Volume 23, Issue 10, October 2011

We present a search for conformal invariance in vorticity isolines of twodimensional compressible turbulence. The vorticity is measured by tracking the motion of particles that float at the surface of a turbulent tank of water. The threedimensional turbulence in the tank has a Taylor microscale Re _{λ} ≃ 160. The conformal invariance theory being tested here is related to the behavior of equilibrium systems near a critical point. This theory is associated with the work of Löwner, Schramm and others and is usually referred to as SchrammLöwner evolution (SLE). The system was exposed to several tests of SLE. The results of these tests suggest that zerovorticity isolines exhibit noticeable departures from this type of conformal invariance.
 LETTERS


Drag reduction of a hairy disk
View Description Hide DescriptionWe investigate experimentally the hydrodynamics of a hairy disk immersed in a twodimensional flowing soap film. Drag force is measured as a function of hair length, density, and coating area. An optimum combination of these parameters yields a drag reduction of 17%, which confirms previous numerical predictions (15%). Flow visualization indicates the primary mechanism for drag reduction is the bending, adhesion, and reinforcement of hairs trailing the disk, which reduces wake width and traps “dead water.” Thus, the use of hairy coatings can substantially reduce an object’s drag while negligibly increasing its weight.

Confined shocks inside isolated liquid volumes: A new path of erosion?
View Description Hide DescriptionThe unique confinement of shock waves inside isolated liquid volumes amplifies the density of shock–liquid interactions. We investigate this universal principle through an interdisciplinary study of shockinduced cavitation inside liquid volumes, isolated in 2 and 3 dimensions. By combining highspeed visualizations of ideal water drops realized in microgravity with smoothed particle simulations, we evidence strong shockinduced cavitation at the focus of the confined shocks. We extend this analysis to groundobservations of jets and drops using an analytic model and argue that cavitation caused by trapped shocks offers a distinct mechanism of erosion in highspeed impacts.

Negative turbulent production during flow reversal in a stratified oscillating boundary layer on a sloping bottom
View Description Hide DescriptionThreedimensional direct numerical simulations are performed to model an internal tidal beam at nearcritical slope, and the phase dependence of turbulent processes is investigated. Convective instability leads to density overturns that originate in the upper flank of the beam and span the beam width of 6 m during flow reversal from downslope to upslope boundary motion. During this flow reversal event, negative turbulent production is observed signaling energy transfer from velocity fluctuations to the mean flow. In this note, we explain the mechanism underlying negative production.

Lagrangian time correlations of vorticity alignments in isotropic turbulence: Observations and model predictions
View Description Hide DescriptionMotivated by results from recent particle tracking experiments in turbulence Xu et al. [Nature Phys. 7, 709 (2011)], we study the Lagrangian time correlations of vorticity alignments with the three eigenvectors of the deformationrate tensor. We use data from direct numerical simulations (DNS), and explore the predictions of a Lagrangianmodel for the velocity gradient tensor. We find that the initial increase of correlation of vorticity direction with the most extensive eigendirection observed by Xu et al. is reproduced accurately using the Lagrangianmodel, as well as the evolution of correlation with the other two eigendirections. Conversely, time correlations of vorticity direction with the eigenframe of the pressure Hessian tensor show differences with the model.

An apparent symmetry property of the mean velocity gradient in turbulent Poiseuille flows and its implications
View Description Hide DescriptionA novel feature of the mean velocity gradient in turbulent parallel Poiseuille flows has been found using the results available in databases of direct numerical simulations at moderately high frictionReynolds numberR _{τ}, up to 2000. The computed turbulence statistics show that the logarithm of the mean velocity gradient, normalized by its value at the quarterchannel height, is very close to symmetric with respect to this position. At this location, the ratio of the viscous transport term to the viscous stress is a minimum. The range of validity of this property increases with the Reynolds number and is between y/h = 0.1 and y/h = 0.9 for R _{τ} = 2000. This property is a convenient tool in channel flow analysis since the velocity profile in the wall region can be accurately predicted from values much further away. We explore in some detail the properties that follow from this discovery.

 ARTICLES

 Biofluid Mechanics

Optimal feeding is optimal swimming for all Péclet numbers
View Description Hide DescriptionCells swimming in viscous fluids create flow fields which influence the transport of relevant nutrients, and therefore their feeding rate. We propose a modeling approach to the problem of optimal feeding at zero Reynolds number. We consider a simplified spherical swimmer deforming its shape tangentially in a steady fashion (socalled squirmer). Assuming that the nutrient is a passive scalar obeying an advectiondiffusion equation, the optimal use of flow fields by the swimmer for feeding is determined by maximizing the diffusive flux at the organism surface for a fixed rate of energy dissipation in the fluid. The results are obtained through the use of an adjointbased numerical optimization implemented by a Legendre polynomialspectral method. We show that, to within a negligible amount, the optimal feeding mechanism consists in putting all the energy expended by surface distortion into swimming—socalled treadmill motion—which is also the solution maximizing the swimming efficiency. Surprisingly, although the rate of feeding depends strongly on the value of the Péclet number, the optimal feeding stroke is shown to be essentially independent of it, which is confirmed by asymptotic analysis. Within the context of steady actuation, optimal feeding is therefore found to be equivalent to optimal swimming for all Péclet numbers.
 Micro and Nanofluid Mechanics

Alternating current electroosmotic flow of the Jeffreys fluids through a slit microchannel
View Description Hide DescriptionUsing the method of separation of variables, semianalytical solutions are presented for the time periodic EOFflow of linear viscoelastic fluids between microparallel plates. The linear viscoelastic fluids used here are described by the Jeffreys model. The solution involves solving the PoissonBoltzmann (PB) equation, together with the Cauchy momentum equation and the Jeffreys constitutive equation considering the depletion effect produced by the interaction between macromolecules of the Jeffreys fluid and the channel surface. The overall flow is divided into depletion layer and bulk flow outside of depletion layer. The velocity expressions of these two layers were obtained, respectively. By numerical computations, the influence of oscillating Reynolds number, Re, normalized retardation time, λ _{2} ω, and normalized wall zeta potential, , on velocity amplitude is presented. Results show that the magnitude of the velocity amplitude becomes smaller with the increase of retardation time for small and intermediate Re. For large Re, the velocity is almost unchanged near the EDL with retardation time. Moreover, high zeta potential results in larger the magnitude of EOF velocity no matter whether the Re is large or not, especially within the depletion layer.

Electrokinetic flows through a parallelplate channel with slipping stripes on walls
View Description Hide DescriptionLongitudinal and transverse electrohydrodynamicflows through a plane channel, of which the walls are micropatterned with a periodic array of stripes, are considered. One unit of wall pattern consists of a slipping stripe and a nonslipping stripe, each with a distinct zeta potential. The problems are solved by a semianalytical method, where the basic solutions satisfying the electrohydrodynamic equations are expressed by eigenfunction expansions, and the coefficients are determined numerically by point collocation satisfying the mixed stickslip boundary conditions. In the regime of linear response, the Onsager relations for the fluid and current fluxes are deduced as linear functions of the hydrodynamic and electric forcings. The phenomenological coefficients are explicitly expressed as functions of the channel height, the Debye parameter, the slipping area fraction of the wall, the intrinsic slip length, and the zeta potentials. Attention is paid to some particular kinds of patterns, with a view to revisit and to generalize the theoretical limits made in previous studies on electrokineticflow over an inhomogeneously slipping surface. One should be cautious when applying the theoretical limits. We show that when a surface is not 100% uniformly slipping but has a small fraction of area being covered by noslip slots, the electroosmotic enhancement can be appreciably reduced. We also show that when the electric double layer is only moderately thin, slippinguncharged regions on a surface will have finite inhibition effect on the electroosmotic flow.

Production of monodisperse submicron drops of dielectric liquids by chargeinjection from highly conducting liquids
View Description Hide DescriptionWhen ions or electrons are injected into an insulating liquid, they migrate towards its free surface, destabilize it, and form a charged jet. The jet then breaks into uniform dropscharged at an approximately constant fraction of the Rayleigh limit, which relates the drop diameter D_{D} to the flow rate of dielectric liquid Q_{D} and the injected current I as D_{D} ∼ (Q_{D}/I)^{2/3}. We have previously studied the analogous problem where the ions are substituted by nanodrops produced by a Taylor cone of a highly conducting ionic liquid (EMIBF_{4}) immersed in heptane or decane. This yielded hydrocarbon droplets with diameters as small as 4 μm [C. Larriba and J. Fernández de la Mora, Phys. Fluids 22, 1 (2010)], with only incidental barriers to reaching smaller sizes. Here, we overcome these barriers via silica capillaries with smaller bores. These achieve substantially smaller Q_{D} and Q_{D}/I values, resulting in drops well below the ∼12 μm measurable with a phase Doppler anemometer. Extrapolating the D_{D} ∼ (Q_{D}/I)^{2/3} scaling to the smallest Q_{D}/I obtained yields calculated drop diameters of 280 nm. The current is studied as a function of Q_{D} and the ionic liquid flow rate Q_{IL}. The usual law applies here only at small Q_{D} and high Q_{IL}. An unusual dependence appears at low Q_{D}, in contrast with the previously expected approximate independence of I on Q_{D}. This effect results from the acceleration of the dielectric jet at decreasing Q_{D} due to an increase in current given by the removal of the space charge and leading to an overall decrease in Q_{D}/I. An anomalous behavior is observed at low Q_{D} and high Q_{IL} in which the dropcharge appears to exceed the Rayleigh limit. A plausible explanation is proposed based on the injection into the gas of anomalously small secondary drops and/or ions. We also investigate the injection of ionic liquid nanodrops into a quiescent liquid bath. The observed algebraic dependence of the current I ∼ V^{2}ɛ_{o}/L on tip voltage V and tip to collector distance L is interpreted as resulting from two things: a current limited by space charge and an almost constant mobility Z of the nanodrops.

Curvatureinduced secondary microflow motion in steady electroosmotic transport with hydrodynamic slippage effect
View Description Hide DescriptionIn order to exactly understand the curvatureinduced secondary flow motion, the steady electroosmotic flow(EOF) is investigated by applying the full PoissonBoltzmann/NavierStokes equations in a whole domain of the rectangular microchannel. The momentum equation is solved with the continuity equation as the pressurevelocity coupling achieves convergence by employing the advanced algorithm, and generalized Navier’s slip boundary conditions are applied at the hydrophobic curved surface. Two kinds of channels widely used for labonchips are explored with the glass channel and the heterogeneous channel consisting of glass and hydrophobic polydimethylsiloxane, spanning thin to thick electric double layer(EDL) problem. According to a sufficiently low Dean number, an inward skewness in the streamwise velocity profile is observed at the turn. With increasing EDL thickness, the electrokinetic effect gets higher contribution in the velocity profile. Simulation results regarding the variations of streamwise velocity depending on the electrokinetic parameters and hydrodynamic fluid slippage are qualitatively consistent with the predictions documented in the literature. Secondary flows arise due to a mismatch of streamline velocity between fluid in the channel center and nearwall regions. Strengthened secondary flow results from increasing the EDL thickness and the contribution of fluid inertia (i.e., electric field and channel curvature), providing a scaling relation with the same slope. Comparing with and between the cases enables us to identify the optimum selection in applications of curved channel for enhanced EOF and stronger secondary motion relevant to the mixing effect.
 Interfacial Flows

Spin coating of an evaporating polymer solution
View Description Hide DescriptionWe consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the onedimensional development of a thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent mass fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system. The main practical interest is in controlling the appearance and development of a “skin” on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. In practice, a fast and uniform drying of the film is required. The critical parameters controlling this behaviour are found to be the ratio of the diffusion to advection time scales , the ratio of the evaporation to advection time scales δ and the ratio of the diffusivity of the pure polymer and the initial mixture exp(−1/γ). In particular, our analysis shows that for very small evaporation with skin formation can be prevented.

Bubble growth by injection of gas into viscous liquids in cylindrical and conical tubes
View Description Hide DescriptionThe effect of partial confinement on the shape and volume of bubbles generated by injection of a constant flow rate of gas into a very viscousliquid is studied numerically and experimentally. Numerical solutions of the Stokes equations for the liquid and the evolution equation for the surface of a bubble, and experiments with two different liquids, show that cylindrical and conical walls concentric with a gas injection orifice in the horizontal bottom of the liquid may strongly affect the shape and volume of the bubbles, and can be used to control the size of the generated bubbles without changing the flow rate of gas. A wellknown scaling law for the volume of the bubbles generated by injection of a high flow rate of gas in a very viscous unconfined liquid is extended to take into account the presence of cylindrical or conical walls around the injection orifice.

A perturbation method for solving the microregion heat transfer problem
View Description Hide DescriptionA perturbation method is proposed and used to model the twodimensional equations governing evaporation in the microregion of a meniscus on a heated substrate. The novelty of the method lies in the choice of the physical quantities which are used to describe the hydrodynamic and heat transfer phenomena. The chosen quantities are the pressure jump function across the liquidvapor interface and a modifiedshape function. The problem is thus transformed into a set of decoupled initialvalue subproblems that can be solved recursively from lower to higher orders. This approach represents many advantages compared with existing theories. The model is then applied, accounting for the effect of gravity, to describe the microregion shape and heat transfer. The results obtained following this approach are then validated against those given in literature. The comparison demonstrated the validity of the developed model as well as its wider range of applicability. The influence of the interaction between liquid, vapor, and the solid substrates (mainly through the dispersion constant) as well as gravity on heat transfer and meniscus shape is also discussed. In particular, it is found that although gravity affects the shape of the microregion and the apparent contact angle, it has no significant effect on the magnitude and distribution of the evaporation flux.

Inertial modes of a periodically forced buoyant drop attached to a capillary
View Description Hide DescriptionA drop of heptane attached to a capillary tip immersed in water is submitted to small amplitude volume oscillations. Its interface is imaged by means of a highspeed camera and its shape decomposed into spherical harmonics. The forcing frequency is varied over a large range including the frequencies of resonance of the three first modes of inertial shape oscillations. For a small drop, which remains almost spherical at rest, the geometrical constraint imposed by the attachment on the capillary tip causes the oscillation modes to be very different from those of a free drop. Surprisingly, the resonance of large drops is observed at the frequency predicted for a free, pure, and neutrally buoyant drop and mainly involves a single spherical harmonic; only the damping rate is observed to be moderately larger. Since it gives rise to oscillations close to this ideal case, the present experimental method could be used, complementary to quasistatic oscillation of a pendant drop, to investigate dynamic interfacial tension at high frequency of various fluid systems.

Bubble dynamics atop an oscillating substrate: Interplay of compressibility and contact angle hysteresis
View Description Hide DescriptionWe consider a sessile hemispherical bubble sitting on the transversally oscillating bottom of a deep liquid layer and focus on the interplay of the compressibility of the bubble and the contact angle hysteresis. In the presence of contact angle hysteresis, the compressible bubble exhibits two kinds of terminal oscillations: either with the stickslip motion of the contact line or with the completely immobile contact line. For the stickslip oscillations, we detect a double resonance, when the external frequency is close to eigenfrequencies of both the breathing mode and shape oscillations. For the regimes evolving to terminal oscillations with the fixed contact line, we find an unusual transient resembling modulated oscillations.

Features of the rupture of free hanging liquid film under the action of a thermal load
View Description Hide DescriptionWe consider a deformation and a rupture of a thin liquid film which is hanging between two solid flat walls under the action of concentrated thermal load action. A twodimensional model is applied to describe the motion of thin layers of viscous nonisothermal liquid under microgravity conditions. For flow simulation, twodimensional Navier–Stokes equations are used. A computational analysis of the influence of thermal loads on the deformation and the rupture behavior of the thin freely hanging film is carried out. It is shown that the rupture of the thin film with generation of a droplet can occur under the thermal beam of specific width acting on the free surface of the film. The results of the model problem solutions are presented.

The effect of air leakage and heat exchange on the decay of entrapped air pocket slamming oscillations
View Description Hide DescriptionThe phenomenon studied in this work is that of an air pocket entrapped by a free surfacewater wave inside a rectangular tank at a high filling level. The wave, which is a gravity wave, is caused by forced horizontal motion which is constructed in a particular way, in order to entrap an air pocket as it approaches the upper left corner of the tank. As the wave touches the roof, the air is compressed and starts to oscillate. The oscillations resemble, to some extent, the free oscillations of an underdamped massspring system, where the mass is related to the generalized added mass effect of the waterpressure associated with the air pocket oscillations. The stiffness is due to the compressibility of the air. The reason for the damping or, more generally, the decay of the air pocket oscillations is less understood. Air leakage has been proposed as one possible reason for this decay. In this work, the role of air leakage is found not to be the reason for the decay of the air pocket oscillations, because it is not present during major parts of the impact. However, by drilling holes in the roof of the tank, the effect of leakage during the oscillations is proven to cause decay. To explain the physical source of the decay of the oscillations, damping due to heat transfer to and from the air pocket is investigated through an analytical onedimensional steadystate model. The damping due to heat transfer is observed to play an important role. The obtained understanding of the mechanisms causing the decay of the airpocket impact at the upper corner is believed to be relevant to other types of impacts, particularly the entrapment of air pockets on walls by breaking waves.
 Particulate, Multiphase, and Granular Flows

On the dynamics and breakup of a bubble rising in a turbulent flow
View Description Hide DescriptionExperimental investigations of the dynamics of a deformable bubble rising in a uniform turbulent flow are reported. The turbulence is characterized by fast particle imagevelocimetry. Timeresolved evolutions of bubble translation, rotation, and deformation are determined by threedimensional shape recognition from three perpendicular camera views. The bubble dynamics involves three mechanisms fairly decoupled: (1) average shape is imposed by the mean motion of the bubble relative to liquid; (2) wake instability generates almost periodic oscillations of velocity and orientation; (3) turbulence causes random deformations that sometimes lead to breakup. The deformation dynamics is radically different from that observed in the absence of a significant sliding motion due to buoyancy. Large deformations that lead to breakup are not axisymmetric and correspond to elongations in the horizontal direction. The timescale of decay of shape oscillations is of the same order as their natural frequency f _{2}, so that breakup always results from the interaction with a single turbulenteddy. This overdamping causes the statistics of large deformations and the statistics of breakup identical to the statistics of turbulence. The bubble response time however controls the duration of individual breakup events.

From streamline jumping to strange eigenmodes: Bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows
View Description Hide DescriptionThrough a combined computational–experimental study of flow in a slowly rotating quasitwodimensional container, we show several new aspects related to the kinematics of granular mixing. In the Lagrangian frame, for small numbers of revolutions, the mixing pattern is captured by a model termed “streamline jumping.” This minimal model, arising at the limit of a vanishingly thin surface flowing layer, possesses no intrinsic stretching or streamline crossing in the usual sense, yet it can lead to complex particle trajectories. Meanwhile, for intermediate numbers of revolutions, we show the presence of naturally persistent granular mixing patterns, i.e., “strange” eigenmodes of the advectiondiffusion operator governing the mixing process in Eulerian frame. Through a comparative analysis of the structure of eigenmodes and the corresponding Poincaré section and finitetime Lyapunov exponent field of the flow, the relationship between the Eulerian and Lagrangian descriptions of mixing is highlighted. Finally, we show how the mapping method for scalar transport can be modified to include diffusion. This allows us to examine (for the first time in a granular flow) the change in shape, lifespan, and eventual decay of eigenmodes due to diffusive effects at larger numbers of revolutions.
 Laminar Flows

Added mass of a pair of discs
View Description Hide DescriptionIrrotational flow of inviscid incompressible fluid around a pair of coaxial discs of zero thickness, moving along their common axis, is considered. The added mass of an accelerating disc is modified by the presence of a second, coaxial disc. It is reduced if both discs accelerate equally in the same direction along their common axis and increased if their accelerations are equal in magnitude but in opposite directions. Other accelerations can be treated as linear combinations of these two cases. Analytic predictions are obtained when the disc separation 2z _{0} is large compared with the disc radius a and also when the discs are close . Results for intermediate separations are obtained as numerical solutions of a pair of dual integral equations.