Volume 24, Issue 6, June 2012

We use Lagrangian measures, depicted by finitetime Lyapunov exponents, to characterize transport patterns of inertial pollutant particles formed in urban flows. Motivated by actual events we focus on flows in realistic urban geometry. Both deterministic and stochastic particle transport patterns have been identified, as inertial Lagrangian coherent structures. For the deterministic case, the organizing structures are welldefined and we extract them at different hours of a day to reveal the variability of coherent patterns. For the stochastic case, we use a random displacement model for fluid particles and derive the governing equation for inertial particles to examine the change in organizing structures due to “zerothorder” random noise. We find that, (1) the Langevin equation for inertial particles can be reduced to a random displacement model; (2) using random noise based on inhomogeneous turbulence, whose diffusivity is derived from k − ε models, major coherent structures survive to organize local flow patterns and weaker structures are smoothed out due to random motion.
 LETTERS


Sensitivity of 2D turbulent flow past a Dshaped cylinder using global stability
View Description Hide DescriptionWe use adjointbased gradients to analyze the sensitivity of turbulent wake past a Dshaped cylinder at Re = 13000. We assess the ability of a much smaller control cylinder in altering the shedding frequency, as predicted by the eigenfrequency of the most unstable global mode to the mean flow. This allows performing beforehand identification of the sensitive regions, i.e., without computing the actually controlled states. Our results obtained in the frame of 2D, unsteady Reynoldsaveraged Navier–Stokes compare favorably with experimental data reported by Parezanović and Cadot [J. Fluid Mech.693, 115 (2012)] and suggest that the control cylinder acts primarily through a local modification of the mean flow profiles.

On integral length scales in anisotropic turbulence
View Description Hide DescriptionWe found experimentally a dependence of the integral length scales of correlation functions measured in different directions in a turbulent flow on the velocity fluctuation anisotropy in those same directions. We derive invariants for anisotropic turbulence that is locally isotropic, and so a relationship between the velocity and length scales. The results emphasize the importance of defining the Reynolds number, which was about 480, in terms of scalar quantities instead of these scales. We also find that the normalized energy dissipation rate was approximately independent of the anisotropy.

Libration driven elliptical instability
View Description Hide DescriptionThe elliptical instability is a generic instability which takes place in any rotating flow whose streamlines are elliptically deformed. Up to now, it has been widely studied in the case of a constant, nonzero differential rotation between the fluid and the elliptical distortion with applications in turbulence, aeronautics, planetology, and astrophysics. In this letter, we extend previous analytical studies and report the first numerical and experimental evidence that elliptical instability can also be driven by libration, i.e., periodic oscillations of the differential rotation between the fluid and the elliptical distortion, with a zero mean value. Our results suggest that intermittent, spacefilling turbulence due to this instability can exist in the liquid cores and subsurface oceans of socalled synchronized planets and moons.
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 ARTICLES

 Biofluid Mechanics

The effects of flagellar hook compliance on motility of monotrichous bacteria: A modeling study
View Description Hide DescriptionA crucial structure in the motility of flagellated bacteria is the hook, which connects the flagellum filament to the motor in the cell body. Early mathematical models of swimming bacteria assume that the helically shaped flagellum rotates rigidly about its axis, which coincides with the axis of the cell body. Motivated by evidence that the hook is much more flexible than the rest of the flagellum, we develop a new model that allows a naturally straight hook to bend. Hook dynamics are based on the Kirchhoff rod model, which is combined with a boundary element method for solving viscous interactions between the bacterium and the surrounding fluid. For swimming in unbounded fluid, we find good support for using a rigid model since the hook reaches an equilibrium configuration within several revolutions of the motor. However, for effective swimming, there are constraints on the hook stiffness relative to the scale set by the product of the motor torque with the hook length. When the hook is too flexible, its shape cannot be maintained and large deformations and stresses build up. When the hook is too rigid, the flagellum does not align with the cell body axis and the cell “wobbles” with little net forward motion. We also examine the attraction of swimmers to noslip surfaces and find that the tendency to swim steadily close to a surface can be very sensitive to the combination of the hook rigidity and the precise shape of the cell and flagellum.

Depletion layer formation in suspensions of elastic capsules in Newtonian and viscoelastic fluids
View Description Hide DescriptionMotivated by observations of the effects of dragreducing polymer additives on various aspects of blood flow,suspensions of fluidfilled elastic capsules in Newtonian fluids and dilute solutions of high molecular weight (dragreducing) polymers are investigated during plane Couette flow in a slit geometry. A simple model is presented to describe the crossstream distribution of capsules as a balance of shearinduced diffusion and wallinduced migration due to capsule deformability. The model provides a theoretical prediction of the dependence of capsuledepleted layer thickness on the capillary number. A computational approach is then used to directly study the motion of elastic capsules in a Newtonian fluid and in polymer solutions. Capsule membranes are modeled using a neoHookean constitutive model and polymer molecules are modeled as beadspring chains with finitely extensible nonlinearly elastic springs, with parameters chosen to loosely approximate 4000 kDa poly(ethylene oxide). Simulations are performed with a Stokes flow formulation of the immersed boundary method for the capsules, combined with Brownian dynamics for the polymer molecules. Results for an isolated capsule near a wall indicate that the wallinduced migration depends on the capillary number and is strongly reduced by addition of polymer. Numerical simulations of suspensions of capsules in Newtonian fluid illustrate the formation of a capsuledepleted layer near the walls. The thickness of this layer is found to be strongly dependent on the capillary number. The shearinduced diffusivity of the capsules, on the other hand, shows only a weak dependence on capillary number. These results thus indicate that the mechanism of wallinduced migration is the primary source for determining the capillary number dependence of the depletion layer thickness. Both the wallinduced migration and the shearinduced diffusive motion of the capsules are attenuated under the influence of polymer; reduction of migration dominates, however, so the net effect of polymers on the capsule suspension is to reduce the thickness of the capsuledepleted layer. This prediction is in qualitative agreement with experimental observations.

Noninvasive determination of external forces in vortexpaircylinder interactions
View Description Hide DescriptionExpressions for the conserved linear and angular momenta of a dynamically coupled fluid + solid system are derived. Based on the knowledge of the flow velocity field, these expressions allow the determination of the external forces exerted on a body moving in the fluid such as, e.g., swimming fish. The verification of the derived conserved quantities is done numerically. The interaction of a vortex pair with a circular cylinder in various configurations of motions representing a generic test case for a dynamically coupled fluid + solid system is investigated in a weakly compressible NavierStokes setting using a Cartesian cutcell method, i.e., the moving circular cylinder is represented by cut cells on a moving mesh. The objectives of this study are twofold. The first objective is to show the robustness of the derived expressions for the conserved linear and angular momenta with respect to bounded and discrete data sets. The second objective is to study the coupled dynamics of the vortex pair and a neutrally buoyant cylinder free to move in response to the fluid stresses exerted on its surface. A comparison of the vortexbody interaction with the case of a fixed circular cylinder evidences significant differences in the vortex dynamics. When the cylinder is fixed strong secondary vorticity is generated resulting in a repeating process between the primary vortex pair and the cylinder. In the neutrally buoyant cylinder case, a stable structure consisting of the primary vortex pair and secondary vorticity shear layers stays attached to the moving cylinder. In addition to these fundamental cases, the vortexpaircylinder interaction is studied for locomotion at constant speed and locomotion at constant thrust. It is shown that a similar vortex structure like in the neutrally buoyant cylinder case is obtained when the cylinder moves away from the approaching vortex pair at a constant speed smaller than the vortex pair translational velocity. Finally, the idealized symmetric settings are complemented by an asymmetric interaction of a vortex pair and a cylinder. This case is discussed for a fixed and a neutrally buoyant cylinder to show the validity of the derived relations for multidimensional body dynamics.
 Micro and Nanofluid Mechanics

Using sharp transitions in contact angle hysteresis to move, deflect, and sort droplets on a superhydrophobic surface
View Description Hide DescriptionIn order to make an effective dropletbased microfluidic device, one must be able to precisely control a number of key processes including droplet positioning, motion, coalescence, mixing, and sorting. In a typical threedimensional device, these processes are well understood. However, for planar or open microfluidic devices, many of these processes have yet to be demonstrated. In this paper, a series of superhydrophobicsurfaces created by sanding Teflon are used as the microfluidics platform. The superhydrophobicsurfaces used in this study all have advancing contact angles of 150° but have contact angle hysteresis that were varied smoothly from 3° to 30° as the grit size of the sandpaper is changed. Dropmotion was initiated by placing the surface on an inclined plane. To deflect and move droplets along the surface, single and multiple transition lines in receding contact angle were created by spatially varying the surface roughness of the Teflon. The degree of droplet deflection was studied as a function of droplet size, droplet speed, and the angle that the transition line in contact angle hysteresis made with the principle direction of dropletmotion.Droplet deflections across a single transition as large as 140% the droplet diameter were observed. The droplet deflection was found to increase with increasing difference in contact angle hysteresis across the transition and increasing transition angles up to about 40°. The largest deflections were observed over a very narrow range of droplet velocities corresponding to a range in Weber numbers between 0.1 and 0.2. This narrow range in Weber number suggests that transitions in receding contact angle can be used to sort drops based on velocity, size or wetting properties with a strong degree of selectivity. The direction of deflection was observed to change depending on whether the drops transitioned from a region of low to high or high to low contact angle hysteresis. In a transition from low to high hysteresis, a large portion of the drop's kinetic energy is converted into interfacial energy as the receding contact line of the drop is deformed. Alternatively, a transition from high to low hysteresis results in some of the drop's interfacial energy converted into kinetic energy as the deformation of the droplet is reduced. The result is either a reduction or increase in the droplet's velocity normal to the line of transition depending on the sign of the transition in contact angle hysteresis. Finally, single and multiple stripes of different contact angle hysteresis are also shown to be effective at deflecting droplets.

Investigation on heat transfer between two coaxial cylinders for measurement of thermal accommodation coefficient
View Description Hide DescriptionThe heat flux between two coaxial cylinders was measured in the range from the free molecular to the early transitional flow regimes for extraction of the thermal accommodation coefficient using an approximate relation on the pressure dependence of the heat flux. The experimental coaxial cylinders' geometry has been traditionally implemented for the measurement of the thermal accommodation coefficient using the lowpressure method; however, the actual experimental setup was characterized by large temperature difference and large cylinders' radius ratio. Compared to the original lowpressure method, much higher pressure range was applied. In order to verify assumptions in the accommodation coefficient extraction, the heat flux under measurement conditions was simulated numerically by the nonlinear Smodel kinetic equation. Very good agreement was found between the measured and the simulated heat flux. The proposed procedure of the thermal accommodation coefficient extraction was discussed in detail and verified. The temperature dependence of the thermal accommodation coefficient was also found.
 Interfacial Flows

Measurements of gas/oil free surface deformation caused by parallel gas flow
View Description Hide DescriptionFlowinduced dynamic freesurface deformations are experimentally studied in a confined liquid volume of 5cSt silicone oil (Prandtl number Pr = 68). The geometry of the problem is a liquid column concentrically surrounded by an annular gas channel. A gas stream entering the duct from the top or bottom entrains the motionless liquid. The dynamic deformation of the gas–liquid interface is caused by a steady axisymmetric sheardriven flow. The experiments are performed in normal gravity conditions and the static deformation of a liquid bridgeinterface is unavoidable. The magnitude and shape of the dynamic surfacedeformation are analyzed using optical measurements with a comprehensive treatment of the images. The deviation of the free surface shape from the corresponding equilibrium profile is determined with an uncertainty of about 0.1 μm. The order of magnitude of the interfacedeformation is proportional to the capillary number, which is defined as the ratio of the viscous force per unit area to the capillary pressure. The study is performed for a large range of volumes and aspect ratios as well as for different gas velocities. As a general trend, the dynamic deformation grows with the gas velocity, which plays the role of a driving force, but a linear dependence is not observed for all volume ratios, despite the small Reynolds numbers, 280 < Re _{ g } < 560. The dynamic deformation displays a strong dependence on the liquid volume ratio and the direction of the gas stream parallel to the interface. When the gas flow is directed against gravity, the largest interfacedeformations are observed at the smallest volumes among the analyzed ones. In contrast, when the gas stream is aligned with gravity, then the deformations decrease with a decrease in the volume ratio, at a certain value attaining zero (vanishing) and then changing sign.

Drop trapping in axisymmetric constrictions with arbitrary contact angle
View Description Hide DescriptionThe differential YoungLaplace equations are solved numerically with an iterative solution using the method of steepest descent to determine the shape of a drop trapped under gravity in an axisymmetric ring constriction. Prior work for nonwetting drops with a contact angle of π is extended to arbitrary values of the contact angle at the threephase contact lines. The critical Bond number, representing a dimensionless ratio of gravitational and interfacial forces, and separating static trapping at lower Bond numbers from dynamic squeezing at higher Bond numbers, decreases with decreasing contact angle, indicating that drop squeezing occurs more easily at smaller contact angle. Indeed, a critical contact angle, which depends only on the droptohole and ringcrosssectiontohole size ratios, is found, below which all drops squeeze through the hole.

A model for the human tear film with heating from within the eye
View Description Hide DescriptionA model for tear filmdynamics and cooling during the interblink period is formulated that includes heat transfer from the interior of the eye. Lubrication theory is used to derive an equation for the thickness of the film; the nonlinear partial differential equation for the thickness is solved subject to either a fixed temperature at the substrate or with heat diffusion from within two different model rectangular domains. The model domains are simplified geometries that represent the anterior eye and that may include the cornea and some aqueous humor; one model domain is asymptotically thin (thin substrate) and the other has finite thickness (thick substrate). The thick substrate case captures temperature decreases that are observed in vivo, while the thin substrate and fixed temperature models do not. Parameters to reproduce observed temperature decreases are found.

The effect of surfactant convection and diffusion on the evolution of an axisymmetric pendant droplet
View Description Hide DescriptionIn this work, we consider the evolution of a single axisymmetric droplet as it grows from an initially uniform thin liquid film on the underside of a solid, horizontal substrate. We consider the effects of an insoluble surfactant on the free surface, the concentration of which changes due to convection and diffusion. Employing the lubrication approximations, we derive a linear solution valid in the initial stages of drop growth, and a full nonlinear solution, which we solve numerically using finite differences. Both a linear analysis and a numerical solution show that for sufficiently thin films, corresponding to low Bond numbers, diffusion of surfactant can effectively negate the effects of surfactantconvection, and the drop evolves as if there is no surfactant on the free surface. For sufficiently thick films, corresponding to high Bond numbers, gravitational forces are stronger than surface tension gradient forces, and again the droplet evolves as if there is no surfactant present. For intermediate Bond numbers, the convection of surfactant can significantly slow the growth rate and must be included in the analysis to accurately model the time evolution of a typical pendant droplet. The interactions between coating height, flux, and surface velocity are used to explain the physics of this behavior.

Addition of dispersive terms to the method of averaged Lagrangian
View Description Hide DescriptionWhitham's method of averaged Lagrangian is applied to the problem of Stokes waves on the surface of a layer of ideal fluid. We derive a Lagrangian which contains additional terms with and aa _{ xx } besides nonlinear terms of Whitham with a ^{2} and a ^{4}, a being the wave amplitude. These terms with derivatives appear after taking into account (1) additional terms in the Stokes expansions with the same approximation as the one was taken into consideration by Whitham; (2) supplementary slow (in comparison to the rapid phase oscillations) dependence of the amplitudes of harmonics on coordinates and time. We demonstrate the need for the account of such terms in the Lagrangian for obtaining the correct coefficients of dispersive terms of evolution equations from the corresponding variational equations.

Effect of neighboring perturbations on drop coalescence at an interface
View Description Hide DescriptionCoalescence at a quiescent silicone oil/water glycerine interface was investigated for water/glycerine drops with Bond number ∼7 and Ohnesorge number = 0.01 using highspeed imaging and timeresolved tomographic particle image velocimetry. In addition to a single drop case, three perturbation cases were considered corresponding with a second drop, a solid particle wetted in oil, and a solid particle wetted in water/glycerine placed adjacent to the coalescingdrop. Each perturbing object caused an initial tilting of the drop, influencing its rupture location and eventual collapse behavior. Once tilted, drops typically ruptured near their lowest vertical position which was located either toward or away from the perturbing object depending on the case. The initial retraction speed of the ruptured film was higher for drops initially tilted at significant angles, and the local variations in retraction speed correlated well with the expected variations in local film thickness. The drop fluid always collapsed away from the drop axis in the direction of the rupture location in all unperturbed or perturbed cases. In the case of a drop next to a particle wetted in water/glycerine, the collapsing fluid travelled away from the particle, and the downward propagating vortex ring which developed was similar to that resulting from an unperturbed drop rupture. By contrast, the drop fluid collapsed toward either a second drop or a particle wetted in oil. The resulting vortex rings were more asymmetric, and viscous interaction between the particle and collapsing fluid hindered the downward motion of the associated ring.

Characteristics of tsunami motion and energy budget during runup and rundown processes over a plane beach
View Description Hide DescriptionMotions of tsunamiwaves during runup and rundown processes on the uniform sloping beach are studied numerically by the fully nonlinear and highly dispersive Boussinesq equations. We first study the leadingdepression Nwave defined by solitary theory. The shoreline movement and its moving speed are analyzed during the wave approaching the coastal area. Details on the flow field and the energy transformation are obtained in terms of the reconstruction of the full velocity field by Boussinesq equations. In addition, solitary wave is studied as a comparison. The different energy budget explains the phenomenon that the Nwave leads much larger runup than the solitary wave in some specific situations. The investigation is then extended to the patterns of the energy transformation of Nshape waves, humplike waves, and sinusoidal long waves. These waves are of the same order in scale with the recent giant tsunamis. The results show that the potential energy of the Nshape tsunamiwaves nearly reaches the maximum while that of the humplike tsunamiwaves does not reach the maximum at the maximum runup, meanwhile the kinetic energy of both waves does not go to zero. For the sinusoidal wave train, however, its potential energy reaches the maximum and its kinetic energy goes to zero exactly at the maximum runup. To understand the responses caused by the variation of the waveforms, effects of the nonlinearity and the dispersion on the energy budget are studied in the geophysical tsunamis order. Moreover, the regularities of the energy budget of some general cases including the leadingdepression Nwaves (LDNs), the leadingelevation Nwave, and generalized LDNs are investigated extensively. The mechanism of the energy budget of these waves is quite different as just considering the potential energy at the maximum runup.

Experimental and numerical investigation of binary coalescence: Liquid bridge building and internal flow fields
View Description Hide DescriptionIn this work, we present the results of experimental and numerical investigation of the liquid bridge building and internal flows during a binary dropletdroplet coalescence event. A system consisting of two rising toluene droplets with different droplet radii that coalesce in surrounding water is considered. The growth of the liquid bridge between the two droplets during the coalescence event is found to be in the inertial regime, whereas the bridge radius grows in proportion to the square root of time. Further, the internal mixing of the two droplets is investigated. It is observed that when two droplets with the same radius coalesce, there is nearly no internal mixing within the first 40 ms after the coalescence start. In contrast, when two droplets with different radii coalesce, a liquid jet directed from the smaller to the larger droplet appears during the coalescence event. A good agreement between the experimental and numerical data is established.

Decomposition driven interface evolution for layers of binary mixtures. III. Twodimensional steady films with flat and modulated surfaces
View Description Hide DescriptionWe study twodimensional steady concentration and film thickness profiles for isothermal free surfacefilms of a binary liquid mixture on a solid substrate employing modelH that couples the diffusive transport of the components of the mixture (convective CahnHilliard equation) and the transport of momentum (NavierStokesKorteweg equations). The analysis is based on minimising the underlying free energy equivalent to solving the static limit of modelH. Additionally, the linear stability (in time) of relevant layered films is analyzed. This allows for a comparison of the position of certain branching points in the bifurcation diagrams of steady solutions with the value predicted as onset of a linear instability. Results are presented for the cases of (i) a flat film without energetic bias at the free surface, (ii) a flat film with energetic bias, (iii) a heightmodulated film without energetic bias, and (iv) a heightmodulated film with energetic bias. In all cases we discuss symmetries of the various steady solutions allowing us to order them and to infer properties of solution branches and relations between them.
 Viscous and NonNewtonian Flows

Effect of oxidation on the mechanical properties of liquid gallium and eutectic galliumindium
View Description Hide DescriptionLiquid metals exhibit remarkable mechanical properties, in particular large surface tension and low viscosity. However, these properties are greatly affected by oxidation when exposed to air. We measure the viscosity,surface tension, and contact angle of gallium and a eutectic galliumindium alloy while controlling such oxidation by surrounding the metals with an acid bath of variable concentration. Rheometry measurements reveal a yield stress directly attributable to an oxide skin that obscures the intrinsic behavior of the liquid metals. We demonstrate how the intrinsic viscosity can be obtained with precision through a scaling technique that collapses low and highReynolds number data. Measuring surface tension with a pendant drop method, we show that the oxide skin generates a surface stress that mimics surface tension and develop a simple model to relate this to the yield stress obtained from rheometry. We find that yield stress,surface tension, and contact angle all transition from solidlike to liquid behavior at the same critical acid concentration, thereby quantitatively confirming that the wettability of these liquid metals is due to the oxide skin.
 Particulate, Multiphase, and Granular Flows

Resuspension onset and crater erosion by a vortex ring interacting with a particle layer
View Description Hide DescriptionThis paper presents results from an experimental investigation of the interaction of a vortex ring with a particle layer. The flowdynamics during the onset of particle resuspension are analysed using particle image velocimetry, while a light attenuation method provides accurate measurements of the final eroded crater shape. This work is a continuation of the research described in R. J. Munro, N. Bethke, and S. B. Dalziel, “Sediment resuspension and erosion by vortex rings,” Phys. Fluids21, 046601 (2009)10.1063/1.3083318, which focussed on the general resuspension onset dynamics and initial crater formation. Here, we analyse the velocity induced by the vortex ring on the particle layer surface during the resuspension of particles for different particle sizes, and the shape and size of the final craters that are formed by the impact of the vortex ring. We find that the boundary condition is characterised by a quasislip velocity at the particle layer surface, independent of the particle size. The particle diameter, and thus bed permeability, is found to have a significant effect on the final crater characteristics.

The geometry of inertial particle mixing in urban flows, from deterministic and random displacement models
View Description Hide DescriptionWe use Lagrangian measures, depicted by finitetime Lyapunov exponents, to characterize transport patterns of inertial pollutant particles formed in urban flows. Motivated by actual events we focus on flows in realistic urban geometry. Both deterministic and stochastic particle transport patterns have been identified, as inertial Lagrangian coherent structures. For the deterministic case, the organizing structures are welldefined and we extract them at different hours of a day to reveal the variability of coherent patterns. For the stochastic case, we use a random displacement model for fluid particles and derive the governing equation for inertial particles to examine the change in organizing structures due to “zerothorder” random noise. We find that, (1) the Langevin equation for inertial particles can be reduced to a random displacement model; (2) using random noise based on inhomogeneous turbulence, whose diffusivity is derived from k − ε models, major coherent structures survive to organize local flow patterns and weaker structures are smoothed out due to random motion.