Volume 26, Issue 9, September 2014
 ARTICLES

 Biofluid Mechanics

Cilia beating patterns are not hydrodynamically optimal
View Description Hide DescriptionWe examine the hydrodynamic performance of two cilia beating patterns reconstructed from experimental data. In their respective natural systems, the two beating patterns correspond to: (A) pumpingspecialized cilia, and (B) swimmingspecialized cilia. We compare the performance of these two cilia beating patterns as a function of the metachronal coordination in the context of two model systems: the swimming of a ciliated cylinder and the fluid pumping by a ciliated carpet. Three performance measures are used for this comparison: (i) average swimming speed/pumping flow rate; (ii) maximum internal moments generated by the cilia; and (iii) swimming/pumping efficiencies. We found that, in both models, pattern (B) outperforms pattern (A) in almost all three measures, including hydrodynamic efficiency. These results challenge the notion that hydrodynamic efficiency dictates the cilia beating kinematics, and suggest that other biological functions and constraints play a role in explaining the wide variety of cilia beating patterns observed in biological systems.
 Micro and Nanofluid Mechanics

Thermooptical pressure difference in onecomponent gas
View Description Hide DescriptionA new phenomenon—thermooptical pressure difference in the gas (TOPD) is regarded. This effect is the steady state of the second order which arises in the gas located in a closed capillary in the presence of a fixed temperature gradient and a resonant optical radiation. TOPD is the result of imposition thermal transpiration and lightinduced drift of gas in a capillary. The problem is solved on the basis of the linearized Boltzmann kinetic equations for excited and unexcited gaseous particles. Expressions for the kinetic coefficients and pressure drop in gas at the ends of the closed capillary are obtained. Possible cases of the steady state are regarded for atoms and molecules. Numerical estimates of this effect for atomic and molecular gases in the whole range of Knudsen numbers are given.

Wall heat transfer effects on the hydro/thermal behaviour of Poiseuille flow in micro/nanochannels
View Description Hide DescriptionWe investigate effects of wall heat transfer on the structure of pressure driven flow in micro/nanochannels using the Direct Simulation Monte Carlo method. The effects of nonzero wall heat flux on the pressure distribution, velocity profiles, heat flow patterns, and the mass flow rate are reported. The simulation results show that cooling decreases slip at the wall and pressure along the channel. Cooling changes the heat flow direction along the channel while heating does not. At higher degree of rarefaction, the direction of the heat flow is mainly axial along the channel. An existence of coldtohot heat transfer process is demonstrated in the cooling wall case. Cooling can also create a heat singularity point in the domain. There is a critical Knudsen number about unity for which heating or cooling does not affect the mass flow rate through the channel. Below the critical Knudsen number, heating decreases and cooling increases the mass flow rate. Above it, heating increases and cooling decreases the mass flow rate.

Anisotropic mobility of particles near the interface of two immiscible liquids
View Description Hide DescriptionThreedimensional mobility of colloidal particles in the close vicinity of a liquidliquid interface is experimentally quantified and compared with established theories. Evanescent wavebased particle tracking velocimetry is used to measure the Brownian motion of fluorescent spheres near an interface between water and nonpolar oil. The experimental results confirm that the mobility of particles suspended in the less viscous liquid is anisotropically suppressed. The measured hindered mobility are in agreement with theoretical models.
 Interfacial Flows

A force balance model for the motion, impact, and bounce of bubbles
View Description Hide DescriptionA force balance model has been developed to predict the terminal velocity of a submillimetric bubble as its rises in water under buoyancy. The dynamics of repeated collisions and rebounds of the bubble against a horizontal solid surface is modeled quantitatively by including forces due to buoyancy, added mass, drag, and hydrodynamic lubrication—the last arises from the drainage of water trapped in the thin film between the solid surface and the surface of the deformable bubble. The result is a selfcontained, parameterfree model that is capable of giving quantitative agreement with measured trajectories and observed collisions and rebounds against a solid surface as well as the spatiotemporal evolution of the thin film during collision as measured by interferometry.

Moving towards the cold region or the hot region? Thermocapillary migration of a droplet attached on a horizontal substrate
View Description Hide DescriptionWe study computationally thermocapillary migration of a twodimensional droplet attached on a horizontal substrate with a constant temperature gradient. A levelset approach is employed to track the droplet interface and a Navier slip boundary condition is imposed to alleviate a stress singularity at the moving contact lines. The present numerical model allows us to consider droplets with large contact angles and to take into account effects of the fluid outside the droplet, both have not been well studied so far. In the limits of a zero contact angle hysteresis and a small viscosity ratio of the fluids outside and inside the droplet (μ out /μ in ⩽ 0.1), we find the droplet finally migrates towards the cold region, and both the steady migration speed and the velocity field inside the droplet obtained from numerical simulation agree very well with the lubrication theory of Ford and Nadim [“Thermocapillary migration of an attached drop on a solid surface,” Phys. Fluids6, 3183–3185 (1994)] when the contact angles are small (⩽45°). Beyond this regime, increasing the contact angles leads to increased deviations between numerical simulation and the lubrication theory, and the steady migration speed of the droplet towards the cold side decreases with the contact angles. The simulation results show that the droplet could fall in a motionless regime when its contact angles are around 100° even without any contact angle hysteresis. It is very interesting to find that a droplet with even larger contact angles migrates towards the hot region in a steady speed. We also find the transition of the migration direction of a droplet could strongly depend on the viscosity ratio. With increasing the viscosity of the external fluid, the transition could happen at much smaller values of contact angles. We summarize the results in a phase diagram and discuss the effects of other system parameters, including the contact angle hysteresis, the effective Marangoni number, the Prandtl number, and the slip length, on thermocapillary migration of the droplet.

Spreading of droplet with insoluble surfactant on corrugated topography
View Description Hide DescriptionThe flow of microscale fluid on a topography surface is a key to further development of MEMS, nanoscience and technology. In the present paper, a theoretical model of the droplet spreading with insoluble surfactant over corrugated topography is established with the lubrication theory, and the evolution equations of film thickness and surfactant concentration in base state and disturbance state are formulated. The droplet dynamics, the nonlinear stability based on nonmodal stability theory, and the effects of topography structure and Marangoni stress are numerically simulated with PDECOL scheme. Results show that the impact of topographical surface is strengthened apparently while the Marangoni stress driven by surfactant concentration is weakened in the midlate stages of the spreading. The droplet radius on the topography advances faster and the lowest height of liquid/gas interface near the droplet edge reduces remarkably in the intermediate stage compared with those on the flat wall. The quantity of the wavelet similar to the topography increases gradually, with the characteristics of wavelet crest height with time exhibiting a singlehump feature. The spreading stability is enhanced under the disturbance wavenumber of 4, however, is to deteriorate and even to transform into instability when wavenumber increases further. In addition, the reductive Marangoni number, enhancive capillary number, modest Peclet number, the low height of the topography as well as small wavenumber of topography can make contributions to the evident stability of droplet spreading.

Dynamic spreading on pillararrayed surfaces: Viscous resistance versus molecular friction
View Description Hide DescriptionThe dynamic spreading of a liquid droplet on micropillararrayed surfaces is experimentally investigated. A theoretical model is proposed to include energy dissipations raised from both the viscous resistance at mesoscale and the molecular friction at microscale in the triplephase region. The scaling laws and spreading shape of the droplet change with the variation of the liquid viscosity because of the competition between these two mechanisms of energy dissipations at the moving contact line. The Laplace pressures at the interior corner and at the wavy contact line are the answers to the excess driving energy and the superwetting on pillararrayed surfaces. The formation and evolution of the bulk and the fringe are also analyzed in detail. Our results may help to understand the wetting dynamics on microtextured surfaces and assist the future design of engineered surfaces in practical applications.

Asymmetric flexuralgravity lumps in nonuniform media
View Description Hide DescriptionHere we show that asymmetric fully localized flexuralgravity lumps can propagate on the surface of an inviscid and irrotational fluid covered by a variablethickness elastic material, provided that the thickness varies only in one direction and has a local minimum. We derive and present equations governing the evolution of the envelope of flexuralgravity wave packets allowing the flexing material to have small variations in the transverse (to propagation) direction. We show that the governing equation belongs to the general family of DaveyStewartson equations, but with an extra term in the surface evolution equation that accounts for the variable thickness of the elastic cover. We then use an iterative NewtonRaphson scheme, with a numerical continuation procedure via Lagrange interpolation, in a search to find fully localized solutions of this system of equations. We show that if the elastic sheet thickness has (at least) a local minimum, flexuralgravity lumps can propagate near the minimum thickness, and in general have an asymmetric bellshape in the transverse to the propagation direction. In applied physics, flexuralgravity waves describe for instance propagation of waves over the icecovered bodies of water. Ice is seldom uniform, nor is the seafloor, and in fact near the boundaries (iceedges, shorelines) they typically vary only in one direction (toward to edge), and are uniform in the transverse direction. This research suggests that fully localized waves are not restricted to constant icethickness/waterdepth areas and can exist under much broader conditions. Presented results may have implications in experimental generation and observation of flexuralgravity (as well as capillarygravity) lumps.

Wake pattern and wave resistance for anisotropic moving disturbances
View Description Hide DescriptionWe present a theoretical study of gravity waves generated by an anisotropic moving disturbance. We model the disturbance by an elliptical pressure field of given aspect ratio . We study the wake pattern as a function of and the longitudinal hull Froude number , where V is the velocity, g is the acceleration of gravity, and L is the size of the disturbance in the direction of motion. For large hull Froude numbers, we analytically show that the rescaled surface profiles for which is kept constant coincide. In particular, the angle outside which the surface is essentially flat remains constant and equal to the Kelvin angle, and the angle corresponding to the maximum amplitude of the waves scales as , thus showing that previous work on the wake's angle for isotropic objects can be extended to anisotropic objects of given aspect ratio. We then focus on the wave resistance and discuss its properties in the case of an elliptical Gaussian pressure field. We derive an analytical expression for the wave resistance in the limit of very elongated objects and show that the position of the speed corresponding to the maximum wave resistance scales as .
 Viscous and NonNewtonian Flows

Viscoplastic sculpting
View Description Hide DescriptionViscoplastic lubrication (VPL) has been established as a method for reliably suppressing interfacial instabilities and enhancing flow stability for multilayer systems. Here we extend this methodology to the formation of shaped interfaces in multifluid coreannular configurations. We study multilayer VPL flows in which we perform both experiment and computation with oscillating the flow rates of the individual phases. According to the flow rate variations we succeed in freezing in a range of different interfacial patterns. Experiments performed with carbopol as lubricating fluid, and with xanthan and polyethylene oxide solutions as core fluid, serve to illustrate the potential of the method. We show that single pulsed changes in the imposed inflow rates can result in small interface indentations that remain frozen into the interface as it propagates downstream. Repeated pulses produce periodically patterned interfaces. We are able to control the frequency and amplitude of the interfacial patterns, but not directly the shape. Inelastic core fluids have been observed to produce rounded bulges whereas elastic core fluids have produced diamond shapes. Moreover, numerical simulations extend the range of shapes achievable and give us interesting insights into the forming process.

Effect of pore geometry and interfacial tension on wateroil displacement efficiency in oilwet microfluidic porous media analogs
View Description Hide DescriptionUsing oilwet polydimethylsiloxane (PDMS) microfluidic porous media analogs, we studied the effect of pore geometry and interfacial tension on wateroil displacement efficiency driven by a constant pressure gradient. This situation is relevant to the drainage of oil from a bypassed oilwet zone during water flooding in a heterogeneous formation. The porosity and permeability of analogs are 0.19 and 0.133–0.268 × 10^{−12} m^{2}, respectively; each analog is 30 mm in length and 3 mm in width, with the longer dimension aligned with the flow direction. The pore geometries include three random networks based on Voronoi diagrams and eight periodic networks of triangles, squares, diamonds, and hexagons. We found that among random networks both pore width distribution and vugs (large cavities) decreased the displacement efficiency, among the periodic networks the displacement efficiency decreased with increasing coordination number, and the random network with uniform microfluidic channel width was similar to the hexagon network in the displacement efficiency. When vugs were present, displacement was controlled by the sequence of vugfilling and the structure of intervug texture was less relevant. Surfactant (0.5 wt. % ethoxylated alcohol) increased the displacement efficiency in all geometries by increasing the capillary number and suppressing the capillary instability.
 Particulate, Multiphase, and Granular Flows

The patterning behaviour and accumulation of spherical particles in a vibrated nonisothermal liquid
View Description Hide DescriptionA completely new phenomenon of particle accumulation in vibrated nonisothermal monodisperse suspensions of solid spheres (in a liquid) is analyzed. For the first time evidence is provided for this case that even in situations in which particleparticle hydrodynamic interactions are negligible (dilute systems), intriguing nonlinear effects can lead to the irreversible formation of welldefined particulate structures over “long” temporal scales, i.e., times much larger than the period of the applied vibrations. The longrange translational ordering is produced by the delicate interplay between convective effects (of thermovibrational nature) and the (inertial) response of each isolated particle to the timeperiodic acceleration. A new family of particle attractors in the physical space is identified with the topological dimension being essentially a function of the “symmetry properties” of the considered vibrated system and related geometrical constraints.

The motion of solid spherical particles falling in a cellular flow field at low Stokes number
View Description Hide DescriptionWe present a jointed experimental and numerical study examining the influence of vortical structures on the settling of solid spherical particles under the action of gravity at low Stokes numbers. The twodimensional model experiment uses electroconvection to generate a twodimensional array of controlled vortices which mimics a simplified vortical flow. Particle imagevelocimetry and tracking are used to examine the motion of the particles within this vortical flow. Particle motion is compared to the predictions of a numerical simulation inspired by the model equation developed by Maxey [“The motion of small spherical particles in a cellular flow field,” Phys. Fluids30, 1915 (1987)].
 Laminar Flows

Forced response of a laminar shockinduced separation bubble
View Description Hide DescriptionThe source of unsteadiness in shockwave/boundarylayer interactions is currently disputed. This paper considers a twodimensional separation bubble induced by an oblique shock wave interacting with a laminar boundary layer at a freestream Mach number of 1.5. The global response of the separated region to white noise forcing is analyzed for different interaction strengths, which generate small and large separation bubbles. Forcing location and amplitude effects have been examined. For both interaction strengths and for forcing both upstream and inside the bubble, the wallpressure spectra downstream of the separation show a highfrequency peak that is demonstrated to be a KelvinHelmholtz instability. A lowfrequency response at the separation point is also found when the separation bubble is only forced internally, therefore with a disturbancefree upstream boundary layer. For lowamplitude internal forcing, the lowfrequency response at the separation point and downstream of the bubble is linear. However, when forced upstream the lowfrequency unsteadiness of the large separation bubble is found to be driven by nonlinearities coming from the downstream shedding. The same nonlinear behavior is found when the separation bubble is internally forced over a narrow band around the shedding frequency, without lowfrequency disturbances. This analysis for a laminar interaction is used to interpret the lowfrequency unsteadiness found at the foot of the shock of turbulent interactions. Here, the lowfrequency unsteadiness occurs in the absence of upstream disturbances and a linear relationship is found between the internal forcing and the response near the separation point. When lowfrequencies are not present in the forcing they are generated from weak nonlinearities of the shearlayer instability modes.

Scaling and dimensional analysis of acoustic streaming jets
View Description Hide DescriptionThis paper focuses on acoustic streaming free jets. This is to say that progressive acoustic waves are used to generate a steady flow far from any wall. The derivation of the governing equations under the form of a nonlinear hydrodynamics problem coupled with an acoustic propagation problem is made on the basis of a time scale discrimination approach. This approach is preferred to the usually invoked amplitude perturbations expansion since it is consistent with experimental observations of acoustic streaming flows featuring hydrodynamic nonlinearities and turbulence. Experimental results obtained with a plane transducer in water are also presented together with a review of the former experimental investigations using similar configurations. A comparison of the shape of the acoustic field with the shape of the velocity field shows that diffraction is a key ingredient in the problem though it is rarely accounted for in the literature. A scaling analysis is made and leads to two scaling laws for the typical velocity level in acoustic streaming free jets; these are both observed in our setup and in former studies by other teams. We also perform a dimensional analysis of this problem: a set of seven dimensionless groups is required to describe a typical acoustic experiment. We find that a full similarity is usually not possible between two acoustic streaming experiments featuring different fluids. We then choose to relax the similarity with respect to sound attenuation and to focus on the case of a scaled water experiment representing an acoustic streaming application in liquid metals, in particular, in liquid silicon and in liquid sodium. We show that small acoustic powers can yield relatively high Reynolds numbers and velocity levels; this could be a virtue for heat and mass transfer applications, but a drawback for ultrasonic velocimetry.
 Instability and Transition

Penetrative RayleighBénard convection in water near its maximum density point
View Description Hide DescriptionThe presence of a density maximum in water near 4 °C significantly modifies the nature and onset conditions of convective flows due to imposed temperature differences. In the present study, vertical temperature gradients are imposed upon a horizontal, rectangular layer of water, with the top and bottom surfaces maintained above and below the maximum density temperature, respectively. In such an arrangement, convection beginning in the lower, unstable portion of the layer (as small as 1/3 of the layer height) may penetrate into the upper, stable region. The resulting convection patterns are visualized using schlieren or shadowgraph techniques along multiple visual axes. The measured onset conditions and observed patterns are discussed in the context of preceding predictions and experimental observations in similar penetrative systems. As expected from the nonBoussinesq nature of water in this temperature range, convection sets in at temperature differences below those predicted by linear stability theory when the unstable portion of the layer is sufficiently small. The conductionconvection transition is also hysteretic in nature. At onset, the convection pattern consists of parallel, transverse rolls due to the boundary conditions of the fluid chamber. When the unstable portion of the layer is significantly less than half of the fluid layer height, the convective motion is found to penetrate only partway into the upper stable region, within which weakly counterrotating motions are driven. At higher Rayleigh numbers, the fluid undergoes secondary transitions to either hexagonal cellular or longitudinal roll states which are visualized for the first time. Pattern heights and wavenumbers were measured in some instances, establishing qualitative (in general) and quantitative (over some parameter ranges) agreement with linear theory.

Mechanisms for the transition to waviness for Taylor vortices
View Description Hide DescriptionBuilding on the weakly nonlinear amplitude equation of the saturated Taylor vortices developing in a Taylor–Couette cell with a rotating inner cylinder and a fixed outer one, the physical mechanism underlying the destabilization of these vortices resulting in azimuthal waviness is addressed using Floquet analysis. For narrow gap configurations, analysis and direct numerical simulations together with existing experimental results support the idea that the waviness is generated by the axial shear in the azimuthal velocity due to the alternate advection by the Taylor vortices of azimuthal momentum between the cylinders. For wide gap configurations, this mechanism is no longer able to drive the azimuthal waviness and a different mechanism tends to select a subharmonic instability.

Transition to finger convection in doublediffusive convection
View Description Hide DescriptionFinger convection is observed experimentally in an electrodeposition cell in which a destabilizing gradient of copper ions is maintained against a stabilizing temperature gradient. This doublediffusive system shows finger convection even if the total density stratification is unstable. Finger convection is replaced by an ordinary convection roll if convection is fast enough to prevent sufficient heat diffusion between neighboring fingers, or if the thermal buoyancy force is less than 1/30 of the compositional buoyancy force. At the transition, the ion transport is larger than without an opposing temperature gradient.

Effects of weak noise on oscillating flows: Linking quality factor, Floquet modes, and Koopman spectrum
View Description Hide DescriptionMany fluid flows, such as bluff body wakes, exhibit stable selfsustained oscillations for a wide range of parameters. Here we study the effect of weak noise on such flows. In the presence of noise, a flow with selfsustained oscillations is characterized not only by its period, but also by the quality factor. This measure gives an estimation of the number of oscillations over which periodicity is maintained. Using a recent theory[P. Gaspard, J. Stat. Phys.106, 57 (2002)], we report on two observations. First, for weak noise the quality factor can be approximated using a linear Floquet analysis of the deterministic system; its size is inversely proportional to the innerproduct between first direct and adjoint Floquet vectors. Second, the quality factor can readily be observed from the spectrum of evolution operators. This has consequences for Koopman/Dynamic mode decomposition analyses, which extract coherent structures associated with different frequencies from numerical or experimental flows. In particular, the presence of noise induces a damping on the eigenvalues, which increases quadratically with the frequency and linearly with the noise amplitude.