Volume 17, Issue 5, May 2005
 LETTERS


Effective slip on textured superhydrophobic surfaces
View Description Hide DescriptionWe study fluid flow in the vicinity of textured and superhydrophobically coated surfaces with characteristic texture sizes on the order of . Both for droplets moving down an inclined surface and for an external flow near the surface (hydrofoil), there is evidence of appreciable drag reduction in the presence of surface texture combined with superhydrophobic coating. On textured inclined surfaces, the drops roll faster than on a coated untextured surface at the same angle. The highest drop velocities are achieved on surfaces with irregular textures with characteristic feature size . Application of the same texture and coating to the surface of a hydrofoil in a water tunnel results in drag reduction on the order of 10% or higher. This behavior is explained by the reduction of the contact area between the surface and the fluid, which can be interpreted in terms of changing the macroscopic boundary condition to allow nonzero slip velocity.

Inducedcharge electrophoresis of nonspherical particles
View Description Hide DescriptionThe electrophoreticmotion of a conducting particle, driven by an inducedcharge mechanism, is analyzed. The dependence of the motion upon particle shape is embodied in two tensorial coefficients that relate the particle translational and rotational velocities to the externally applied electric field. The coefficients are represented as surface integrals of the electric potential over the particle boundary, thereby eliminating the need to solve the flow field. Nonspherical particles may translate and∕or rotate in response to the imposed field, even if their net electric charge vanishes.

Variance reduction for Monte Carlo solutions of the Boltzmann equation
View Description Hide DescriptionWe show that by considering only the deviation from equilibrium, significant computational savings can be obtained in Monte Carlo evaluations of the Boltzmann collision integral for flow problems in the small Mach number (Ma) limit. The benefits of this variance reduction approach include a significantly reduced statistical uncertainty when the deviation from equilibrium is small, and a flowvelocity signaltonoise ratio that remains approximately constant with Ma in the limit. This results in stochastic Boltzmannsolution methods whose computational cost for a given signaltonoise ratio is essentially independent of Ma for ; our numerical implementation demonstrates this for Mach numbers as low as . These features are in sharp contrast to current particlebased simulation techniques in which statistical sampling leads to computational cost that is proportional to , making calculations at small Ma very expensive.

Identification of the wind in Rayleigh–Bénard convection
View Description Hide DescriptionUsing a symmetryaccounting ensembleaveraging method, we have identified the wind in unbounded Rayleigh–Bénard convection. This makes it possible to distinguish the wind from fluctuations and to identify dynamic features of each. We present some results from processing five independent threedimensional direct numerical simulations of a aspectratio domain with periodic side boundaries at and . It is found that the wind boundary layer scales linearly very close to the wall and has a logarithmic region further away. Despite the still noticeable molecular effects, the identification of log scaling and significant velocity and temperature fluctuations well within the thermal boundary layer clearly indicate that the boundary layer cannot be classified as laminar.
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 ARTICLES

 Interfacial Flows

A joint experimental and numerical study of mechanisms associated to instability of partial cavitation on twodimensional hydrofoil
View Description Hide DescriptionThe present work was carried out in the scope of a numericalexperimental collaborative research program, whose main objective is to understand the mechanisms of instabilities in partial cavitating flow. Experiments were conducted in the configuration of a rectangular foil located in a cavitation tunnel. Partial cavitation was investigated by multipoint wallpressure measurements together with lift and drag measurements and numerical videos. The computations were conducted on twodimensional hydrofoil section and are based on a single fluid model of cavitation: the liquid/vapor mixture is considered as a homogeneous fluid whose composition is regulated by a barotropic state law. The algorithm of resolution is derived from the SIMPLE approach, modified to take into account the high compressibility of the medium. Several physical features were pointed out by this joint approach. Particularly two distinct cavity selfoscillation dynamics characterized by two different frequencies (dynamics 1 and dynamics 2) were obtained experimentally and numerically at the angles of incidence of 6° and 8°. In both cases, the reentrant jet was found to be mainly responsible for the cavity breakdown. Dynamics 2 corresponds to the “classical” cavity breakdown and resulting cloud cavitation. A more complex flow pattern was evidenced for dynamics 1. In this case the growth/breakdown cycle of the cavity was observed at a lower Strouhal number than dynamics 2 . Moreover, the mechanism is composed of two successive steps: (i) an interaction between the reentrant jet and the cavity interface in the closure region leading to the periodic shedding of secondary cavitation clouds before the main cloud detachment occurs, and (ii) a shock wave induced by the collapse of the main cloud, which influences the growth of the residual cavity.

The flow of thin liquid films over spinning disks: Hydrodynamics and mass transfer
View Description Hide DescriptionWe study the hydrodynamics and mass transfer associated with gas absorption into a thin liquid film flowing over a spinning disk. We use the thinlayer approximation in conjunction with the Karman–Polhausen method to derive evolution equations for the film thickness and the volumetric flow rates in the radial and azimuthal directions. We also use the integral balance method to derive evolution equations for the thickness of the diffusion boundary layer as well as the concentration of solute at the disk surface. Numerical solutions of these partial differential equations, which govern the hydrodynamics and the associated mass transfer, reveal the formation of large finiteamplitude waves and elucidate their significant effect on the masstransfer characteristics. We illustrate this dependence quantitatively by examining the effect of system parameters on the timeaveraged and spatially averaged Sherwood numbers. The results are assessed by comparison with computations of the parabolized convective diffusionequation and experimental data.

Instability of miscible interfaces in a cylindrical tube
View Description Hide DescriptionWe report experimental results on the displacement of a less viscous liquid by a more viscous,miscible liquid in a vertical tube of small diameter. The liquids used are silicone oils of various viscosities. The more viscous fluid has a higher density than the less viscous fluid. Both upward displacement and downward displacement have been studied. Our observations show that for downward displacement, the “interface” between the liquids can be unstable and exhibits an asymmetric, sinuous shape. During upward displacement, the interface forms an axisymmetric finger of the intruding liquid. At the tip of the main finger, a spike of the more viscous liquid is observed under some conditions. The Reynolds number is negligibly small in these experiments and the Peclet number for mass transfer is . The influence of gravity is characterized by the dimensionless parameter where is the acceleration due to gravity, is the density difference between the fluids, is the tube diameter, is the viscosity of the more viscous fluid, and is the displacement speed. A plot of versus the viscosity ratio of the fluids is constructed to delineate the stable and unstable regimes of flow.

Experimental investigation of selfinduced thermocapillary convection for an evaporating meniscus in capillary tubes using micro–particle image velocimetry
View Description Hide DescriptionThe present paper reports an experimental investigation of the selfinduced liquid convection for an evaporating meniscus in small capillary tubes. The strong evaporative cooling at the triple contact line leads to a variation in temperature along the liquid–vapor interface, which generates a gradient of surface tension that in turn drives the observed convection. Ethanol and methanol in three tube sizes (600, 900, and 1630 μm) were investigated in this study. The flow pattern in the liquid phase has been characterized using a micro–particle image velocimetry (PIV) technique with a vector spatial resolution of 640 nm. Thermocapillary Marangoni convection is observed in horizontal diametrical sections of the horizontally oriented capillary tubes as two contrarotating vortices of similar strength, whereas in vertical diametrical sections a single clockwise vortex is mostly present. This distortion of the flow pattern could be attributed to gravity. The distortion and loss of symmetry in the vertical section is found to exhibit an oscillatory behavior. The convection (represented by the vorticity) is found to be stronger for more volatile liquids and smaller tube sizes. The vorticity normalized with the convective time scale is found to be higher for the less volatile liquid and to increase with the tube radius. Therefore, a further correction of the normalized vorticity using a dimensionless liquid saturated vapor pressure leads to a parameter that is found independent of the tube size and the liquid properties, suggesting that the phenomena described here are universal and dictated by the local conditions near the triple line.

Viscous contributions to the pressure for potential flow analysis of capillary instability of two viscous fluids
View Description Hide DescriptionCapillary instability of a liquid cylinder immersed in another liquid is analyzed using viscouspotential flow. An effect of viscosity on the irrotational motion may be introduced by evaluating the viscous normal stress at the liquid–liquid interface on the irrotational motions. In a second approximation, the explicit effects of the discontinuity of the shear stress and tangential component of velocity which cannot be resolved pointwise in irrotational flows, can be removed in the mean from the power of traction integrals in the energy equation by the selection of two viscous corrections of the irrotational pressure. The actual resolution of these discontinuities presumably takes place in a boundary layer which is not computed or needed. We include the irrotational stress and pressure correction in the normal stress balance and compare the computed growth rates to the growth rates of the exact viscousflowsolution. The agreement is excellent when one of the liquids is a gas; for two viscous liquids, the agreement is good to reasonable for the maximum growth rates but poor for long waves. Calculations show that good agreement is obtained when the vorticity is relatively small or the irrotational part is dominant in the exact viscoussolution. We show that the irrotational viscousflow with pressure corrections gives rise to exactly the same dispersion relation as the dissipation method in which no pressure at all is required and the viscous effect is accounted for by evaluating the viscous dissipation using the irrotational flow.

Inertial instability of a liquid film inside a rotating horizontal cylinder
View Description Hide DescriptionWe examine the dynamics of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating about this axis. The stability of the film has been previously explored using the leadingorder lubrication approximation, under which it was found to be neutrally stable. In the present paper, we examine how the stability of the film is affected by higherorder corrections, such as inertia (described by the material derivatives in the Navier–Stokes equations), surface tension, and the hydrostatic pressure gradient. Assuming that these effects are weak, we derive an asymptotic equation which takes them into account as perturbations. The equation is used to examine the stability of the steadystate distribution of film around the cylinder (rimming flow) with respect to linear disturbances with harmonic dependence on time (normal modes). It is shown that hydrostatic pressure gradient does not affect those at all, and the effect of surface tension is weak—whereas inertia always causes instability. The inertial instability, however, can be inhibited by viscosity, which can make the characteristic time of growth so large that the film would be effectively stable.

Kelvin–Helmholtz instability in a HeleShaw cell: Large effect from the small region near the meniscus
View Description Hide DescriptionIn an attempt to improve the poor prediction of our previous theory, we examine corrections from the small region in a HeleShaw cell near the meniscus where the flow is three dimensional. At larger Reynolds numbers, we find an change to the effective boundary condition for mass conservation which is to be applied to the large scale flow outside the small region.
 Viscous and NonNewtonian Flows

Stretching distributions in chaotic mixing of droplet dispersions with unequal viscosities
View Description Hide DescriptionThe stretching behavior of droplet dispersions with viscosity different from the matrix fluid is examined in chaotic and regular flows, in the limit of zero interfacial tension. Computations use a Lagrangian particle method, with the microstructure for each particle based on an exact solution for ellipsoidal droplets in the dilute limit. Two closed, twodimensional timeperiodic flows are considered: flow between eccentric cylinders and the sine flow. In regular flows with viscosity ratio of five or greater, many droplets display oscillatory motion and never experience large stretching. The global average stretch grows linearly in a regular flow at a rate that decreases as viscosity ratio increases. In contrast, chaotic flows gradually stretch and orient highviscosity droplets, such that the droplets asymptotically follow the stretching of the underlying flow. Consequently, for long times, droplet stretching statistics display the universal features shown by passive fluid elements in a chaotic flow: the geometric mean stretch grows exponentially at the rate of the Lyapunov exponent, and the log of the principal stretch ratio, scaled by its mean and standard deviation, settles to an invariant global probability distribution and an invariant spatial distribution. These results demonstrate that chaotic flows are highly effective at stretching microstructures that do not stretch readily in regular flows, and show that the stretching ability of a chaotic flow can be concisely described, independent of the viscosity ratio of the dispersion that is being mixed.

Stretching and mixing of nonNewtonian fluids in timeperiodic flows
View Description Hide DescriptionThe stretching of fluid elements and the dynamics of mixing are studied for a variety of polymer solutions in nearly twodimensional magnetically driven flows, in order to distinguish between the effects of viscoelasticity and shear thinning.Viscoelasticity alone is found to suppress stretching and mixing mildly, in agreement with some previous experiments on timeperiodic flows. On the other hand, the presence of shear thinningviscosity (especially when coupled with elasticity) produces a dramatic enhancement in stretching and mixing compared to a Newtonian solution at the same Reynolds number. In order to understand this observation, we study the velocity field separately in the sheared and elongational regions of the flow for various polymer solutions. We demonstrate that the enhancement is accompanied by a breaking of timereversal symmetry of the particle trajectories, on the average. Finally, we discuss possible causes for the time lags leading to this temporal symmetry breaking, and the resulting enhanced mixing.
 Particulate, Multiphase, and Granular Flows

Pressure and relative motion in colloidal suspensions
View Description Hide DescriptionWe examine the nature of relative motion in colloidalsuspensions. By distinguishing carefully between the thermodynamic pressure of a mixture, defined by Gibbs, and the pressure measured by Darcy in porous media, we resolve apparent contradictions between the results and interpretations of different experiments. We show that Fick’s and Darcy’s laws, two empirical equations thought to describe different and complementary physical phenomena, are in fact particular limits of a single, unifying thermodynamic equation which can be used more generally to describe transport in colloidal systems. Importantly, this equation relates macroscopically measurable quantities. We use it to provide new interpretations of experiments in ultrafiltration.
 Laminar Flows

Penetration of a negatively buoyant jet in a miscible liquid
View Description Hide DescriptionWe report experimental results on the evolution of a laminar liquid jet injected with negatively buoyant condition in a miscible surrounding liquid. Since molecular diffusion is negligible, the only significant miscible effect is the absence of any surface tension. After an initial intrusion phase, the jet reaches a steadystate characterized by a constant penetration depth. A simple theoretical model is derived which successfully predicts the transient phase as well as the subsequent steady state in terms of stationary penetration depth and jet’s profile. All the experimental points collapse on a master curve involving two dimensionless numbers: the densimetric Froude number Fr and , a number comparing viscous friction to buoyancy. Finally, this curve obtained for laminar flows is compared to classical results on turbulent fountains.

Axisymmetric Stokes flow through a circular orifice in a tube
View Description Hide DescriptionAxisymmetric Stokes flow through a circular tube with a thin orifice inside is investigated. At far upstream and downstream from the orifice, the Hagen–Poiseuille flow exists in the tube. The problem is investigated by considering Stokes equation analytically using a complex eigenfunction. Flow properties such as stream function and pressure distribution are determined. From the results, the streamline pattern and pressure distribution are drawn. The excess pressure drop due to the orifice and the force exerted on the orifice are calculated as functions of the radius of the orifice.
 Instability and Transition

Selfsimilarity of Rayleigh–Taylor mixing rates
View Description Hide DescriptionWe establish a renormalized selfsimilar scaling law for fluid mixing in the deeply compressible regime. Compressibility introduces a new length scale into the mixing but our time dependent analysis of the density contrast largely removes the effects of this length scale, so that selfsimilarity is maintained. Dynamically induced density changes lead to a dynamic Atwood number to measure density contrasts. We improve previous density renormalizations to allow a unified treatment of mass diffusion and compressible density stratification in a range of weakly to strongly compressible threedimensional multimode Rayleigh–Taylor simulations. Some of these simulations use front tracking to prevent numerical interfacial mass diffusion, while the others are untracked and diffusive. Using the dynamic Atwood number as opposed to the customary Atwood number to define growth rate constants, approximate universality of the mixing rate constant is obtained at low compressibility. Furthermore, earlier results giving consistent simulation, experiment and theory for nearly incompressible mixing, are now extended to show renormalized selfsimilar scaling with increases in the mixing rates for simulations of highly compressible mixing. The renormalized (i.e., dynamic Atwood number corrected) mixing rates of the diffusive and nondiffusive simulations agree, and show very similar compressibility dependence, with selfsimilar mixing rates tripling as compressibility becomes strong.

The stability analysis of two layers in a supercritical pure fluid: Rayleigh–Taylorlike instabilities
View Description Hide DescriptionWe numerically investigate the linear stability of two superposed near critical isobar fluid layers of variable thickness initially at two different temperatures. The very large compressibility and the very low heat diffusivity of near critical pure fluids induce very large density gradients which lead to a Rayleigh–Taylorlike (RTL) gravitational instability of the heat diffusion layer when the top layer temperature is some mK cooler than the bottom one. This instability in a onephase fluid seems to be similar to that which occurs in between two miscible liquids where the species diffusion is replaced by the heat diffusion coefficient. We find that this RTL configuration becomes stable when the heat diffusion length on the time scale of the faster unstable mode becomes larger than the bottom hot layer thickness.

Viscous fingering in shear thickening silica suspensions
View Description Hide DescriptionWe make an experimental study of the viscous fingering behavior of air displacing shear thickeningsilicasuspensions in linear HeleShaw cells with different cell gaps as a function of the injection pressure. The imposed shear rate defined by the ratio of the finger tip velocity and the half of a cell gap, at which the onset of the viscous fingering instability is observed, is close to the critical shear rates of the corresponding shear thickeningsilicasuspensions, irrespective of the cell gap and the injection pressure. The modified Darcy’s law, where the constant viscosity is replaced by the shear dependent viscosity, gives good agreement with the experiments when the imposed shear rate is less than the critical shear rate. When the imposed shear rate is beyond the critical shear rate, the shear thickeningsilicasuspensions give the more negative deviation from the modified Darcy’s law, irrespective of the injection pressure and the cell gap. The relative finger width can be related with nonNewtonian behavior of the silicasuspensions.

Bénard–Marangoni convection in a differentially heated cylindrical cavity
View Description Hide DescriptionThe work described in this paper concerns the study of a Bénard–Marangoni convection problem in a differentially heated cylindrical cavity. The study had two main aims; first to justify from a numerical point of view the transitions that have been reported in several experiments as the aspect ratio is varied and, second, to study both theoretically and experimentally the role of vertical and horizontal temperature differences in lateral heating convection. Initially, we analyzed the role of the aspect ratio in layers where a dynamic flow is imposed through a nonzero temperature gradient at the bottom. The basic solutions are linear or return flows depending on different parameters. Depending on the vertical temperature difference and other heatrelated parameters, the problem bifurcates either to stationary or oscillatory structures. Competing solutions at codimension two bifurcation points were found: stationary radial rolls with different wavenumbers and radial rolls together with hydrothermal waves. For small aspect ratios it was found that the Biot number does not influence the bifurcations, whereas for large aspect ratios it does. In the second part we present experimental results obtained at larger aspect ratios and for stronger surface tension effects. The role of horizontal gradients to determine the type of bifurcation both in experiments and in numerics approaching experimental conditions are discussed along with the role of vertical temperature gradients in comparison with previous theoretical works. Good agreement was obtained in terms of patterns, bifurcation sequences, and thresholds between theory, where eigenfunctions are obtained by tuning two parameters in a linear stability analysis, and experiments, where patterns are due to a nonlinear secondary bifurcation sequence caused by increasing one of the parameters.