Volume 22, Issue 11, November 2010

Junction flows may suffer from secondary flows such as horseshoe vortices and corner separations that can dramatically impair the performances of aircrafts. The present article brings into focus the unsteady aspects of the flow at the intersection of a wing and a flat plate. The simplified junction flow test case is designed according to a literature review to favor the onset of a corner separation. The salient statistical and fluctuating properties of the flow are scrutinized using large eddy simulation and wind tunnel tests, which are carried out at a Reynolds number based on the wing chord and the free stream velocity of . As the incoming boundary layer at ( being the boundary layer momentum thickness onehalf chord upstream the junction) experiences the adverse pressure gradient created by the wing, a three dimensional separation occurs at the nose of the junction leading to the formation of a horseshoe vortex. The low frequency, large scale bimodal behavior of the horseshoe vortex at the nose of the junction is characterized by multiple frequencies within (where is the boundary layer thickness onehalf chord upstream the wing). Downstream of the bimodal region, the meandering of the core of the horseshoe vortex legs in the crossflow planes is scrutinized. It is found that the horseshoe vortex oscillates around a mean location over an area covering almost 10% of the wing chord in the tranverse plane at the trailing edge at normalized frequencies around . This socalled meandering is found to be part of a global dynamics of the horseshoe vortex initiated by the bimodal behavior. Within the corner, no separation is observed and it is shown that a high level of anisotropy (according to Lumley’s formalism) is reached at the intersection of the wing and the flat plate, which makes the investigated test case challenging for numerical methods. The conditions of apparition of a corner separation are eventually discussed and we assume that the vicinity of the horseshoe vortex suction side leg might prevent the corner separation. It is also anticipated that higher Reynolds number junction flows are more likely to suffer from such separations.
 ARTICLES

 Biofluid Mechanics

Efficiency optimization and symmetrybreaking in a model of ciliary locomotion
View Description Hide DescriptionA variety of swimming microorganisms, called ciliates, exploit the bending of a large number of small and densely packed organelles, termed cilia, in order to propel themselves in a viscous fluid. We consider a spherical envelope model for such ciliary locomotion where the dynamics of the individual cilia are replaced by that of a continuous overlaying surface allowed to deform tangentially to itself. Employing a variational approach, we determine numerically the timeperiodic deformation of such surface which leads to lowReynolds locomotion with minimum rate of energy dissipation (maximum efficiency). Employing both Lagrangian and Eulerian points of views, we show that in the optimal swimming stroke, individual cilia display weak asymmetric beating, but that a significant symmetrybreaking occurs at the organism level, with the whole surface deforming in a wavelike fashion reminiscent of metachronal waves of biological cilia. This wavemotion is analyzed using a formal modal decomposition, is found to occur in the same direction as the swimming direction, and is interpreted as due to a spatial distribution of phase differences in the kinematics of individual cilia. Using additional constrained optimizations, as well as a constructed analytical ansatz, we derive a complete optimization diagram where all swimming efficiencies, swimming speeds, and amplitudes of surface deformation can be reached, with the mathematically optimal swimmer, of efficiency onehalf, being a singular limit. Biologically, our work suggests therefore that metachronal waves may allow cilia to propel cells forward while reducing the energy dissipated in the surrounding fluid.

Effect of wing inertia on hovering performance of flexible flapping wings
View Description Hide DescriptionInsect wings in flight typically deform under the combined aerodynamic force and wing inertia; whichever is dominant depends on the mass ratio defined as , where is the surface density of the wing, is the density of the air, and is the characteristic length of the wing. To study the differences that the wing inertia makes in the aerodynamic performance of the deformable wing, a twodimensional numerical study is applied to simulate the flowstructure interaction of a flapping wing during hovering flight. The wing section is modeled as an elastic plate, which may experience nonlinear deformations while flapping. The effect of the wing inertia on lift production, drag resistance, and power consumption is studied for a range of wing rigidity. It is found that both inertiainduced deformation and flowinduced deformation can enhance lift of the wing. However, the flowinduced deformation, which corresponds to the lowmass wing, produces less drag and leads to higher aerodynamic power efficiency. In addition, the wing deformation has a significant effect on the unsteady vortices around the wing. The implication of the findings on insect flight is discussed.

Hydrodynamic performance of a fishlike undulating foil in the wake of a cylinder
View Description Hide DescriptionThe hydrodynamic performances of a fishlike undulating foil in the wake of a Dsection cylinder are numerically investigated by using a modified immersed boundary method. The results regarding the effects of various controlling parameters, including the distance between the foil and the Dcylinder, the frequency and the phase angle of foil’s undulation, and the phase angle of heaving motion on the thrust and the input power, are reported. It is observed that the foil without undulation in the vortex street can gain a thrust, as a result of the fact that the passing vortices produce reverse flows with respect to the mainstream in vicinity of the foil surface. When an undulating foil is placed at different distances behind the Dsection cylinder, different wake structures form behind the cylinder. The wake area can be divided into three domains: suction domain, thrust enhancing domain, and weak influence domain. The undulation of the foil can inhibit the rollup instability of the shear layers and vortex shedding from the cylinder and consequently significantly enlarge the suction domain, compared to the foilfree case or the stationary foil case. The thrust on the foil first increases and then decreases, as the distance between the foil and the cylinder increases. The undulation plays a negative role in the foil propulsion when the foil is located near the cylinder (largely in the suction domain) and a positive role when the distance between the foil and the cylinder is beyond a critical value. The mean thrusts do not vary significantly with the undulation phase angle when the heaving motion is not considered and the undulation amplitude studied is relatively small, instead, they are significantly affected by the phase angle of the heaving motion. The foil bypassing the vortices undergoes both minimum thrust and input power, whereas the one passing through vortices experiences a larger thrust. The phase angle difference between the heave and the undulation is important.
 Micro and Nanofluid Mechanics

Rarefied gas flows through a curved channel: Application of a diffusiontype equation
View Description Hide DescriptionRarefied gasflows through a curved twodimensional channel, caused by a pressure or a temperature gradient, are investigated numerically by using a macroscopic equation of convectiondiffusion type. The equation, which was derived systematically from the Bhatnagar–Gross–Krook model of the Boltzmann equation and diffusereflection boundary condition in a previous paper [K. Aoki et al., “A diffusion model for rarefied flows in curved channels,” Multiscale Model. Simul.6, 1281 (2008)], is valid irrespective of the degree of gas rarefaction when the channel width is much shorter than the scale of variations of physical quantities and curvature along the channel. Attention is also paid to a variant of the Knudsen compressor that can produce a pressure raise by the effect of the change of channel curvature and periodic temperature distributions without any help of moving parts. In the process of analysis, the macroscopic equation is (partially) extended to the case of the ellipsoidalstatistical model of the Boltzmann equation.

Multiscale molecular simulations of argon vapor condensation onto a cooled substrate with bulk flow
View Description Hide DescriptionA hybrid simulation method is employed to study the condensation of saturated argon vapor flowing tangentially across a stationary cooled substrate, at nanoscale resolution. The method combines a direct simulation Monte Carlo treatment of the bulk vapor phase with a nonequilibrium molecular dynamics treatment of the condensed liquid and interphase regions; it provides an efficient simulation procedure for a heterogeneous system with a large ratio of vapor to liquid length scales. Starting from a bare, crystalline solid wall, the condensation process evolves from a transient unsteady state to a quasisteady state, where interfacial properties and heat and mass transfer parameters are analyzed. The Knudsen layer structure from the hybrid simulation is compared with kinetic theory predictions from a modified moment method analysis and from pure DSMC simulation. The effects of condensation strength and a tangential flow velocity that is on the order of the condensation velocity are examined. A comparison is made between the nonequilibrium results and equilibrium results for the interphase transition between liquid and vapor. The results reveal the structure of the interphase for such phenomena as inverted temperature, drift flux, and heat transfer. Heat transfer phenomena at the substrate surface are also described.

Fieldamplified sample stacking and focusing in nanofluidic channels
View Description Hide DescriptionNanofluidic technology is gaining popularity for bioanalytical applications due to advances in both nanofabrication and design. One major obstacle in the widespread adoption of such technology for bioanalytical systems is efficient detection of samples due to the inherently low analyte concentrations present in such systems. This problem is exacerbated by the push for electronic detection, which requires an even higher sensorlocal sample concentration than optical detection. This paper explores one of the most common preconcentration techniques, fieldamplified sample stacking, in nanofluidic systems in efforts to alleviate this obstacle. Holding the ratio of background electrolyte concentrations constant, the parameters of channel height, strength of electric field, and concentration are varied. Although in micron scale systems, these parameters have little or no effect on the final concentration enhancement achieved, nanofluidicexperiments show strong dependencies on each of these parameters. Further, nanofluidic systems demonstrate an increased concentration enhancement over what is predicted and realized in microscale counterparts. Accordingly, a depthaveraged theoretical model is developed that explains these observations and furthermore predicts a novel focusing mechanism that can explain the increased concentration enhancement achieved. Specifically, when the electric double layer is sufficient in size relative to the channel height, negatively charged analyte ions are repelled from negatively charged walls, and thus prefer to inhabit the centerline of the channels. The resulting induced pressure gradients formed due to the high and low electrical conductivity fluids in the channel force the ions to move at a slower velocity in the lowconductivity region, and a faster velocity in the highconductivity region, leading to focusing. A simple singlechannel model is capable of predicting key experimental observations, while a model that incorporates the details of the fluid inlet and outlet ports allows for more detailed comparisons between model and experiment.

An extended macroscopic transport model for rarefied gas flows in long capillaries with circular cross section
View Description Hide DescriptionPressuredriven and thermally driven rarefied gasflows in long capillaries with circular cross sections are investigated. For both Poiseuille and thermal transpiration flows, a unified theoretical approach is presented based on the linear form of regularized 13moment (R13) equations. The captured nonequilibrium effects in the processes are compared to available kinetic solutions, and the shortcomings of classical hydrodynamics, i.e., the Navier–Stokes–Fourier equations, are highlighted. Breakdown of Onsager’s symmetry is proposed as a criterion to determine the range of applicability of extended macroscopic models. Based on Onsager’s reciprocity relation it is shown that linearized R13 equations provide agreement with kinetic data for moderate Knudsen numbers, . Twoway flow pattern and thermomolecular pressure difference in simultaneous pressuredriven and temperaturedriven flows are analyzed. Moreover, secondorder boundary conditions for velocity slip and temperature jump are derived for the Navier–Stokes–Fourier system. The proposed boundary conditions effectively improve classical hydrodynamics in the transition flow regime.
 Interfacial Flows

Nonlocal description of evaporating drops
View Description Hide DescriptionWe present a theoretical study of the evolution of a drop of pure liquid on a solid substrate, which it wets completely. In a situation where evaporation is significant, the drop does not spread, but instead the drop radius goes to zero in finite time. Our description couples the viscous flow problem to a selfconsistent thermodynamic description of evaporation from the drop and its precursor film. The evaporation rate is limited by the diffusion of vapor into the surrounding atmosphere. For flat drops, we compute the evaporation rate as a nonlocal integral operator of the drop shape. Together with a lubrication description of the flow, this permits an efficient numerical description of the final stages of the evaporation problem. We find that the drop radius goes to zero like , where has value close to 1/2, in agreement with experiment.

Shockwave solutions in twolayer channel flow. I. Onedimensional flows
View Description Hide DescriptionWe study the dynamics of an interface separating two immiscible layers in an inclined channel. Lubrication theory is used to derive an evolution equation for the interface position that models the twodimensional flow in both co and countercurrent configurations. This equation is parameterized by viscosity and density ratios, and a total dimensionless flow rate; the system is further characterized by the height of the interface at the channel inlet and outlet, which are treated as additional parameters. In the present work, which corresponds to part I of a twopart paper, we focus on onedimensional flows. We use an entropyflux analysis to delineate the existence of various types of shocklike solutions, which include compressive Lax shocks, pairs of Lax and undercompressive shocks, and rarefaction waves.Flows characterized by unstably stratified layers are accompanied by the formation of propagating, largeamplitude interfacial waves, which are not shocklike in nature. The results of our transient numerical simulations accord with our analytical predictions and elucidate the mechanisms underlying spatiotemporal development of the various types of waves; the stability of these waves to spanwise perturbations is investigated in part II of this work.

Threedimensional linear instability in pressuredriven twolayer channel flow of a Newtonian and a Herschel–Bulkley fluid
View Description Hide DescriptionThe threedimensional linear stability characteristics of pressuredriven twolayer channel flow are considered, wherein a Newtonian fluid layer overlies a layer of a Herschel–Bulkley fluid. We focus on the parameter ranges for which Squire’s theorem for the twolayer Newtonian problem does not exist. The modified Orr–Sommerfeld and Squire equations in each layer are derived and solved using an efficient spectral collocation method. Our results demonstrate the presence of threedimensional instabilities for situations where the square root of the viscosity ratio is larger than the thickness ratio of the two layers; these “interfacial” mode instabilities are also present when density stratification is destabilizing. These results may be of particular interest to researchers studying the transient growth and nonlinear stability of twofluid nonNewtonian flows. We also show that the “shear” modes, which are present at sufficiently large Reynolds numbers, are most unstable to twodimensional disturbances.

Influence of surfactant on drop deformation in an electric field
View Description Hide DescriptionThe deformation of a surfactantcovered, viscousdrop suspended in a viscous fluid under the influence of an electric field is investigated using numerical simulations. The full Navier–Stokes equations are solved in both fluid phases, and the motion of the interface and the interfacial discontinuities are handled using the levelset method. The leakydielectric model is used to take into account the effect of an electric field. The surfactant is assumed to be insoluble, and an evolution equation for the motion of surfactant is solved along the dropsurface. The surfactant concentration and the interfacial tension are coupled through a nonlinear equation of state. The numerical results show that the effect of surfactant strongly depends on the relative permittivity and conductivity between the fluids. The presence of surfactant can both increase and reduce the deformation, depending on the shape of the deformation and the direction of the electrically induced circulation.

Spatial instability of coflowing liquidgas jets in capillary flow focusing
View Description Hide DescriptionConsidering both the first nonaxisymmetric and the axisymmetric disturbances, a viscousspatialinstability analysis of coflowing liquidgasjets in capillary flow focusing is carried out. A detailed parametric study is performed to explore characteristics of the spatially amplified branch in a convectively unstable regime. The numerical results show that the Weber number and the velocity at the interface have significant influences on the transition between axisymmetric and nonaxisymmetric instabilities, whereas the other parameters such as the Reynolds number, the slope of the liquid velocity profile at the interface, the density ratio, and the viscosity ratio hardly change the transition. Nonaxisymmetric disturbances grow faster than axisymmetric ones for relatively high Weber numbers. Particularly, the comparison of the theoretical prediction with the experimental results reported by Si et al. [J. Fluid Mech.629, 1 (2009)] indicates that the spatialinstability analysis is in better agreement with experiments than the temporal instability analysis for moderate and high Weber numbers.

Nominally twodimensional waves in inclined film flow in channels of finite width
View Description Hide DescriptionTraveling waves in inclined filmflow in channels of finite width are never truly twodimensional (2D) because of a longrange effect of sidewalls. The present study documents the characteristics of the first waves that are observed beyond the primary instability (termed nominally 2D) by taking measurements in a 3000 mm long inclined facility with adjustable width up to 450 mm using a fluorescence imaging technique. It is observed that nominally 2D waves are very persistent structures with their crests attaining a parabolic shape, which is symmetric with respect to the channel centerplane irrespective of the 3D content of the inlet forcing. The apex curvature of the parabola varies inversely with channel width and Reynolds number. The wave height is maximum at the centerplane and decreases to zero at the sidewalls, irrespective of the wetting properties of the system. The linear phase velocity of nominally 2D waves is always lower than predicted by the theory for small amplitude, 2D waves, and significantly in narrow channels and/or small inclinations. The above characteristics are shown to explain discrepancies between theory and observations, in particular the recently reported deviation of the onset of the primary instability from the classical prediction [M. Vlachogiannis et al., Phys. Fluids22, 012106 (2010)].

Electrodispersion of a liquid of finite electrical conductivity in an immiscible dielectric liquid
View Description Hide DescriptionOrderofmagnitude estimates and numerical computations are used to analyze an electrospray operating in the conejet mode in a bath of an immiscible dielectric liquid. In agreement with experimental results in the literature, the analysis predicts that the electric current carried by the jet increases as the square root of the flow rate of dispersed liquid in a wide range of conditions of the flow. The characteristics of the current transfer region determining the electric current are estimated taking into account the viscous drag of the dielectric liquid that surrounds the jet. The electric current is predicted to depart from the square root law for small flow rates, when charge relaxation effects become important in the current transfer region, and also when the flow rate increases to values of the order of , where and are the permittivity and viscosity of the dielectric liquid, is the electrical conductivity of the dispersed liquid, is the radius of the capillary needle through which this liquid is injected, and is the interfacial tension of the liquid pair. When the flow rate becomes of order , the meniscus at the tip of the capillary ceases to resemble a Taylor cone, the current transfer region ceases to be short compared to the size of the meniscus, the electric current levels to a constant value, and the stationary jet cannot extend very far downstream of the meniscus.

Nonsimilar solutions of the viscous shallow water equations governing weak hydraulic jumps
View Description Hide DescriptionThe steady viscous shallow water equations are often used for the study of hydraulic jumps. We cast these as a single parametric ordinary differential equation with global continuity as a constraint. The solution provides both the local velocity profile and the downstream evolution of the film height. Moreover this is an exact approach, in contrast with existing approaches which encounter a closure problem and need modeling. There is only one solution which is supercritical initially. This shows a jumplike behavior at a Froude number close to unity, in consonance with predictions of inviscid theory. At low Froude number, it is shown that two solutions are possible, one with a separated profile and one without. Flow downstream of a real hydraulic jump must switch to the second solution, calling into question the validity of the shallow water approach in resolving the region of the switch. A series solution of the velocity profile shows that the first correction to a streamwisevarying parabolic profile is a quartic term. Circular and planar solutions are qualitatively similar.

Electrodiffusiophoresis: Particle motion in electrolytes under direct current
View Description Hide DescriptionColloidal particles in electrolytes move in response to electric fields(electrophoresis) and salt concentration gradients (diffusiophoresis), and related flows also occur at fixed surfaces (electroosmosis and diffusioosmosis, respectively). In isolation, these electrokinetic phenomena are well understood, e.g., electrophoresis without farfield concentration gradients and diffusiophoresis without applied electric fields. When the electrolyte passes direct current, however, concentration gradients accompany the bulk electric field (concentration polarization) and the resulting particle motion, called “electrodiffusiophoresis,” involves a nonlinear combination of electrophoresis and diffusiophoresis, depending on ion transference numbers and particle properties. In this work, we analyze the electrodiffusiophoresis of spherical particles in the limit of thin double layers, neglecting surface conduction and convection , considering both nonpolarizable (fixed charge) and ideally polarizable (inducedcharge) surfaces. Via asymptotic approximations and numerical solutions, we develop a physical picture to guide potential applications in electrochemicalcells, such as analyte focusing, electrophoretic deposition, and microfluidic mixing near membranes or electrodes. By controlling the mean salt concentration, particle size, current, and concentration gradient, significant motion of particles (or fluid) is possible toward either electrode and toward high or low concentration.

Electrohydrodynamics of drops in strong uniform dc electric fields
View Description Hide DescriptionDrop deformation in an uniform dc electric field is a classic problem. The pioneering work of Taylor demonstrated that for weakly conducting media, the dropfluid undergoes a toroidal flow and the drop adopts a prolate or oblate spheroidal shape, the flow and shape being axisymmetrically aligned with the applied field. However, recent studies have revealed a nonaxisymmetric rotational flow in strong fields, similar to the rotation of solid dielectric particles observed by Quincke in the 19th century. We present a systematic experimental study of this phenomenon, which highlights the importance of charge convection along the dropsurface. The critical electric field,drop inclination angle, and rate of rotation are measured. We find that for small, high viscositydrops, the threshold field strength is well approximated by the Quincke rotation criterion. Reducing the viscosity ratio shifts the onset for rotation to stronger fields. The drop inclination angle increases with field strength. The rotation rate is approximately given by the inverse Maxwell–Wagner polarization time. Novel features are also observed such as a hysteresis in the tilt angle for large lowviscosity drops.

Axisymmetric oscillation modes of a double droplet system
View Description Hide DescriptionA double droplet system (DDS) consists of a sessile and a pendant drop that are coupled through a liquid filled cylindrical hole in a plate of thickness . For a small hole radius , equilibrium shapes of both drops are sections of spheres. While DDSs have a number of applications in microfluidics, a DDS oscillating about its equilibrium state can be used as a fast focusing liquid lens. Here, a DDS consisting of an isothermal, incompressible Newtonian fluid of constant density and constant viscosity that is surrounded by a gas is excited by oscillating in time (a) the pressure in the gas surrounding either drop (pressure excitation), (b) the plate perpendicular to its plane (axial excitation), and (c) the hole radius (radial excitation). In contrast with previous works that assumed transient drop shapes are spherical, they are determined here by simulation and used to identify the natural modes of axisymmetric oscillations from resonances observed during frequency sweeps with DDSs for which the combined volume of the two drops is less than . Pressure and axial excitations are found to have identical responses but axial and radial excitations are shown to excite different modes. These modes are compared to those exhibited by single pendant (sessile) drop systems. In particular, while a single pendant (sessile) drop has one additional oscillation mode compared to a free drop, a DDS is found to exhibit roughly twice as many oscillation modes as a pendant (sessile) drop. The effects of dimensionless volume , dimensionless plate thickness , and Ohnesorge number , where is the surface tension of the DDSgas interface, on the resonance frequencies are also investigated.

Capillary wave motion excited by high frequency surface acoustic waves
View Description Hide DescriptionThis paper presents a numerical and experimental study of capillary wave motion excited by high frequency surface acoustic waves(SAWs). The objective of this study is to provide insight into the dynamic behavior of the fluid free surface and its dependence on the excitation amplitude. A twodimensional numerical model that couples the motion of the piezoelectric substrate to a thin liquid layer atop the substrate is constructed. A perturbation method, in the limit of smallamplitude acoustic waves, is used to decompose the equations governing fluid motion to resolve the widely differing time scales associated with the high frequency excitation. While this model focuses on the free surface dynamics in the lowamplitude flow regime, the experimental study focuses on the highamplitude flow regime. Transformation of time series data from both experiments and simulations into the frequency domain reveals that, in the lowamplitude regime, a fundamental resonant frequency and a superharmonic frequency are found in the frequency spectra. The former is found to be identical to that of the applied SAW, and the free surface displacement magnitude is comparable to that of the substrate displacement. Our numerical results also confirm previous speculation that the separation distance between two displacement antinodal points on the free surface is for a film and for a drop, where and denote the SAW wavelength and the acoustic wavelength in the fluid, respectively. Finally, in the highamplitude regime, strong nonlinearities shift the acoustic energy to a lower frequency than that of the SAW; this lowfrequency broadband response, quite contrary to the subharmonic halffrequency capillary wave excitation predicted by the classical linear or weakly nonlinear Faraday theories, is supported by a scaling analysis of the momentum equations.

Depinning of evaporating liquid films in square capillary tubes: Influence of corners’ roundedness
View Description Hide DescriptionIn this paper, evaporation of a volatile, perfectly wettingliquid confined in an initially filled capillary tube of square internal cross section is studied, when conditions are such that liquid films develop along the tube internal corners under the effect of capillary forces, as the bulk meniscus recedes inside the tube. More precisely, the emphasis is on the moment when the liquid film tips depin from the tube top once they have reached a critical length, a phenomenon observed in experiments. A model taking into account liquid corner flow and phase change at the film tip is proposed in order to predict the critical film length at depinning. The model is found to be in good agreement with experimental data and highlights that the critical film length depends strongly on the degree of roundedness of the tube internal corners. Thus, it is crucial to take into account this purely geometrical factor when modeling evaporation in polygonal capillary tubes or, more generally, corner flows in a rounded wedge.