Volume 24, Issue 10, October 2012

Experiments to understand the effect of surface wettability on impact characteristics of water drops onto solid dry surfaces were conducted. Various surfaces were used to cover a wide range of contact angles (advancing contact angle from 48° to 166°, and contact angle hysteresis from 5° to 56°). Several different impact conditions were analyzed (12 impact velocities on 9 different surfaces, among which 2 were superhydrophobic). Results from impact tests with millimetric drops show that two different regimes can be identified: a moderate Weber number regime (30 < We < 200), in which wettability affects both drop maximum spreading and spreading characteristic time; and a high Weber number regime (We > 200), in which wettability effect is secondary, because capillary forces are overcome by inertial effects. In particular, results show the role of advancing contact angle and contact angle hysteresis as fundamental wetting parameters to allow understanding of different phases of drop spreading and beginning of recoiling. It is also shown that drop spreading on hydrophilic and superhydrophobicsurfaces occurs with different time scales. Finally, if the surface is superhydrophobic, eventual impalement, i.e., transition from Cassie to Wenzel wetting state, which might occur in the vicinity of the dropimpact area, does not influence drop maximum spreading.
 SPECIAL TOPIC: THE 14TH BIENNIAL CENTER FOR TURBULENCE RESEARCH SUMMER PROGRAM


The 14th biennial Center for Turbulence Research Summer Program
View Description Hide DescriptionThis is a brief account of the 14th biennial Summer Program of the Center for Turbulence Research, which was held at Stanford University from June 25 to July 20, 2012. Key accomplishments achieved during the summer program were presented at the conclusion of the program, and a few highlights are reported here. More detailed results are to be reported at the annual American Physical Society–Division of Fluid Dynamics meeting in San Diego, November 1820, 2012, and in Proceedings of the 2012 Summer Program to be published in December 2012 by the Center for Turbulence Research of Stanford University. A repeating theme of the 2012 Summer Program was applications of the dynamic mode decomposition analysis, which revealed important dynamical information of flow physics. Other new findings are highlighted.
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 LETTERS


Can flexibility help you float?
View Description Hide DescriptionWe consider the role of flexibility in the weightbearing characteristics of bodies floating at an interface. Specifically, we develop a theoretical model for a twodimensional thin floating plate that yields the maximum stable plate load and optimal stiffness for weight support. Plates small relative to the capillary length are primarily supported by surface tension, and their weightbearing potential does not benefit from flexibility. Above a critical size comparable to the capillary length, flexibility assists interfacial flotation. For plates on the order of and larger than the capillary length, deflection from an initially flat shape increases the force resulting from hydrostatic pressure, allowing the plate to support a greater load. In this large plate limit, the shape that bears the most weight is a semicircle, which displaces the most fluid above the plate for a fixed plate length. Exact results for maximum weightbearing plate shapes are compared to analytic approximations made in the limits of large and small plate sizes. The value of flexibility for floating to a number of biological organisms is discussed in light of our study.
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 ARTICLES

 Biofluid Mechanics

Why and how does collective red blood cells motion occur in the blood microcirculation?
View Description Hide DescriptionThe behaviour of red blood cells (RBCs), modelled as vesicles, in Poiseuille flow, mimicking the microvasculature, is studied with numerical simulations in two dimensions. RBCs moving in the centre of the Poiseuille flow (as in blood capillaries) are shown to attract each other and form clusters only due to hydrodynamic interactions, provided that their distance at a given time is below a certain critical value. This distance depends on physical parameters, such as the flow strength. Our simulations reveal that clusters are unstable above a threshold value in the number of forming RBCs, beyond which one or few cells escape the pack by a selfregulating mechanism that select the marginally stable size. This size selection depends on the flow strength as well as on the RBC swelling ratio. The results are interpreted via the analysis of the perturbation of the flow field induced by the vesicles and the interplay with bending and tension forces. This sheds a novel light on the process of collective motion of RBCs observed in vivo.

Inertial squirmer
View Description Hide DescriptionAlthough the propulsion of microorganisms has been extensively studied in the literature, current studies have mainly focused on their propulsion in the absence of inertia. Here in this paper, we quantify the effects of convective inertial forces in the limit of small, but nonzero, Reynolds number regime. We analytically quantify the role of inertia on swimming speed, energy expenditure, and flow signature of an archetypal swimming model “squirmer”. Our results suggest that pushers, generating thrust behind their body, have a competitive advantage in swimming due to higher motility in the inertial regime. In contrast, those organisms that generate thrust in front of their body, pullers, have more efficient foraging in the inertial regime compared to their counterparts in the Stokes regime. Inertia enhances the swimming speed of a pusher swimmer and hinders it for a puller, potentially affecting a broad range of abundant millimeter to centimetersize organisms living in oceans and lakes.

Thrust performance of a flexible lowaspectratio pitching plate
View Description Hide DescriptionWe numerically investigate the hydrodynamic performance of an elastic plate of aspect ratio 0.54 and mass ratio 0.1 that pitches around its leading edge in a free stream at Reynolds number 640. It is found that for the rigid plate and the flexible plate with the firstmode deformation, the thrust coefficient nearly collapses onto the same curve when plotted against the Strouhal number that is defined using the tail excursion. Exceptions are found when the plate is overly flexible and higher deformation modes take place. On the other hand, the flexible plate has significantly higher power efficiency than the rigid plate at the same Strouhal number. Like the rigid plate, wake transition is observed for the flexible plate as the Strouhal number is varied. Specifically, at low Strouhal numbers (St < 0.28) the wake consists of a chain of interlocked horseshoeshape vortices; at high Strouhal numbers (St > 0.28), the wake turns into two trains of closed vortex rings with hairpin legs.
 Micro and Nanofluid Mechanics

On the flow field about an electrophoretic particle
View Description Hide DescriptionThe flow field about an electrophoretic body is theoretically investigated by analytical methods. An effective boundary condition for the electric potential at particle surface is derived. This condition, which generalizes the one obtained by Levich [Physicochemical Hydrodynamics (PrenticeHall, Englewood Cliffs, 1962), Chap. 9, p. 475], captures the effect of (convective and electromigratory) surfacecurrent in the Debye layer and is valid as far as it is legitimate to neglect ionconcentration gradient in the bulk liquid. Conditions for negligible concentration gradients are also presented and discussed. The effect of surfacecurrent determines a deviation from Morrison's “classical” theory, which predicts irrotational flowfield for any particle shape with electrophoretic velocity given by the wellknown Smoluchowski formula and always directed along the applied electric field. It is shown here that in the presence of the above effect the irrotationality of the flow field is not preserved if the particle surface has nonuniform curvature. However, irrotational flowfield still subsists for a sphere and a cylinder and is analytically determined in terms of a new nondimensional parameter, referred to as the electrophoretic number. The case of spheroidal objects is also examined in detail. In this case the flow field, though not strictly irrotational, is shown to be nearly approximated by an irrotational flowfield, which is also determined over wide ranges of electrophoretic number and spheroid aspect ratio. The quality of this approximation is expressed as a relative error on the HelmholtzSmoluchowski condition and numerically evaluated both in longitudinal and transverse configuration. The limiting cases of spheroid degenerating into a needle and a disk are also addressed. In all above cases the respective mobilities deviate from Smoluchowski's formula and depend on the electrophoretic number. An important effect of surface iontransport in the double layer is anisotropy of electrophoretic mobility for nonspherical objects. That always bears a bias of the electrophoretic velocity with respect to the applied electric field when the latter is not collinear with a symmetry axis of the body. For a cylinder, bias is always toward the axis. For a spheroid, it is generally toward the polar axis; however, bias toward the equatorial axis is predicted for moderately oblate spheroids. In general, the bias angle is remarkable, which is of potential consequence in technical applications of electrophoresis. Comparison of results of the present theory with experimental work from the literature is presented and discussed.

Fluidstructure interaction in deformable microchannels
View Description Hide DescriptionA polydimethylsiloxanemicrofluidic device composed of a single microchannel with a thin flexible layer present over a short length along one side of the channel was fabricated and modelled in order to investigate the complex fluidstructure interaction that arises between a flowing fluid and a deformable wall. Experimental measurements of thin layer deformation and pressure drop are compared with predictions of two and threedimensional computational models that numerically solve the coupled set of equations governing both the elasticity of the thin layer and the fluid. It is shown that the twodimensional model, which assumes the flexible thin layer comprises an infinitely wide elastic beam of finite thickness, reasonably approximates a threedimensional model, and is in excellent agreement with experimental observations of the thin layer profile when the width of the thin layer is beyond a critical value, roughly twice the length of the thin layer.
 Interfacial Flows

Numerical simulation of drop deformations and breakup modes caused by direct current electric fields
View Description Hide DescriptionA drop suspended in another fluid shows different dynamic behaviors in an electric field that depends on its physical properties. The phenomenon of dropdeformation under the application of an electric field, in the absence of a net volume charge, is simply caused by the surface stresses. Therefore, an accurate method is required for numerical modeling of the electric driving force at the interface to handle all of the discontinuities involved in the model. For this purpose, in this study the level set method is used along with the ghost fluid method to investigate the responses of three types of drop in the presence of an electric field. Moreover, to demonstrate the accuracy of the method, the breakup modes of each electric model are carefully simulated. Finally, the results of the simulations are compared with similar numerical and experimental results from the literature. The simulation results indicate the accuracy of the method for modeling of the phenomenon over a wide range of electric capillary numbers, and particularly for the capture of the drop profile at the instant of disintegration.

Breakup of a poorly conducting liquid thread subject to a radial electric field at zero Reynolds number
View Description Hide DescriptionWe study the breakup of an axisymmetric viscous liquid thread with finite conductivity immersed in another viscous fluid, which are confined to a concentrically placed cylindrical electrode that is held at a constant voltage potential. The annular fluid between the core thread and the electrode is assumed to be insulating. The flow then is driven by a radial electric field together with capillary and viscous forces. A linear stability analysis is carried out when the perturbation on the thread interface is small and nonlinear evolution and satellite drop formation near pinchoff are investigated by direct numerical simulations based on boundary integral method. The numerical results reveal that satellite formation as well as breakup time is affected significantly when the effect of charge convection is important compared with electric conduction. For large conduction, the evolutions of the thread are close to those obtained for a perfectly conducting core fluid. Finally, we show numerically that the local dynamics may be altered when the conduction is weak compared to the perfect conductor limit. New scalings near breakup are obtained from a long wave model.

Analysis of laminar jet impingement and hydraulic jump on a horizontal surface with slip
View Description Hide DescriptionThis paper explores the influence surface slip, uniform in all directions with constant slip length, exerts on the physics of laminar jet impingement on a flat horizontal surface. Slip exists on superhydrophobicsurfaces, and due to the relatively thin filmdynamics associated with the growth of the laminar jet after impingement, its influence on the fluid physics is significant. An analysis based on momentum considerations is presented that allows prediction of the relevant thin film parameters as a function of radial position from the impingement point, jet Reynolds number, and constant relative slip length of the surface. Further, the analysis allows determination of the hydraulic jump location in terms of laminar jet characteristics and imposed downstream liquid depth. The results reveal that at a given radial location, the boundary layer growth and thin film thickness decrease, while the surface velocity of the thin film increases with increasing slip at the surface. The departure from classical noslip behavior is quantified over a range of realizable slip conditions. Increasing slip length also leads to formation of hydraulic jumps at increasing radial location. An expression based on the results is presented that allows prediction of the hydraulic jump location as a function of the magnitude of the slip and all other influencing variables.

Drop impact and wettability: From hydrophilic to superhydrophobic surfaces
View Description Hide DescriptionExperiments to understand the effect of surface wettability on impact characteristics of water drops onto solid dry surfaces were conducted. Various surfaces were used to cover a wide range of contact angles (advancing contact angle from 48° to 166°, and contact angle hysteresis from 5° to 56°). Several different impact conditions were analyzed (12 impact velocities on 9 different surfaces, among which 2 were superhydrophobic). Results from impact tests with millimetric drops show that two different regimes can be identified: a moderate Weber number regime (30 < We < 200), in which wettability affects both drop maximum spreading and spreading characteristic time; and a high Weber number regime (We > 200), in which wettability effect is secondary, because capillary forces are overcome by inertial effects. In particular, results show the role of advancing contact angle and contact angle hysteresis as fundamental wetting parameters to allow understanding of different phases of drop spreading and beginning of recoiling. It is also shown that drop spreading on hydrophilic and superhydrophobicsurfaces occurs with different time scales. Finally, if the surface is superhydrophobic, eventual impalement, i.e., transition from Cassie to Wenzel wetting state, which might occur in the vicinity of the dropimpact area, does not influence drop maximum spreading.

Conservation laws of surfactant transport equations
View Description Hide DescriptionEquations of the interfacial convection and convectiondiffusion describing the transport of surfactants, and more general interfacial balance laws, in the context of a threedimensional incompressible twophase flow are considered. Here, the interface is represented implicitly by a zero level set of an appropriate function. All interfacial quantities and operators are extended from the interface to the threedimensional domain. In both convection and convectiondiffusion settings, infinite families of conservation laws that essentially involve surfactant concentration are derived, using the direct construction method. The obtained results are also applicable to the construction of the general balance laws for other excess surface physical quantities. The system of governing equations is subsequently rewritten in a fully conserved form in the threedimensional domain. The latter is essential for simulations using modern numerical methods.

Curvature singularity in the asymmetric breakup of an underwater air bubble
View Description Hide DescriptionThe presence of slight azimuthal asymmetry in the initial shape of an underwater bubble entirely alters the final breakup dynamics. Here, I examine the influence of initial asymmetry on the final breakup by simulating the bubble surface evolution as a Hamiltonian evolution corresponding to an inviscid, twodimensional, planar implosion. I find two types of breakups: a previously reported coalescence mode in which distant regions along the airwater surface curve inwards and eventually collide with finite speed, and a hitherto uncharacterized cusplike mode where the surface develops sharp tips whose radii of curvature are much smaller than the average neck radius. I present three sets of results that characterize the nature of this cusp mode. First, I show that the cusplike mode corresponds to a phase space trajectory that passes close to a saddlenode structure. In other words, an evolution towards a crosssection shape with sharp tips invariably later evolves away from it. In phase space, this saddlenode separates coalescence modes whose coalescence planes lie along different spatial orientations. Second, I show that the formation of the sharp tips can be interpreted as a weakly firstorder transition which becomes secondorder, corresponding to the formation of a finitetime curvature singularity, in the limit that the initial perturbation amplitude approaches zero. Third, I show that, as the curvature singularity is approached, the maximum surface curvature diverges approximately as (t _{ c } − t)^{−0.8}, where t _{ c } is the onset time of the singularity and the maximum velocity diverges approximately as (t _{ c } − t)^{−0.4}. In practice, these divergences imply that viscous drag and compressibility of the gas flow, two effects not included in my analysis, become significant as the interface evolves towards the curvature singularity.

Thermodynamically consistent description of the hydrodynamics of free surfaces covered by insoluble surfactants of high concentration
View Description Hide DescriptionIn this paper, we propose several models that describe the dynamics of liquid films which are covered by a high concentration layer of insoluble surfactant. First, we briefly review the “classical” hydrodynamic form of the coupled evolution equations for the film height and surfactant concentration that are well established for small concentrations. Then we reformulate the basic model as a gradient dynamics based on an underlying free energy functional that accounts for wettability and capillarity. Based on this reformulation in the framework of nonequilibrium thermodynamics, we propose extensions of the basic hydrodynamic model that account for (i) nonlinear equations of state, (ii) surfactantdependent wettability, (iii) surfactantphase transitions, and (iv) substratemediated condensation. In passing, we discuss important differences to most of the models found in the literature.

The influence of the horizontal component of the temperature gradient on nonlinear convective oscillations in twolayer systems
View Description Hide DescriptionThe influence of the horizontal component of the temperature gradient on nonlinear oscillatory convective regimes, developed under the joint action of buoyant and thermocapillary effects in the 47 v2 silicone oilwater system, is investigated. Cavities with different lengths have been considered. Transitions between oscillatory flow regimes with different symmetry properties and steady flows have been studied. It is shown that under the action of the horizontal component of the temperature gradient, specific asymmetric oscillatory flow develops in the system.

Stability of developing film flow down an inclined surface
View Description Hide DescriptionFilm flows on inclined surfaces are often assumed to be of constant thickness, which ensures that the velocity profile is halfPoiseuille. It is shown here that by shallow water theory, only flows in a portion of Reynolds numberFroude number (Re–Fr) plane can asymptotically attain constant film thickness. In another portion on the plane, the constant thickness solution appears as an unstable fixed point, while in other regions the film thickness seems to asymptote to a positive slope. Our simulations of the NavierStokes equations confirm the predictions of shallow water theory at higher Froude numbers, but disagree with them at lower Froude numbers. We show that different regimes of film flow show completely different stability behaviour from that predicted earlier. Supercritical decelerating flows are shown to be always unstable, whereas accelerating flows become unstable below a certain Reynolds number for a given Froude number. Subcritical flows on the other hand are shown to be unstable above a certain Reynolds number. In some range of parameters, two solutions for the base flow exist, and the attached profile is found to be more stable. All flows except those with separation become more stable as they proceed downstream.

A study of pressuredriven displacement flow of two immiscible liquids using a multiphase lattice Boltzmann approach
View Description Hide DescriptionThe pressuredriven displacement of two immiscible fluids in an inclined channel in the presence of viscosity and density gradients is investigated using a multiphase lattice Boltzmann approach. The effects of viscosity ratio, Atwood number, Froude number, capillary number, and channel inclination are investigated through flow structures, front velocities, and fluid displacement rates. Our results indicate that increasing viscosity ratio between the fluids decreases the displacement rate. We observe that increasing the viscosity ratio has a nonmonotonic effect on the velocity of the leading front; however, the velocity of the trailing edge decreases with increasing the viscosity ratio. The displacement rate of the thinlayers formed at the later times of the displacement process increases with increasing the angle of inclination because of the increase in the intensity of the interfacialinstabilities. Our results also predict the front velocity of the lockexchange flow of two immiscible fluids in the exchange flow dominated regime. A linear stability analysis has also been conducted in a threelayer system, and the results are consistent with those obtained by our lattice Boltzmann simulations.

Capillarydriven flow induced by a stepped perturbation atop a viscous film
View Description Hide DescriptionThin viscous liquid films driven by capillarity are well described in the lubrication theory through the thin film equation. In this article, we present an analytical solution of this equation for a particular initial profile: a stepped perturbation. This initial condition allows a linearization of the problem making it amenable to Fourier analysis. The solution is obtained and characterized. As for a temperature step in the heat equation, selfsimilarity of the first kind of the full evolution is demonstrated and a longterm expression for the excess free energy is derived. In addition, hydrodynamical fields are described. The solution is then compared to experimental profiles from a model system: a polystyrene nanostep above the glass transition temperature which flows due to capillarity. The excellent agreement enables a precise measurement of the capillary velocity for this polymeric liquid, without involving any numerical simulation. More generally, as these results hold for any viscous system driven by capillarity, the present solution may provide a useful tool in hydrodynamics of thin viscous films.

Electrified freesurface flow of an inviscid liquid past topography
View Description Hide DescriptionThe flow of an electrified liquid layer moving over a prescribed topography is studied with the aim of determining the shape of the free surface. The steady flow is assumed to be inviscid, incompressible, and irrotational. The liquid is assumed to act as a perfect conductor and the air above the layer is assumed to act as a perfect dielectric. The electric field is produced by placing one or more charged electrodes at a distance above the free surface. A weakly nonlinear onedimensional analysis is used to classify the possible solutions and nonlinear solutions are obtained numerically by boundary integral equation methods. It is found that the shape of the liquid layer's surface can be manipulated (using charged electrodes) to become wavefree.
 Viscous and NonNewtonian Flows

Viscous heating in largeamplitude oscillatory shear flow
View Description Hide DescriptionWhen measuring rheological properties in oscillatoryshear flow, one worries about experimental error due to the temperature rise in the sample that is caused by viscous heating. For polymeric liquids, for example, this temperature rise causes the measured values of the components of the complex viscosity to be systematically low. For such linear viscoelastic property measurements, we use an analytical solution by Ding et al. [J. NonNewtonian Fluid Mech.86, 359 (1999)10.1016/S03770257(99)00004X] to estimate the temperature rise. However, for largeamplitude oscillatoryshear flow, no such analytical solution is available. Here we derive an analytical solution for the temperature rise in a corotational Maxwell fluid (a model with just two parameters: η_{0} and λ) subject to largeamplitude oscillatoryshear flow. This result can then be generalized to a superposition of corotational Maxwell models for a quantitative estimate of the temperature rise. We chose the corotational Maxwell model because, when generalized for multiple relaxation times, it gives an accurate prediction for molten plastics in largeamplitude oscillatoryshear flow. We identify three relevant pairs of thermal boundary conditions: (i) both plates isothermal, (ii) with heat loss by convection from both plates, and (iii) one plate isothermal, the other with heat loss by convection. We find that the timeaveraged viscous heating increases as an even power series of the dimensionless shear rate amplitude (Weissenberg number), and that it decreases with the dimensionless imposed frequency (Deborah number). We distinguish between the dimensionless timeaveraged temperature rise, , and the oscillating part, , where . We solve analytically for the profile through the sample thickness for all three pairs of thermal boundary conditions. For the worst case, two adiabatic walls, we derive an expression for the oscillating part of the temperature rise, . We find this to be a Fourier series of even harmonics whose contribution to the temperature rise can be as important as . If both plates are adiabatic, then the sample temperature rises without bound. Otherwise, it does not.