Taylor numbers. Absolutely unstable axisymmetric modes occur for inward radial flows, while helical absolute instability modes having a helicity identical to that of the base flow occur at high enough axial Reynolds numbers for outward radial flow.
 LETTERS


A simple walllayer model for large eddy simulation with immersed boundary method
View Description Hide DescriptionA walllayer model is proposed for large eddy simulation of high Reynolds numberturbulent flows in conjunction with immersed boundaries. The model is based on two main steps: the reconstruction of the velocity field at the first grid point off the immersed body and the modelization of the actual wall shear stress at the immersed boundary through imposition of a Reynolds averaged Navier–Stokeslike eddyviscosity obtained by means of analytical considerations. The model is tested in a turbulent plane channel flow with walls reproduced by immersed boundaries considering both Cartesian and curvilinear grids. Even with coarse and distorted grids the proposed methodology is able to reproduce accurately both first and secondorder turbulent statistics.

Tube geometry can force switchlike transitions in the behavior of propagating bubbles
View Description Hide DescriptionMicroscale process engineering requires precise control of bubbles and droplets. We investigate geometryinduced control and find that a centered constriction in the cross section of rectangular tubes can lead to new families of steadily propagating bubbles, which localize in the leastconstricted regions of the cross section. Tuning the constriction geometry can cause a switchlike transition from centered to localized bubbles at a critical value of the flow rate: a mechanism for flowratedriven bubble control. The accompanying large change in bubble volume could be significant for liquid recovery applications.

Exact and asymptotic conditions on traveling wave solutions of the Navier–Stokes equations
View Description Hide DescriptionWe derive necessary conditions that traveling wave and other solutions of the Navier–Stokes equations must satisfy in the pipe, Couette, and channel flow geometries. Some conditions are exact and must hold for any traveling wave solution or periodic solution irrespective of the Reynolds number (Re). Other conditions are asymptotic in the limit . For the pipe flow geometry, we give computations up to showing the connection of our asymptotic conditions to critical layers that accompany vortex structures at high Re.

Response modes of a flexible filament in the wake of a cylinder in a flowing soap film
View Description Hide DescriptionFlow induced vibrations are observed in many engineering applications. A flexible body located in the wake of an obstacle is usually forced to vibrate by the periodic vortices shedding from the obstacle. Here we focus on the response of the flexible body in the wake. Soap film tunnels are used to provide twodimensional flow. Cylinders and flexible filaments are employed as obstacles and flexible bodies, respectively. The filaments exhibit lockin behavior to the wake. Three response modes are found by changing the distance between the filaments and cylinders. The observations are illuminated in terms of waving plate theory.

Chaos in a cylinder wake due to forcing at the Strouhal frequency
View Description Hide DescriptionIn this letter we show that a cylinder oscillating harmonically in line with an incoming flow at a frequency equal to the natural frequency of vortex shedding induces for certain amplitudes of oscillation a chaotic state in the flow, characterized by an aperiodic lift force. The result is obtained through numerical simulation of the Navier–Stokes equations for twodimensional flow. The chaos is attributed to the competition between two modes: the natural mode of the wake and the mode forced by the moving cylinder, which have entirely different spatial structures.
 Top

 ARTICLES

 Biofluid Mechanics

On intra and intersubject variabilities of airflow in the human lungs
View Description Hide DescriptionThe effects of intra and intersubject variabilities in airway geometry on airflow in the human lungs are investigated by large eddy simulation. The airway models of two human subjects consisting of extra and intrathoracic airways are reconstructed from CT images. For intrasubject study, airflows at two inspiratory flow rates are simulated on the airway geometries of the same subject with four different levels of truncation. These airway models are the original complete geometry and three geometries obtained by truncating the original one at the subglottis, the supraglottis, and the laryngopharynx, respectively. A comparison of the airflows in the complete geometry model shows that the characteristics of the turbulentlaryngeal jet in the trachea are similar regardless of Reynolds number in terms of mean velocities, turbulence statistics, coherent structures, and pressure distribution. The truncated airway models, however, do not produce the similar flow structures observed in the complete geometry. An improved inlet boundary condition is then proposed for the airway model truncated at the laryngopharynx to improve the accuracy of solution. The new boundary condition significantly improves the mean flow. The spectral analysis shows that turbulent characteristics are captured downstream away from the glottis. For intersubject study, although the overall flow characteristics are similar, two morphological factors are found to significantly affect the flows between subjects. These are the constriction ratio of the glottis with respect to the trachea and the curvature and shape of the airways.
 Micro and Nanofluid Mechanics

Entrainment of a film on a surface from the meniscus of a liquid wedge during coating
View Description Hide DescriptionThe shape evolution of an entrained film from the meniscus of a liquid wedge is studied, both experimentally and theoretically. The liquid wedge is formed by a droplet of liquid injected between a substrate and a tilted plate. When the substrate moves relative to the tilted plate with a constant velocity, a film of a constant slope is entrained on it, while another film remains on the tilted plate. The numerical and analytical investigation of the process provides the dependence of the length and slope of the entrained film after the end of drawing process, as well as the maximum thickness of the film on the tilted plate, on the capillary number. The length of the entrained film was found to be minimal for infinitely large capillary numbers when the surface tension effects are negligibly small. Experimental data confirm the predicted characteristic geometry of the film for capillary numbers up to 0.75.

Transport properties of Brownian particles confined to a narrow channel by a periodic potential
View Description Hide DescriptionWe investigate the transport of Brownian particles in a twodimensional potential moving under the action of an external force or convected by a flow field. The potential is periodic in one direction and confines the particles to a narrow channel of varying cross section in the other direction. We apply the standard longwave asymptotic analysis in the narrow dimension and show that the leading order term is equivalent to that obtained previously from a direct extension of the Fick–Jacobs approximation. We also show that the confining potential has similar effects on the transport of Brownian particles to those induced by a solid channel. Finally, we compare the analytical results with Brownian dynamics simulations in the case of a sinusoidal variation of the width of a parabolic potential in the cross section. We obtain excellent agreement for the marginal probability distribution, the average velocity of the Brownian particles, and the asymptotic dispersion coefficient over a wide range of Péclet numbers.

Nucleation threshold and deactivation mechanisms of nanoscopic cavitation nuclei
View Description Hide DescriptionThe acoustic nucleation threshold for bubbles trapped in cavities has theoretically been predicted within the crevice theory by Atchley and Prosperetti [“The crevice model of bubble nucleation,” J. Acoust. Soc. Am.86, 1065 (1989)]. Here, we determine this threshold experimentally, by applying a single pressure pulse to bubbles trapped in cylindrical nanoscopic pits (“artificial crevices”) with radii down to 50 nm. By decreasing the minimum pressure stepwise, we observe the threshold for which the bubbles start to nucleate. The experimental results are quantitatively in good agreement with the theoretical predictions of Atchley and Prosperetti. In addition, we provide the mechanism which explains the deactivation of cavitation nuclei: gas diffusion together with an aspherical bubble collapse. Finally, we present superhydrophobic nuclei which cannot be deactivated, unless with a highspeed liquid jet directed into the pit.

Measurements of tangential momentum accommodation coefficient for various gases in plane microchannel
View Description Hide DescriptionMass flow rate measurements in a single silicon microchannel were carried out for various gases in isothermal steady flows. The results obtained from hydrodynamic to near free molecular regime by using a powerful experimental platform allowed us to deduce interesting information, notably about the reflection/accommodation process at the wall. In the 0–0.3 Knudsen range, a continuum analytic approach was derived from the NS equations, associated with first or second order slip boundary conditions. Identifying the experimental mass flow rate curves to the theoretical ones the tangential momentum accommodation coefficient (TMAC) of various gases was extracted. Over the full Knudsen range [0–30] the experimental results were compared with theoretical values calculated from the kinetic approaches: using variable accommodation coefficient values as fitting parameter, the theoretical curves were fitted to the experimental ones. Whatever the Knudsen range and whatever the theoretical approach, the TMAC values are found decreasing when the molecular weights of the gas increase (as long as the different gases are compared using the same approach). Moreover, the values of the various accommodation coefficients are rather close to one another but sufficiently smaller than unity indicating that the full accommodation modeling is not satisfactory to describe the gas/wall interaction.
 Interfacial Flows

Nonmodal and nonlinear dynamics of a volatile liquid film flowing over a locally heated surface
View Description Hide DescriptionThe stability of a thin, volatile liquid film falling under the influence of gravity over a locally heated, vertical plate is analyzed in the noninertial regime using a model based on longwave theory. The model is formulated to account for evaporation that is either governed by thermodynamic considerations at the interface in the onesided limit or limited by the rate of mass transfer of the vapor from the interface. The temperature gradient near the upstream edge of the heater induces a gradient in surface tension that opposes the gravitydriven flow, and a pronounced thermocapillary ridge develops in the streamwise direction. Recent theoretical analyses predict that the ridge becomes unstable above a critical value of the Marangoni parameter, leading to the experimentally observed rivulet structure that is periodic in the direction transverse to the bulk flow. An oscillatory, thermocapillary instability in the streamwise direction above the heater is also predicted for films with sufficiently large heat loss at the free surface due to either evaporation or strong convection in the adjoining gas. This present work extends the recent linear stability analysis of such flows by Tiwari and Davis [Phys. Fluids21, 022105 (2009)] to a nonmodal analysis of the governing nonselfadjoint operator and computations of the nonlinear dynamics. The nonmodal analysis identifies the most destabilizing perturbations to the film and their maximum amplification. Computations of the nonlinear dynamics reveal that small perturbations can be sufficient to destabilize a linearly stable film for a narrow band of wave numbers predicted by the nonmodal, linearized analysis. This destabilization is linked to the presence of stable, discrete modes that appear as the Marangoni parameter approaches the critical value at which the film becomes linearly unstable. Furthermore, the thermocapillary instability leads to a new, timeperiodic base state. This transition corresponds to a Hopf bifurcation with increasing Marangoni parameter. A linear stability analysis of this timeperiodic state reveals further instability to transverse perturbations, with the wave number of the most unstable mode about 50% smaller than for the rivulet instability of the steady base state and exponential growth rate about three times larger. The resulting film behavior is reminiscent of inertial waves on locally heated films, although the wave amplitude is larger in the present case near the heater and decays downstream where the Marangoni stress vanishes. The film’s heat transfer coefficient is found to increase significantly upon the transition to the timeperiodic flow.

Enhanced slip on a patterned substrate due to depinning of contact line
View Description Hide DescriptionWe perform numerical simulations of a shear flow over a periodically patterned substrate with entrapped gas bubbles. A diffuseinterface model is employed to handle the liquidgas interfacedeformation and the threephase contact line. Depending on the shear rate and the pattern geometry, four flow regimes are observed. The contact lines can be pinned, depinned, or eliminated depending on the competition between the shear force and the surface tension. The effective slip length is found to be dependent on the morphology of the menisci and hence on the shear rate. In particular, the bubbles are transformed into a continuous gas film when the shear rate is larger than a critical value, resulting in a significantly enhanced slip length proportional to the liquidgas viscosity ratio. The present results have interesting implications for effective slip on superhydrophobic surfaces.

Electrowetting with contact line pinning: Computational modeling and comparisons with experiments
View Description Hide DescriptionThis work describes the modeling and simulation of planar electrowetting on dielectric devices that move fluid droplets by modulating surface tension effects. The fluid dynamics are modeled by HeleShaw type equations with a focus on including the relevant boundary phenomena. Specifically, we include contact angle saturation and a contact line force threshold model that can account for hysteresis and pinning effects. These extra boundary effects are needed to make reasonable predictions of the correct shape and time scale of liquid motion. Without them the simulations can predict droplet motion that is much faster than in experiments (up to 10–20 times faster). We present a variational method for our model, and a corresponding finite element discretization, which is able to handle surface tension, conservation of mass, and the nonlinear contact line pinning in a straightforward and numerically robust way. In particular, the contact line pinning is captured by a variational inequality. We note that all the parameters in our model are derived from first principles or from independent experiments except one (the parameter that accounts for the extra resistive effect of contact angle hysteresis and is difficult to measure directly). We quantitatively compare our simulation to available experimental data for four different cases of droplet motion that include splitting and joining of droplets and find good agreement with experiments.
 Viscous and NonNewtonian Flows

Axisymmetric instabilities in electrospinning of highly conducting, viscoelastic polymer solutions
View Description Hide DescriptionIn this paper the axisymmetric instabilities observed during the electrospinning of highly electrically conducting, viscoelastic poly(ethylene oxide) (PEO)/water solutions are investigated. In our theoretical study, a linear stability analysis is coupled with a model for the stable electrospun jet. The combined model is used to calculate the expected bead growth rate and wave number for given electrospinning conditions. In the experimental section of the study, PEO/water solutions are electrospun and the formation of axisymmetric beads is captured using highspeed photography. Experimental values for the bead growth rate and wave number are extracted and compared with the model predictions. An energy analysis is then carried out on the stability results to investigate the mechanism of instability via the coupling between base flow and perturbation. The analysis reveals that the unstable axisymmetric mode for electrically driven, highly conducting jets is not a capillary mode, but is mainly driven by electrical forces due to the interaction of charges on the jet. We note that this axisymmetric, conducting mode often exhibits a growth rate too small to be observed during electrospinning. However, both our experiments and stability analysis demonstrate that the axisymmetric instability with a high growth rate can be seen in practice when the electrical force is effectively coupled with viscoelastic forces.

Coating flow of viscous Newtonian liquids on a rotating vertical disk
View Description Hide DescriptionWe study a Newtonian viscousliquid coating a vertical rotating disk in the creeping flow regime. Experiments were performed at varying disk rotation speeds and liquid volumes, and the thickness profile at steady state was measured. While the maximum liquid supported by the rotating disk varied with rotation rate and liquidviscosity, the numerical value of a dimensionless number signifying the ratio of gravity to viscous forces was the same in all the cases, . A lubrication analysis for the time evolution of the film thickness that accounted for gravity, surface tension, and viscous forces was solved numerically to steady state. The predicted thickness profiles are in quantitative agreement with those obtained experimentally. The lubrication equation at steady state was solved analytically in the absence of surface tension to obtain constant height contours that were circular and symmetric about the horizontal axis. However to obtain a complete solution, knowledge of the height variation across the contours is required, and this is controlled by the surface tension. On including this effect, we derived an asymptotic solution to predict thickness profiles that agree well with measurements for large values of viscosity or rotation rates.

Thermal convection of a viscoplastic liquid with high Rayleigh and Bingham numbers
View Description Hide DescriptionWe consider the effect of yield stress on the Rayleigh–Bénard convection of a viscoplastic material. First we consider the model problem of convection in a differentially heated loop, which is described by the (modified) Lorenz equations. The presence of the yield stress significantly alters the dynamics of the system. In particular, the chaotic motion can stop suddenly (sometimes, after a period of chaotic oscillations). Guided by the model equations we performed direct numerical simulations of convection of the Bingham liquid in a square cavity heated from bellow. Our interest has been concentrated on the situation when both buoyancy and plastic forces are large. The obtained results are in a reasonable agreement with the predictions by the Lorenz equations.
 Particulate, Multiphase, and Granular Flows

Lattice Boltzmann simulation of the rise and dissolution of twodimensional immiscible droplets
View Description Hide DescriptionWe used a coupled multiphase lattice Boltzmann (LB) model to simulate the dissolution of immiscible liquid droplets in another liquid during the rising process resulting from buoyancy. It was found that there existed a terminal rise velocity for each droplet, and there was a power law relationship between the Eötvös (Eo) number and the terminal Reynolds (Re) number. Our simulation results were in agreement with the empirical correlation derived for predicting bubble rise. When more than two identical droplets rose simultaneously in a close proximity, the average terminal rise velocity was lower than that of a single droplet with the same size because of the mutual resistant interactions. The droplet trajectories at the noncentral positions were not straight because of the nonzero net horizontal forces acting on the droplets. The Damkohler (Da) and Peclet (Pe) numbers were varied to investigate the coupling between droplet size, flow field, dissolution at the interface, and solutetransport. For a given Pe, increasing Da led to a higher dissolution rate. For a given Da, increasing Pe led to a higher dissolution rate. For a large Da and a small Pe, the process near the interface was diffusion limited, and the advective flow relative to the droplet resulting from droplet rise was unable to move the accumulated solute away from the interface quickly. In this case, it was favorable to split the single droplet into as many small ones as possible in order to increase the interface area per unit mass and consequently enhance the whole dissolution process. For a small Da and a large Pe, the process was dissolution limited near the interface. The mass of accumulated solute near the interface was little, so the advective flow at the top side of the droplet was able to clean the solute quickly. In this case it was favorable to keep the droplet as a single one in order to obtain a high rise velocity and consequently enhance the whole dissolution process. By studying the coupling between Da and Pe, we qualitatively proposed to construct a DaPe phase plane and found the interface dividing the plane into regions 1 and 2. Region 1 was the collection of points where it was favorable to break down the droplet into as many small ones as possible in order to accelerate dissolution, while region 2 was the collection of points where it was favorable to keep the droplet in a single one for the same purpose. Based on our LB simulations, we found that the interface was an increasing function of Pe. Region 1 was the portion above the interface, while region 2 was the portion below it. In real applications, if both Pe and Da are obtained, it will be easy to judge whether it is favorable to break down the droplet or not in order to accelerate dissolution by checking whether (Pe, Da) falls in region 1 or 2.

Effects of geometric shape on the hydrodynamics of a selfpropelled flapping foil
View Description Hide DescriptionThe hydrodynamics of a free flapping foil is studied numerically. The foil undergoes a forced vertical oscillation and is free to move horizontally. The effect of chordthickness ratio is investigated by varying this parameter while fixing other ones such as the Reynolds number, the density ratio, and the flapping amplitude. Three different flow regimes have been identified when we increase the chordthickness ratio, i.e., leftright symmetry, backandforth chaotic motion, and unidirectional motion with staggered vortex street. It is observed that the chordthickness ratio can affect the symmetrybreaking bifurcation, the arrangement of vortices in the wake, and the terminal velocity of the foil. The similarity in the symmetrybreaking bifurcation of the present problem to that of a flapping body under constraint is discussed. A comparison between the dynamic behaviors of an elliptic foil and a rectangular foil at various chordthickness ratios is also presented.

Effects of viscosity ratio and three dimensional positioning on hydrodynamic interactions between two viscous drops in a shear flow at finite inertia
View Description Hide DescriptionDrops driven toward each other by shear at finite inertia follow two distinct types of trajectories. Type I trajectory is similar to the one in Stokes flow where drops slide past each other. However, at finite inertia, drops display a new type II trajectory, where they reverse their paths. Increasing viscosity ratio results in a transition from type II to type I trajectory. The transition is caused by decreased drop deformation and increased alignment with the flow at higher drop viscosity; both decrease the zone of reversed streamlines that accompanies a drop at finite inertia. The transition is delineated in a phase diagram of Reynolds number and viscosity ratio for different capillary numbers. The critical viscosity ratio, where a type II transitions into type I, increases with Reynolds number except at higher capillary numbers, where the critical viscosity ratio shows a slight nonmonotonic variation with Reynolds number. Also, it is nonmonotonic with capillary numbers in that for a fixed Reynolds number, the critical viscosity ratio first increases with increasing capillary number and then decreases. Similar to the Stokes regime, increased viscosity ratio leads to a decreased postcollision crossstream separation effectively decreasing the shear induced diffusion. Higher viscosity ratio results in an increased separation between drops during encounter, which results in a smaller interaction time. With drops placed initially at different shear planes, drops come under the influence of the reversed flow zone around a single drop that broadens off the central shear plane. Consequently, the trajectory changes from type I to type II as the offset in the vorticity direction increases. The change depends on the initial offset in the shear direction as well. The final displacement in the shear direction varies linearly with the initial offset. The net relative displacement in the shear direction shows a gradual decrease with increasing offset. The net relative displacement in the vorticity direction with increasing offset first increases from a zero value when drops are placed at the same shear plane to a maximum and then decreases. For certain cases, it reaches a negative value.

Sedimentation of a sphere in a fluid channel
View Description Hide DescriptionWe studied both experimentally and numerically the sedimentationvelocity of small solid particles through liquid channels merging at the intersection of three soap films. The wall mobility induces a nontrivial behavior for the particle drag coefficient, providing particular transport properties that are not observed for channels with rigid walls. It is shown that for sufficiently small particles, slow and fast motions are observed for the particle along the channel, depending on the particle position within the channel cross section and the sphere/channel size ratio. The velocity corresponding to fast motions can be as high as twice the Stokes velocity in an unbounded fluid. Moreover, the fast motions are not observed anymore when the size ratio exceeds a critical value, which has been found to be approximately equal to 0.5. As another major difference with the solid wall channel, the sphere velocity does not vanish when the size ratio reaches unity. Instead, the smallest value is found to be of the Stokes velocity.