Volume 22, Issue 3, March 2010

The paper begins by showing how standard results on the average hydrodynamic stress in a uniform fluidparticle system follow from a direct, elementary application of Cauchy’s stress principle. The same principle applied to the angular momentum balance proves the emergence, at the mesoscale, of an antisymmetric component of the volumeaveraged hydrodynamic stress irrespective of the particle Reynolds number. Several arguments are presented to show the physical origin of this result and to explain how the averaging process causes its appearance at the mesoscale in spite of the symmetry of the microscale stress. Examples are given for zero and finite Reynolds number, and for potential flow. For this last case, the antisymmetric stress component vanishes, but the Cauchy principle proves nevertheless useful to derive in a straightforward way known results and to clarify their physical nature.
 LETTERS


On quasiperiodic and subharmonic Floquet wake instabilities
View Description Hide DescriptionThe physical characteristics of bifurcated states in systems with inherent symmetry are constrained in ways that those in systems with broken symmetry are not. Here we examine the issue of quasiperiodic versus subharmonic instability modes of timeperiodic laminar wakes, and how the relationship between them is influenced by weak symmetry breaking. The examples used are the vortex street wake of a circular cylinder, where symmetry is broken by distorting the cylinder into a ring, and the wake of a square cylinder, where symmetry is broken by a small fixed rotation of the cylinder about its axis. In both cases the symmetric wakes exhibit a quasiperiodic instability mode, with a pair of complexconjugate Floquet multipliers and which manifests as a traveling wave. As symmetry is broken these multipliers migrate continuously to the real axis, coalesce, and split into a pair of subharmonic multipliers that move apart along the negative real axis. This behavior resolves an apparent dichotomy between the previously established theoretical results and numerical predictions for the symmetric wake systems, and the predictions and experimental observations for systems with weakly broken symmetry.

Influence of an imposed flow on the stability of a gravity current in a near horizontal duct
View Description Hide DescriptionWe study experimentally the effect of a mean flow imposed on a buoyant exchange flow of two miscible fluids of equal viscosity in a long tube oriented close to horizontal. We measure the evolution of the front velocity as a function of the imposed velocity . At low , an exchangeflow dominated regime is found, as expected, and is characterized here by Kelvin–Helmholtzlike instabilities. With increasing we observed that the flow becomes stable. Here also increases linearly with with slope of . At large we find .

An investigation of string cavitation in a truescale fuel injector flow geometry at high pressure
View Description Hide DescriptionString cavitation has been studied in an optical automotive size fuel injector with truescale flow geometry at injection pressures of up to 2050 bar. The multihole nozzle geometry studied allowed observation of the holetohole vortex interaction and, in particular, that of a bridging vortex in the sac region between the holes. A dependency on Reynolds number was observed in the formation of the visible, vapor filled vortex cores. Above a threshold Reynolds number, their formation and appearance during a 2 ms injection event was repeatable and independent of upstream pressure and cavitation number.
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 ARTICLES

 Biofluid Mechanics

The optimal elastic flagellum
View Description Hide DescriptionMotile eukaryotic cells propel themselves in viscous fluids by passing waves of bendingdeformation down their flagella. An infinitely long flagellum achieves a hydrodynamically optimal lowReynolds number locomotion when the angle between its local tangent and the swimming direction remains constant along its length. Optimal flagella therefore adopt the shape of a helix in three dimensions (smooth) and that of a sawtooth in two dimensions (nonsmooth). Physically, biological organisms (or engineered microswimmers) must expend internal energy in order to produce the waves of deformation responsible for the motion. Here we propose a physically motivated derivation of the optimal flagellum shape. We determine analytically and numerically the shape of the flagellar wave which leads to the fastest swimming for a given appropriately defined energetic expenditure. Our novel approach is to define an energy which includes not only the work against the surrounding fluid, but also (1) the energy stored elastically in the bending of the flagellum, (2) the energy stored elastically in the internal sliding of the polymeric filaments which are responsible for the generation of the bendingwaves (microtubules), and (3) the viscous dissipation due to the presence of an internal fluid. This approach regularizes the optimal sawtooth shape for twodimensional deformation at the expense of a small loss in hydrodynamic efficiency. The optimal waveforms of finitesize flagella are shown to depend on a competition between rotational motions and bending costs, and we observe a surprising bias toward halfinteger wave numbers. Their final hydrodynamic efficiencies are above 6%, significantly larger than those of swimming cells, therefore indicating available room for further biological tuning.

Local instabilities of flow in a flexible channel: Asymmetric flutter driven by a weak critical layer
View Description Hide DescriptionWe examine the linear stability of twodimensional Poiseuille flow in a long channel confined by a rigid wall and a massless dampedtensioned membrane. We seek solutions that are periodic in the streamwise spatial direction and time, solving the homogeneous eigenvalue problem using a Chebyshev spectral method and asymptotic analysis. Several modes of instability are identified, including Tollmien–Schlichting (TS) waves and travelingwave flutter (TWF). The eigenmode for neutrally stable downstreampropagating TWF in the absence of wall damping is shown to have a novel asymptotic structure at high Reynolds numbers, not reported in symmetric flexiblewalled channels, involving a weak but destabilizing critical layer at the channel centerline where the wave speed is marginally greater than the maximum Poiseuille flow speed. We also show that TS instabilities along the lower branch of the neutral curve are modified remarkably little by wall compliance, but can be either stabilized or destabilized by wall damping. We discuss the energy budget underlying TWF and briefly describe the structure of other flowinduced surface instabilities.

Caenorhabditis elegans swimming in a saturated particulate system
View Description Hide DescriptionCaenorhabditis elegans (C. elegans) is a nematode that often swims in saturated soil in nature. We investigated the locomotive behavior of C. elegans swimming in a fluid with particles of various sizes and found that the nematode swims a greater distance per undulation than it does in a fluid without particles. The Strouhal number (a ratio of lateral to forward velocity) of C. elegans significantly decreases in a saturated particulate medium in comparison to a fluid without particles . This result was unexpected due to the generally low performance of a body moving in a high drag medium. In our model, a saturated granular system is approximated as a porous medium where only the hydrodynamic forces on the body are considered. Combining these assumptions with resistive force theory, we find that a porous medium provides more asymmetric drag on a slender body, and consequently that C. elegans locomotes with a greater distance per undulation.
 Micro and Nanofluid Mechanics

Investigation of rarefied supersonic flows into rectangular nanochannels using a threedimensional direct simulation Monte Carlo method
View Description Hide DescriptionThe rarefied flow of nitrogen with speed ratio (mean speed over most probable speed) of , pressure of 10.132 kPa into rectangular nanochannels with height of 100, 500, and 1000 nm is investigated using a threedimensional, unstructured, direct simulation Monte Carlo method. The parametric computational investigation considers rarefaction effects with Knudsen number , geometric effects with nanochannel aspect ratios of from , and backpressure effects with imposed pressures from 0 to 200 kPa. The computational domain features a buffer region upstream of the inlet and the nanochannel walls are assumed to be diffusively reflecting at the free stream temperature of 273 K. The flow analysis is based on the phase space distributions while macroscopic flow variables sampled in cells along the centerline are used to corroborate the microscopic analysis. The phasespace distributions show the formation of a disturbance region ahead of the inlet due to slow particles backstreaming through the inlet and the formation of a density enhancement with its maximum inside the nanochannel. Velocity phasespace distributions show a lowspeed particle population generated inside the nanochannel due to wall collisions which is superimposed with the free stream highspeed population. The mean velocity decreases, while the number density increases in the buffer region. The translational temperature increases in the buffer region and reaches its maximum near the inlet. For nanochannels the gas reaches near equilibrium with the wall temperature. The heat transfer rate is largest near the inlet region where nonequilibrium effects are dominant. For , vacuum back pressure, and , the nanoflow is supersonic throughout the nanochannel, while for , the nanoflow is subsonic at the inlet and becomes sonic at the outlet. For , , and imposed back pressure of 120 and 200 kPa, the nanoflow becomes subsonic at the outlet. For and , the outlet pressure nearly matches the imposed back pressure with the nanoflow becoming sonic at 40 kPa and subsonic at 100 kPa. Heat transfer rates at the inlet and mass flow rates at the outlet are in good agreement with those obtained from theoretical freemolecular models. The flows in these nanochannels share qualitatively characteristics found in microflows and continuum compressible flows in channels with friction and heat loss.

Analysis of electrowettingdriven spreading of a drop in air
View Description Hide DescriptionA set of shape mode equations is derived to describe unsteady motions of a sessile drop actuated by electrowetting. The unsteady, axially symmetric, and linearized flow field is analyzed by expressing the shape of a drop using the Legendre polynomials. A modified boundary condition is obtained by combining the contact angle model and the normal stress condition at the surface. The electrical force is assumed to be concentrated on one point (i.e., threephase contact line) rather than distributed on the narrow surface of the order of dielectric layer thickness near the contact line. Then, the delta function is used to represent the wetting tension, which includes the capillary force, electrical force, and contact line friction. In previous work [J. M. Oh et al., Langmuir24, 8379 (2008)], the capillary forces of the airsubstrate and liquidsubstrateinterfaces were neglected, together with the contactline friction. The delta function is decomposed into a weighted sum of the Legendre polynomials so that each component becomes a forcing term that drives a shape mode of motion. The shape mode equations are nonlinearly coupled between modes due to the contact line friction. The equilibrium contact angle of electrowetting predicted by the present method shows a good agreement with the Lippmann–Young equation and with our experimental results. The present theoretical model is also validated by predicting the spreading of a drop for step input voltages. It shows qualitative agreement with experimental results in temporal evolution of drop shape.
 Interfacial Flows

Nonlinear dynamics of a thin liquid film on an axially oscillating cylindrical surface
View Description Hide DescriptionWe have derived a nonlinear evolution equation describing the dynamics of an axisymmetric liquid film on a cylindrical surface subjected to axial harmonic oscillation. We have found that the capillary longtime film rupture typical for the case of a film on a static cylinder can be arrested if the substrate is forced with a sufficiently high amplitude and/or frequency. The threshold for the rupture prevention is determined by the product of the dimensionless amplitude and frequency of forcing, whereas the value of this product is independent of forcing parameters. This threshold delineates the borderline between the ruptured and nonruptured subdomains. A typical pattern in the nonruptured subdomain consists of a single drop within the periodic domain, whereas the number of drops in the ruptured subdomain varies with the forcing amplitude when the rest of parameters remains fixed. The amplitude of film thickness norm in the parameter domain corresponding to nonruptured states of the system was found to increase with the distance from criticality, which is typical for forward bifurcation.

The water entry of decelerating spheres
View Description Hide DescriptionWe present the results of a combined experimental and theoretical investigation of the vertical impact of lowdensity spheres on a watersurface. Particular attention is given to characterizing the sphere dynamics and the influence of its deceleration on the shape of the resulting air cavity. A theoretical model is developed which yields simple expressions for the pinchoff time and depth, as well as the volume of air entrained by the sphere. Theoretical predictions compare favorably with our experimental observations, and allow us to rationalize the form of waterentry cavities resulting from the impact of buoyant and nearly buoyant spheres.

Numerical prediction of the film thickening due to surfactants in the Landau–Levich problem
View Description Hide DescriptionIn this work numerical solutions of the dip coating problem in the presence of a soluble surfactant are shown. Predictions of film thickening as well as thickening factors are in very good agreement with published experimental data, showing that pure hydrodynamic modeling suffices to mimic the process. Our numerical solutions provide a wealth of information on the functioning of the dip coating system; they show the appearance of a second stagnation point located in the bulk phase near the dynamic meniscus and they give clues about how the flow patterns might change as the surfactant becomes less soluble.
 Viscous and NonNewtonian Flows

Formation of viscoplastic drops by capillary breakup
View Description Hide DescriptionThe process of growth and detachment of drops from a capillary nozzle is studied experimentally by highspeed imaging. Newtonian drops are compared to shearthinning and viscoplasticdrops. Both Newtonian and shearthinningfluid drops grow on the end of the capillary until a maximum supportable tensile stress is reached in the drop neck, after which they become unstable and detach. The critical stress is not influenced by variations in viscosity or in the degree of shear thinning.Viscoplastic fluids show a different behavior: at low values of the yield stress, the critical stability behavior is similar to that of Newtonian and shearthinningdrops. Above a threshold value, characterized in terms of the drop size, surface tension and tensile yieldstress magnitude, yieldstress forces are larger than surface forces, and the maximum tensile stress achievable in the drop neck at the point of critical stability is governed by the von Mises criterion.
 Particulate, Multiphase, and Granular Flows

Orientation, distribution, and deposition of elongated, inertial fibers in turbulent channel flow
View Description Hide DescriptionIn this paper, the dispersion of rigid, highly elongated fibers in a turbulent channel flow is investigated. Fibers are treated as prolate ellipsoidal particles which move according to their inertia and to hydrodynamic drag and rotate according to hydrodynamic torques. The orientational behavior of fibers is examined together with their preferential distribution, nearwall accumulation, and wall deposition: all these phenomena are interpreted in connection with turbulencedynamics near the wall. In this work a wide range of fiber classes, characterized by different elongation (quantified by the fiber aspect ratio, ) and different inertia (quantified by a suitably defined fiber response time, ) is considered. A parametric study in the space confirms that, in the vicinity of the wall, fibers tend to align with the mean streamwise flow direction. However, this aligned configuration is unstable, particularly for higher inertia of the fiber, and can be maintained for rather short times before fibers are set into rotation in the vertical plane. A more complex situation is observed in the spanwise and wallnormal flow directions, where fiber inertia and elongation destabilize nearwall alignment in a nontrivial fashion. Fiber orientational behavior and fiber translational behavior are observed to influence the process of fiber accumulation at the wall. Comparing the behavior of fibers with that of spherical particles, it is observed that the aspect ratio has little or no effect on clustering, preferential distribution, and segregation; yet it does affect the wallward drift velocity of the fibers in such a way that longer fibers tend to deposit at higher rates. No preferential orientation and no significant segregation is observed in the channel centerline, confirming that the role of inertia and, in particular, of elongation becomes less important in and beyond the logarithmic layer.

Experimental study of foam jets
View Description Hide DescriptionWe investigate the flow of a foam injected through a cylindrical inlet into a quiescent liquid which is miscible with the foaming solution. Depending on a Reynolds number, combining inlet diameter, liquid viscosity, and flow rate, the jet disperses into a conical plume, takes a stable cylindrical straight shape whose radius swells with flow rate or disintegrates into blobs. We compare this behavior to that reported for other complex fluid jets and present a simple physical model for the straight jet regime.

Inertial particle resuspension in a turbulent, square duct flow
View Description Hide DescriptionParticle resuspension in a turbulent, square duct flow is studied using large eddy simulation combined with Lagrangian particle tracking under conditions of oneway coupling, with the particle equation of motion solved with the Stokes drag, lift, buoyancy, and gravitational force terms. Here, resuspension is taken to mean the movement of particles in close proximity to the duct walls back in to the mainstream of the flow. The flow considered has a bulk , with four particle sizes ranging from 5 to examined. The results demonstrate that turbulencedriven secondary flows within the duct play an important role in the resuspension process. In the vertical direction, resuspension is promoted by the drag force arising from the secondary flows, which is balanced by the gravitational force, with this effect increasing with decreasing particle size. In the horizontal direction, particle resuspension is promoted by the particle’s inertial force, with this effect increasing with increasing particle size. For resuspension in both directions, the drag force dominates small particle resuspension, while for large particles the lift force is also a contributing factor. In the horizontal direction, the effect of the lift force varies with the direction of the secondary flow and becomes more significant when a particle is large or close to the duct wall. The influence of the lift force is also larger in the vertical than in the horizontal direction due to the effects of gravity.

Effect of rolling friction on binary collisions of spheres
View Description Hide DescriptionFoerster et al. [Phys. Fluids6, 1108 (1994)] proposed a binary collision model for spheres and a way to measure the constant impact parameters in the model. These three parameters were necessary to describe the translational momentum exchange during a collision. An extensive data set for a variety of materials was made possible using these measurements. In recent years many studies have shown that rolling friction between particles has strong effects on the bulk behavior of granular materials. In the present paper, an expansion of the previous collision model is studied to include rolling friction. It is shown that rolling friction does not affect the postcollision linear relative velocities for the conditions used in the impact experiments. Furthermore, it is suggested that the same impact experiment could potentially be used to measure the coefficient of rolling friction.

Complex dynamics of three interacting spheres in a rotating drum
View Description Hide DescriptionNumerous studies have demonstrated the potential for particles in fluids to exhibit complicated dynamical behavior. In this work, we study a horizontal rotating drum filled with pure glycerol and three large, heavy spheres. The rotation of the drum causes the spheres to cascade and tumble and thus interact with each other. We find several different behaviors of the spheres depending on the drum rotation rate. Simpler states include the spheres remaining well separated, or states where two or all three of the spheres come together and cascade together. We also see two more complex states, where two or three of the spheres move erratically. The main signature of this erratic motion is that pairs of spheres intermittently approach each other (sometimes colliding) and then separate; the time between collisions is variable even for a fixed rotation rate. We characterize these disordered states and find a complex phase space with a rich set of behaviors. This experiment serves as a simple model system to demonstrate complex behavior in simple fluid dynamicalsystems.

Physicsbased analysis of the hydrodynamic stress in a fluidparticle system
View Description Hide DescriptionThe paper begins by showing how standard results on the average hydrodynamic stress in a uniform fluidparticle system follow from a direct, elementary application of Cauchy’s stress principle. The same principle applied to the angular momentum balance proves the emergence, at the mesoscale, of an antisymmetric component of the volumeaveraged hydrodynamic stress irrespective of the particle Reynolds number. Several arguments are presented to show the physical origin of this result and to explain how the averaging process causes its appearance at the mesoscale in spite of the symmetry of the microscale stress. Examples are given for zero and finite Reynolds number, and for potential flow. For this last case, the antisymmetric stress component vanishes, but the Cauchy principle proves nevertheless useful to derive in a straightforward way known results and to clarify their physical nature.
 Laminar Flows

The leadingedge vortex and quasisteady vortex shedding on an accelerating plate
View Description Hide DescriptionA computational inquiry focuses on leadingedge vortex (LEV) growth and shedding during acceleration of a twodimensional flat plate at a fixed 10°–60° angle of attack and low Reynolds number. The plate accelerates from rest with a velocity given by a power of time ranging from 0 to 5. During the initial LEV growth, subtraction of the added mass lift from the computed lift reveals an LEVinduced lift augmentation evident across all powers and angles of attack. For the range of Reynolds numbers considered, a universal time scale exists for the peak when , with augmentation lasting about four to five chord lengths of translation. This time scale matches well with the halfstroke of a flying insect. An oscillating pattern of leading and trailingedge vortex shedding follows the shedding of the initial LEV. The nondimensional frequency of shedding and lift coefficient minima and maxima closely match their values in the absence of acceleration. These observations support a quasisteady theory of vortex shedding, where dynamics are determined primarily by velocity and not acceleration. Finally, the nondimensional vortex formation time is found to be a function of the Reynolds number, but only weakly when the Reynolds number is high.
 Instability and Transition

Transient growth analysis of flow through a sudden expansion in a circular pipe
View Description Hide DescriptionResults are presented from a numerical study of transient growth experienced by infinitesimal perturbations to flow in an axisymmetric pipe with a sudden 1–2 diametral expansion. First, the downstream reattachment point of the steady laminar flow is accurately determined as a function of Reynolds number and it is established that the flow is linearly stable at least up to . A direct method is used to calculate the optimal transient energy growth for specified time horizon , Re up to 1200, and loworder azimuthal wavenumber . The critical Re for the onset of growth with different is determined. At each Re the maximum growth is found in azimuthal mode and this maximum is found to increase exponentially with Re. The time evolution of optimal perturbations is presented and shown to correspond to sinuous oscillations of the shear layer. Suboptimal perturbations are presented and discussed. Finally, direct numerical simulation in which the inflow is perturbed by Gaussian white noise confirms the presence of the structures determined by the transient growth analysis.