Volume 24, Issue 2, February 2012

The capillary breakup of a polymer solution evolves via a series of stages. After the initial instability a longlived cylindrical filament is formed, which thins exponentially in time, while the flow is purely extensional. During the final stages of the thinning process, at which the polymers are stretched sufficiently for the filament to become unstable to a Rayleigh–Plateaulike instability, a complex flow pattern develops, which we describe here. Achieving a high spatial resolution well below the optical Rayleigh limit, we describe both the formation of individual droplets as well as that of periodic patterns. Following the periodic instability, a blistering pattern appears, with different generations of smaller droplets. At sufficiently high polymer concentrations, the filament does not break at all, but a solid polymeric fiber with a thickness well below a micron remains. The experiments were performed for various polymer and solvent systems, all of which showed the same qualitative behavior for most of the observed features.
 LETTERS


Reexamining the logarithmic dependence of the mean velocity distribution in polymer drag reduced wallbounded flow
View Description Hide DescriptionA reexamination of the logarithmic dependence of the mean velocity distribution in polymerdrag reduced flows shows that drag reducing polymers modify the von Kármán coefficient and, in channel flow, eradicate the loglayer at high drag reductions. It is also found that the “ultimate profile,” corresponding to the state of maximum drag reduction is not logarithmic.

Oscillatory bubbles induced by geometrical constraint
View Description Hide DescriptionWe show that a simple change in pore geometry can radically alter the behavior of a fluiddisplacing air finger, indicating that models based on idealized pore geometries fail to capture key features of complex practical flows. In particular, partial occlusion of a rectangular cross section can force a transition from a steadily propagating centered finger to a state that exhibits spatial oscillations formed by periodic sideways motion of the interface at a fixed distance behind the moving finger tip. We characterize the dynamics of the oscillations, which suggest that they arise from a global homoclinic connection between the stable and unstable manifolds of a steady, symmetrybroken solution.

Clouds of particles in a periodic shear flow
View Description Hide DescriptionWe have investigated the time evolution of a cloud of nonBrownian particles subjected to a periodic shear flow in an otherwise pure liquid at low Reynolds number. This experiment illustrates the irreversible nature of particulate systems submitted to a shear. When repeating the cycles of shear, we have found that clouds of particles progressively disperse in the flow direction until reaching a threshold critical volume fraction that depends upon the strain amplitude; this critical volume fraction coincides with measurements of the threshold for reversibility found from experiments on homogeneous suspensions in periodic shear. Two distinct patterns, including a “galaxylike” shape, are observed for the evolution of the clouds and the transition between the patterns is identified using a simple scaling analysis. Movies are available with the online version of the paper.

Stokes flow paths separation and recirculation cells in Xjunctions of varying angle
View Description Hide DescriptionFluid and solute transfer in Xjunctions between straight channels is shown to depend critically on the junction angle α in the Stokes flow regime. Experimentally, water and a waterdye solution are injected at equal flow rates in two facing channels of the junction. Planar laser induced fluorescence (PLIF) measurements show that the largest part of each injected fluid “bounces back” preferentially into the outlet channel at the lowest angle to the injection; this is opposite to the inertial case and requires a high curvature of the corresponding streamlines. The proportion of this fluid in the other channel decreases from 50% at α = 90° to 0% at a threshold angle. These counterintuitive features reflect the minimization of energy dissipation for Stokes flows. Finite elements numerical simulations of a 2D Stokes flow of equivalent geometry confirm these results and show that, below the threshold angle α_{ c } = 33.8°, recirculation cells are present in the center part of the junction and separate the two injected flows of the two solutions. Reducing further α leads to the appearance of new recirculation cells with lower flow velocities.

Stokes number effects on particle slip velocity in wallbounded turbulence and implications for dispersion models
View Description Hide DescriptionThe particle slip velocity is adopted as an indicator of the behavior of heavy particles in turbulent channel flow. The statistical moments of the slip velocity are evaluated considering particles with Stokes number, defined as the ratio between the particle response time and the viscous time scale of the flow, in the range 1 < St < 100. The slip velocityfluctuations exhibit a monotonic increase with increasing particle inertia, whereas the fluidparticle velocity covariance is gradually reduced for St ⩾ 5. Even if this covariance equals the particle turbulence intensity, a substantial amount of particle slip may occur. Relevant to twofluid modeling of particleladen flows is the finding that the standard deviation of the slip velocityfluctuations is significantly larger than the corresponding mean slip velocity.
 Top

 ARTICLES

 Micro and Nanofluid Mechanics

Velocity slip coefficients based on the hardsphere Boltzmann equation
View Description Hide DescriptionWe present a kinetic theory derivation of higherorder slip boundary conditions. The situation studied is that of a pressure driven isothermal gas flowing through a plane microchannel. The distribution function is expanded in terms of halfrange Hermite polynomials and the system of moment equations in the expansion coefficients is analytically solved. The velocity slip coefficients, as well as their Knudsenlayer corrections, are obtained by evaluating the solution in the near continuum limit. The proposed approach is accurate and easy to implement. The results are presented for the hardsphere Boltzmann equation and Maxwell's diffusespecular boundary conditions, but can be extended to arbitrary intermolecular interactions and more general scattering kernels.

Microdroplet oscillations during optical pulling
View Description Hide DescriptionIt was recently shown theoretically that it is possible to pull a spherical dielectric body towards the source of a laser beam[J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics5, 531 (2011)], a result with immediate consequences to optical manipulation of small droplets. Optical pulling can be realized, e.g., using a diffractionfree Bessel beam, and is expected to be of great importance in manipulation of microscopic droplets in micro and nanofluidics. Compared to conventional optical pushing, however, the ratio of optical net force to stress acting on a droplet is much smaller, increasing the importance of oscillations. We describe the timedependent surface deformations of a water microdroplet under optical pulling to linear order in the deformation. Shape oscillations have a lifetime in the order of microseconds for droplet radii of a few micrometers. The force density acting on the initially spherical droplet is strongly peaked near the poles on the beam axis, causing the deformations to take the form of jetlike protrusions.
 Interfacial Flows

Bouncing, coalescence, and separation in headon collision of unequalsize droplets
View Description Hide DescriptionThe dynamics of headon collision of unequalsize droplets were experimentally and theoretically investigated, with emphasis on identifying distinct collision outcomes and interpreting the sizeratio dependence. A unified regime diagram in terms of bouncing, permanent coalescence, and separation after coalescence was identified for hydrocarbon and waterdroplets in the parameter space of the size ratio and a collision Weber number. Experimental results show that the transition Weber number, We _{bc}, that separates the bouncing and permanent coalescence regimes, weakly depends on the size ratio, while the transition Weber number, We _{cs}, that separates permanent coalescence and separation regimes, significantly increases with the size ratio. A theoreticalmodel based on energy balance and scaling analysis was developed to explain the sizeratio dependence of We _{cs}. The theoretical results show good agreement with the experimental data for tetradecane and decane droplets, with a moderate discrepancy for waterdroplets.

Thermocapillary motion of a slender viscous droplet in a channel
View Description Hide DescriptionWe extend the previously developed lowcapillarynumber asymptotic theory of thermocapillary motion of a long bubble and a moderately viscousdroplet in a channel [S. K. Wilson, “The effect of an axial temperature gradient on the steady motion of a large droplet in a tube,” J. Eng. Math.29, 205 (1995)10.1007/BF00042854; A. Mazouchi and G. M. Homsy, “Thermocapillary migration of long bubbles in cylindrical capillary tubes,” Phys. Fluids12, 542 (2000)10.1063/1.870260] toward droplets with an arbitrary viscosity. A generalized modified LandauLevichBretherton equation, governing the thickness of the carrier liquid film entrained between the droplet and the channel wall in the transition region between constant thickness film and constant curvature cap, is solved numerically. The resulting droplet velocity is determined applying the mass balance and it is a function of two dimensionless parameters, the modified capillary number, Δσ*, equal to the surface tension variance over a distance of channel halfwidth scaled with the mean surface tension, and the innertoouter liquidviscosity ratio, λ. It is found that the droplet speed decreases with the increase in dropletviscosity, as expected, while this retardation becomes more operative upon the increase in Δσ*.

Observation of collision and oscillation of microdroplets with extremely large shear deformation
View Description Hide DescriptionWe measured the viscosity and surface tension of various liquids under large (∼10^{6} s^{−1}) shear deformation. Oscillation of a 10μm size microdroplet is brought about by the headon collision of two droplets. Since the Reynolds number is as small as 100, the motion of the liquid is stable and the dynamic image is obtained with high reproducibility by the stroboscopic method. By observing and evaluating the mechanical oscillation of the microdroplet, of which frequency ranges typically in 100 – 300 kHz, we found that the viscosity of ethylene glycol and diethylene glycol is smaller than the known literature value, which is considered to be the viscosity at zerofrequency. This phenomena can be attributed to the slow viscous relaxation of associated liquids due to the recombination dynamics of the network of Hbonds.

The physics of aerobreakup. II. Viscous liquids
View Description Hide DescriptionWe extend the work of Theofanous and Li [“On the physics of aerobreakup,” Phys. Fluids20, 052103 (2008)] on aerobreakup physics of waterlike, low viscosity liquid drops, to Newtonian liquids of any viscosity. The scope includes the full range of aerodynamics from near incompressible to high Mach number flows. The key physics of Rayleigh–Taylor piercing (RTP, first criticality) and of shearinduced entrainment (SIE, second and terminal criticality) are verified and quantified by new viscosity and capillaritybased scalings for fluids of any viscosity. The relevance and predictive power of linear stability analysis of the Rayleigh–Taylor and Kelvin–Helmholtz problems (both including viscosity) is demonstrated for the RTP and the SIE regimes, respectively. The advanced stages of breakup and of the resulting particleclouds are observed and clear definition and quantification of breakup times are offered.

Thin films flowing down inverted substrates: Threedimensional flow
View Description Hide DescriptionWe study contact line induced instabilities for a thin film of fluid under destabilizing gravitational force in threedimensional setting. In the previous work [T.S. Lin and L. Kondic, Phys. Fluids22, 052105 (2010)], we considered twodimensional flow, finding formation of surface waves whose properties within the implemented longwave model depend on a single parameter, , where Ca is the capillary number and α is the inclination angle. In the present work we consider fully 3D setting and discuss the influence of the additional dimension on stability properties of the flow. In particular, we concentrate on the coupling between the surface instabilities and the transverse (fingering) instabilities of the film front. We furthermore consider these instabilities in the setting where fluid viscosity varies in the transverse direction. It is found that the flow pattern strongly depends on the inclination angle and the viscosity gradient.

Measurements of liquid film thickness for a droplet at a twofluid interface
View Description Hide DescriptionCoalescence of a droplet at a twofluid interface is studied at Bond numbers larger than one and at three different values of the viscosity ratio. Both the thickness of the liquid film between the rising droplet and the twofluid interface, and the location of film rupture are measured using laser induced fluorescence. Particle image velocimetry was applied to the flow in the film. It is found that the film thins asymetrically, and that the time interval between collision and film rupture is shorter than predicted by commonly used models. The film ruptures at an offcenter location. It can be concluded that asymmetric film drainage speeds up coalescence.

Dip coating with an interaction potential normal to the substrate
View Description Hide DescriptionDip coating in the presence of a substrateliquid interaction potential normal to the substrate, previously theoretically investigated by R. Krechetnikov and G. M. Homsy [Phys. Fluids17, 038101 (2005)], was revisited. Their solution procedure leads to predictions of the entrained film thickness that deviate substantially from the classical LandauLevich law because of the impossibility to identify meniscussolutions satisfying the proper boundary conditions of zero thickness and zero apparent contact angle on the solid substrate (LL BC’s). In contrast, in the present analysis, by choosing a different method of integration and requiring the satisfaction of the boundary condition of flat bath for large, but finite, meniscus thickness, we obtain solutions subject to LL BC's for the same parameter range studied in Krechetnikov and Homsy's paper. Thus, the matching follows a modified LandauLevich law, where is inversely proportional to the meniscus curvature at the substrate. Since the interaction potential changes considerably this curvature, the entrained film significantly thickens for attractive interactions or thins for repulsive ones. Similar results are also found for a potential of the DebyeHückel form.

Analytical study in the mechanism of flame movement in horizontal tubes
View Description Hide DescriptionThe problem of premixed flame propagation in wide horizontal tubes is revisited. Employing the onshell description of flames with arbitrary gas expansion, a nonlinear secondorder differential equation for the front position of steady flame is derived. Solutions to this equation, obtained numerically, reveal two distinct physical regimes of laminar flame propagation controlled by the strong baroclinic effect. They differ by the front shape and flame speed, the ratio of the total consumption rates in the two regimes being 1.4 to 1.8, depending on the value of the gas expansion coefficient. Comparison with the existing experimental data on methaneair flames is made, and explanation of the main trends in the observed flame behavior is given. It is shown, in particular, that the faster (slower) regime of combustion is realized in mixtures close to (far from) the stoichiometric composition, with pronounced changeover in between.

Modeling resistance of nanofibrous superhydrophobic coatings to hydrostatic pressures: The role of microstructure
View Description Hide DescriptionIn this paper, we present a numerical study devised to investigate the influence of microstructural parameters on the performance of fibrous superhydrophobic coatings manufactured via dc and ac electrospinning. In particular, our study is focused on predicting the resistance of such coatings against elevated hydrostaticpressures, which is of crucial importance for submersible applications. In our study, we generate 3D virtual geometries composed of randomly or orthogonally oriented horizontal fibers with bimodal diameter distributions resembling the microstructure of our electrospun coatings. These virtual geometries are then used as the computational domain for performing full morphology numerical simulations to establish a relationship between the coatings’ critical pressure(pressure beyond which the surface may depart from the Cassie state) and their microstructures. For coatings with ordered microstructures, we have also derived analytical expressions for the critical pressure based on the balance of forces acting on the water–air interface. Predictions of our force balance analysis are compared with those of our FM simulations as well as the equations proposed by Tuteja et al. [Proc. Natl. Acad. Sci. U.S.A.105, 18200 (2008)]10.1073/pnas.0804872105, and discussed in detail. Our numerical simulations are aimed at providing useful information with regards to the tolerance of fibrous superhydrophobic coatings against elevated pressures, and helping with the design and optimization of the coatings’ microstructures. Our results show considerably higher pressure tolerance for the case of coatings with orthogonally oriented fibers as compared to those with randomly laid fibers when other microstructural parameters are held constant. Moreover, it is demonstrated that thickness of the coating has less influence on performance in the case of orthogonal microstructures. Coatings’ responses to other variations favor those that yield smallersized interfiber spaces. Studies are also performed investigating the effect of subtle permutations in the layer configurations of our acelectrospun coatings, as well as the use of a hybrid coating that utilizes advantages from both dc and ac electrospinning.

Crosswaves induced by the vertical oscillation of a fully immersed vertical plate
View Description Hide DescriptionCapillary waves excited by the vertical oscillation of a thin elongated plate below an airwater interface are analyzed using timeresolved measurements of the surfacetopography. A parametric instability is observed above a well defined acceleration threshold, resulting in a socalled crosswave, a staggered wavepattern localized near the wavemaker and oscillating at half the forcing frequency. This crosswave, which is stationary along the wavemaker but propagative away from it, is described as the superposition of two almost antiparallel propagating parametric waves making a small angle of the order of 20^{o} with the wavemaker edge. This contrasts with the classical Faraday parametric waves, which are exactly stationary because of the homogeneity of the forcing. Our observations suggest that the selection of the crosswave angle results from a resonant mechanism between the two parametric waves and a characteristic length of the surface deformation above the wavemaker.

Structure of Marangonidriven singularities
View Description Hide DescriptionThis work presents an analytical study of the structure of steady Marangonidriven singularities in the context of chemicalreaction driven tipstreaming, which identifies the conditions when such singularities are observable. As motivated by experimental observations of the conical symmetry of the problem, one can construct selfsimilar solutions of the Stokes equations, which are singular at the tip; these solutions, however, provide no information on the thread structure which is responsible for a resolution of the singularity via tipstreaming. The conetip singularity is resolved here with the help of asymptotic matching of the cone and thread solutions using slender jet approximation, which gives an explicit asymptotic formula for the thread radius and thus of the emitted droplets size as a function of physical parameters governing the problem.
 Viscous and NonNewtonian Flows

The final stages of capillary breakup of polymer solutions
View Description Hide DescriptionThe capillary breakup of a polymer solution evolves via a series of stages. After the initial instability a longlived cylindrical filament is formed, which thins exponentially in time, while the flow is purely extensional. During the final stages of the thinning process, at which the polymers are stretched sufficiently for the filament to become unstable to a Rayleigh–Plateaulike instability, a complex flow pattern develops, which we describe here. Achieving a high spatial resolution well below the optical Rayleigh limit, we describe both the formation of individual droplets as well as that of periodic patterns. Following the periodic instability, a blistering pattern appears, with different generations of smaller droplets. At sufficiently high polymer concentrations, the filament does not break at all, but a solid polymeric fiber with a thickness well below a micron remains. The experiments were performed for various polymer and solvent systems, all of which showed the same qualitative behavior for most of the observed features.

Viscous exchange flows
View Description Hide DescriptionGravitationally driven exchange flows of viscous fluids with different densities are analysed theoretically and investigated experimentally within a horizontal channel. Following initiation from rest when there is a vertical boundary dividing the two fluids, the denser fluid slumps under the less dense along the underlying boundary, while the less dense fluid intrudes along the upper boundary. The motion is driven by the pressure gradients associated with the density differences between the two fluids, resisted by viscous stresses, and mathematically modelled by a similarity solution that depends on the ratio of the viscosities of the two fluids. When the viscosity of the less dense fluid is much smaller than the viscosity of the denser fluid, the shape of the interface between the fluids varies rapidly close to the upper boundary and depends weakly on the viscosity ratio within the interior of the flow. Matched asymptotic expansions are employed in this regime to determine the shape of the interface and the rates of its propagation along the boundaries. The similarity solutions are shown to be linearly stable and thus are expected to represent the intermediate asymptotics of the flow. Experiments confirm the similarity form of solutions and demonstrate close agreement with the theoretical predictions when the viscosities of the fluids are comparable, but exhibit some discrepancies when the viscosities differ more substantially. It is suggested that these discrepancies may be due to mixing between the fluids close to the boundaries, which is induced by the noslip boundary condition. Exchange flows within porous domains are also investigated to determine the shape of the interface as a function of the ratio of the viscosities of the two fluids and using asymptotic analysis, this shape is determined when this ratio is much larger, or smaller, than unity.