Volume 22, Issue 12, December 2010

The swimming behavior of the nematode Caenorhabditis elegans is investigated in aqueous solutions of increasing viscosity. Detailed flowdynamics associated with the nematode’s swimming motion as well as propulsive force and power are obtained using particle tracking and velocimetry methods. We find that C. elegans delivers propulsive thrusts on the order of a few nanonewtons. Such findings are supported by values obtained using resistive force theory; the ratio of normal to tangential drag coefficients is estimated to be approximately 1.4. Over the range of solutions investigated here, the flow properties remain largely independent of viscosity. Velocity magnitudes of the flow away from the nematode body decay rapidly within less than a body length and collapse onto a single master curve. Overall, our findings support that C. elegans is an attractive living model to study the coupling between smallscale propulsion and low Reynolds numberhydrodynamics.
 LETTERS


Flutter instability of rectangle and trapezoid flags in uniform flow
View Description Hide DescriptionAn experiment on the flutter instability of rectangle and trapezoid flags is conducted in a low speed wind tunnel and a physical model is proposed to predict the critical velocity of flutter instability of the flags. The nondimensional second moment of area is used to depict the effect of the flag shape on the flutter. The result shows that the method presented in this paper can be used to predict the lowercritical velocity. The change of the flutter envelope and critical velocity is found to be related to the dominant mode.

The thickness of the turbulent/nonturbulent interface is equal to the radius of the large vorticity structures near the edge of the shear layer
View Description Hide DescriptionDirect numerical simulations at Reynolds numbers ranging from to 160 show that the thickness of the turbulent/nonturbulent (T/NT) interface in planar jets is of the order of the Taylor scale , while in shear free, irrotational/isotropic turbulence is of the order of the Kolmogorov microscale . It is shown that is equal to the radius of the large vorticitystructures (LVSs) in this region, . Thus, the mean shear and the Reynolds number affect the T/NT interface thickness insofar as they define the radial dimension of the LVS near the T/NT interface.

Slip length for longitudinal shear flow over a dilute periodic mattress of protruding bubbles
View Description Hide DescriptionAn analytical formula for the frictional slip length associated with transverse shear flow over a bubble mattress comprising a dilute periodic array of parallel circulararc grooves protruding into the fluid has recently been presented by Davis and Lauga [Phys. Fluids21, 011701 (2009)]. This letter derives an analytical formula for the slip length associated with longitudinal shear flow over the same surface. The formula is in excellent agreement with a phenomenological result based on finite element simulations given by Teo and Khoo [Microfluid. Nanofluid.9, 499 (2010)].

 ARTICLES

 Biofluid Mechanics

Propulsive force measurements and flow behavior of undulatory swimmers at low Reynolds number
View Description Hide DescriptionThe swimming behavior of the nematode Caenorhabditis elegans is investigated in aqueous solutions of increasing viscosity. Detailed flowdynamics associated with the nematode’s swimming motion as well as propulsive force and power are obtained using particle tracking and velocimetry methods. We find that C. elegans delivers propulsive thrusts on the order of a few nanonewtons. Such findings are supported by values obtained using resistive force theory; the ratio of normal to tangential drag coefficients is estimated to be approximately 1.4. Over the range of solutions investigated here, the flow properties remain largely independent of viscosity. Velocity magnitudes of the flow away from the nematode body decay rapidly within less than a body length and collapse onto a single master curve. Overall, our findings support that C. elegans is an attractive living model to study the coupling between smallscale propulsion and low Reynolds numberhydrodynamics.

The impact of uncertainty on shape optimization of idealized bypass graft models in unsteady flow
View Description Hide DescriptionIt is well known that the fluid mechanics of bypass grafts impacts biomechanical responses and is linked to intimal thickening and plaque deposition on the vessel wall. In spite of this, quantitative information about the fluid mechanics is not currently incorporated into surgical planning and bypass graft design. In this work, we use a derivativefree optimization technique for performing systematic design of bypass grafts. The optimization method is coupled to a threedimensional pulsatile Navier–Stokes solver. We systematically account for inevitable uncertainties that arise in cardiovascular simulations, owing to noise in medical image data, variable physiologic conditions, and surgical implementation. Uncertainties in the simulation input parameters as well as shape design variables are accounted for using the adaptive stochastic collocation technique. The derivativefree optimization framework is coupled with a stochastic response surface technique to make the problem computationally tractable. Two idealized numerical examples, an endtoside anastomosis, and a bypass graft around a stenosis, demonstrate that accounting for uncertainty significantly changes the optimal graft design. Results show that small changes in the design variables from their optimal values should be accounted for in surgical planning. Changes in the downstream (distal) graft angle resulted in greater sensitivity of the wallshear stress compared to changes in the upstream (proximal) angle. The impact of cost function choice on the optimal solution was explored. Additionally, this work represents the first use of the stochastic surrogate management framework method for robust shape optimization in a fully threedimensional unsteady Navier–Stokes design problem.
 Micro and Nanofluid Mechanics

Flow regime transition at high capillary numbers in a microfluidic Tjunction: Viscosity contrast and geometry effect
View Description Hide DescriptionFlow regimes obtained as a consequence of two immiscible fluids interacting at a Tjunction are presented for transitional to high capillary numbers and different ratios of the continuous and dispersed phase flow rates and viscosities. Results are presented for the formation of micronsized droplets using simulations performed based on a threedimensional lattice Boltzmann method. The influence of viscosity and geometry of the device on the frequency and volume of droplets formed has been examined and the nondimensional drop size correlated with the capillary number and flow rate ratio. This work reveals two important and new physical features: (a) the transition zone separating droplet and jet flows narrows for high capillary numbers and (b) the critical flow rate ratio separating droplet and parallel flows increases as the dispersed to continuous channel width ratio increases, aspects which have been correlated using a simple relation for both transitions from the dropletatTjunction to dropletinchannel and dropletinchannel to parallel flow. In the dropletatTjunction regime, the dropletformation frequency was recorded as a function of the capillary number, flow rate ratio, and the channel width ratio as well. Results show that the transition to stable jets can be delayed and droplets can be formed even at very high flow rate ratios by significantly increasing the viscosity of the continuous phase.

Influence of streaming potential on the elastic response of a compliant microfluidic substrate subjected to dynamic loading
View Description Hide DescriptionIn the present study, we investigate the effect of streaming potential on the elastic response of a compliant surface subjected to dynamic loading conditions. For illustrating the pertinent physical phenomena, we analyze in particular the dynamical characteristics of the system on squeezing out of a liquid layer through a narrow gap formed between an elasticfluidicsurface and an incipient rigid oscillating sphere. We reveal that the streaming potential effects may amplify the elastic force response of the substrate to a considerable extent. Interestingly and nontrivially, this increment turns out not only to be a function of the pertinent electrokinetic parameters dictating the establishment of the streaming potential, but also a combined consequence of the oscillation frequency and the stiffness of the substrate, consistent with a dynamical interaction between interfacial electrochemicalhydrodynamics and structural responsive characteristics that has hitherto not been emphatically explored.

Breakup of diminutive Rayleigh jets
View Description Hide DescriptionDischarging a liquid from a nozzle at sufficient large velocity leads to a continuous jet that due to capillary forces breaks up into droplets. Here we investigate the formation of microdroplets from the breakup of micronsized jets with ultra highspeed imaging. The diminutive size of the jet implies a fast breakup time scale of the order of 100 ns, and requires imaging at . We directly compare these experiments with a numerical lubrication approximation model that incorporates inertia, surface tension, and viscosity [J. Eggers and T. F. Dupont, J. Fluid Mech.262, 205 (1994); X. D. Shi, M. P. Brenner, and S. R. Nagel, Science265, 219 (1994)]. The lubrication model allows to efficiently explore the parameter space to investigate the effect of jet velocity and liquidviscosity on the formation of satellitedroplets. In the phase diagram, we identify regions where the formation of satellitedroplets is suppressed. We compare the shape of the droplet at pinchoff between the lubrication approximation model and a boundaryintegral calculation, showing deviations at the final moment of the pinchoff. In spite of this discrepancy, the results on pinchoff times and droplet and satellitedroplet velocity obtained from the lubrication approximation agree with the highspeed imaging results.

Transport of Brownian particles confined to a weakly corrugated channel
View Description Hide DescriptionWe investigate the average velocity of Brownian particles driven by a constant external force when constrained to move in twodimensional, weakly corrugated channels. We consider both the geometric confinement of the particles between solid walls as well as the soft confinement induced by a periodic potential. Using perturbation methods we show that the leading order correction to the marginal probability distribution of particles in the case of soft confinement is equal to that obtained in the case of geometric confinement, provided that the (configuration) integral over the crosssection of the confining potential is equal to the width of the solid channel. We then calculate the probability distribution and average velocity in the case of a sinusoidal variation in the width of the channels. The reduction on the average velocity is larger in the case of soft channels at small Péclet numbers and for relatively narrow channels and the opposite is true at large Péclet numbers and for wide channels. In the limit of large Péclet numbers the convergence to bulk velocity is faster in the case of soft channels. The leading order correction to the average velocity and marginal probability distribution agree well with Brownian Dynamics simulations for the two types of confinement and over a wide range of Péclet numbers.

Effects of convection and solid wall on the diffusion in microscale convection flows
View Description Hide DescriptionThe diffusive transport properties in microscale convectionflows are studied by using the direct simulation Monte Carlo method. The effective diffusion coefficient is computed from the mean square displacements of simulated molecules based on the Einstein diffusion equation . Two typical convectionflows, namely, thermal creepconvection and Rayleigh–Bénard convection, are investigated. The thermal creepconvection in our simulation is in the noncontinuum regime, with the characteristic scale of the vortex varying from 1 to 100 molecular mean free paths. The diffusion is shown to be enhanced only when the vortex scale exceeds a certain critical value, while the diffusion is reduced when the vortex scale is less than the critical value. The reason for phenomenon of diffusion reduction in the noncontinuum regime is that the reduction effect due to solid wall is dominant while the enhancement effect due to convection is negligible. A molecule will lose its memory of macroscopic velocity when it collides with the walls, and thus molecules are hard to diffuse away if they are confined between very close walls. The Rayleigh–Bénard convection in our simulation is in the continuum regime, with the characteristic length of 1000 molecular mean free paths. Under such condition, the effect of solid wall on diffusion is negligible. The diffusion enhancement due to convection is shown to scale as the square root of the Péclet number in the steady convection regime, which is in agreement with previous theoretical and experimental results. In the oscillation convection regime, the diffusion is more strongly enhanced because the molecules can easily advect from one roll to its neighbor due to an oscillation mechanism.
 Interfacial Flows

Numerical study of a droplet migration induced by combined thermocapillarybuoyancy convection
View Description Hide DescriptionNumerical computations have been performed to study the effects of thermocapillary convection and buoyancy convection, and free surface deformation induced by gravity on the migration behavior of a liquiddroplet on a horizontal solid surface subjected to a uniform temperature gradient. Investigations are carried out by solving the Navier–Stokes equations coupled with the energy equation through the finite element method. The combined thermocapillary and buoyancy force driven convection produces complex dynamic behavior of fluid motion inside the droplet. The net momentum generated by a pair of asymmetric thermocapillary convectionvortices inside the droplet drives the droplet to move in both small and middle droplet sized regimes. In the small sized regime, the quasisteady migration speed of the droplet is mostly linearly proportional to its size because of the stronger net thermocapillary momentum. When the droplet is in the middle sized regime, its quasisteady migration speed reaches a maximum, but this is gradually reduced as the droplet size increases due to the suppression of the net thermocapillary momentum by the buoyancy force. In the large droplet sized regime, two pairs of convectionvortices exist inside the droplet as a result of the appearance of the buoyancydriven convection accompanying the thermocapillary convection. The quasisteady migration speed quickly diminishes mainly due to the reduction of the net thermocapillary momentum from the stronger buoyancy convection.

Electrohydrodynamic instabilities in thin liquid trilayer films
View Description Hide DescriptionExperiments by Dickey et al. [Langmuir22, 4315 (2006)] and Leach et al. [Chaos15, 047506 (2005)] show that novel pillar shapes can be generated from electrohydrodynamic instabilities at the interfaces of thin polymer/polymer/air trilayerfilms. In this paper, we use linear stability analysis to investigate the effect of free charge and acelectric fields on the stability of trilayer systems. Our work is also motivated by our recent theoretical study [S. A. Roberts and S. Kumar, J. Fluid Mech.631, 255 (2009)] which demonstrates how acelectric fields can be used to increase control over the pillar formation process in thin liquid bilayer films. For perfect dielectric films, the effect of an acelectric field can be understood by considering an equivalent dc field. Leaky dielectric films yield pillar configurations that are drastically different from perfect dielectric films, and ac fields can be used to control the location of free charge within the trilayer system. This can alter the pillar instability modes and generate smaller diameter pillars when conductivities are mismatched. The results presented here may be of interest for the creation of complex topographical patterns on polymer coatings and in microelectronics.

The influence of capillary flow on the fate of evaporating wetted imprint of the sessile droplet in porous medium
View Description Hide DescriptionThe fate of a wetting liquid sessile droplet imbibed by a porous medium is formulated as a multiphase flow problem and a numerical solution is developed using the capillary network model with a microforce balance at the interface. The liquid phase capillary flow and evaporation are solved simultaneously. An exclusive evidence for a multiphase flow is already found in the capillary flow, as a liquid wets a much larger volume of porous medium compared to the wetted volume, calculated by assuming that the medium imbibes the liquid in the singlephase flow. The physics of the multiphase capillary flow includes the formation of local gas clusters and liquid ganglia. The clusters and ganglia distribution is further altered by evaporation. The evaporation tends to shrink the ganglia sizes and open the gas clusters, both due to the liquid mass loss from the porous medium. Still, the capillarity tends to disperse the liquid back into the regions from where the liquid previously evaporated. These changes in the liquid saturation produce the changes in vapor concentration within the porous medium and changes in the mass fluxes. The imprint shape varies, where, for more spherical imprints, the evaporation is enhanced due to the capillary flow. The opposite is true for the elongated imprints for which the capillarity hinders the evaporation rate. Comparing the spherical and elongated imprints, the liquid dispersion differs and the capillary flow the into protrusion direction is pronounced for the elongated imprints. The changes in the liquid dispersion and imprint shape influence the vapor concentration within the porous medium, vapor phase mass fluxes, and liquid persistence time. Finally, the previous behavior is observed for hazardous materials and warfare agents, where predicting the fate of such kind of liquids and their vapors become especially important due to their harmful effects.

Building water bridges in air: Electrohydrodynamics of the floating water bridge
View Description Hide DescriptionThe interaction of electrical fields and liquids can lead to a phenomenon that defies intuition. Some famous examples can be found in electrohydrodynamics as Taylor cones, whipping jets, or noncoalescing drops. A less famous example is the floating water bridge: a slender thread of water held between two glass beakers in which a high voltage difference is applied. Surprisingly, the water bridge defies gravity even when the beakers are separated at distances up to 2 cm. In this paper, experimental measurements and simple models are proposed and discussed for the stability of the bridge and the source of the flow, revealing an important role of polarization forces on the stability of the water bridge. On the other hand, the observed flow can only be explained due to the nonnegligible free charge present in the surface. In this sense, the floating water bridge can be considered as an extreme case of a leaky dielectric liquid [J. R. Melcher and G. I. Taylor, Annu. Rev. Fluid Mech.1, 111 (1969)].

Wavelength selection in the crown splash
View Description Hide DescriptionThe impact of a drop onto a liquid layer produces a splash that results from the ejection and dissolution of one or more liquid sheets, which expand radially from the point of impact. In the crown splash parameter regime, secondary droplets appear at fairly regularly spaced intervals along the rim of the sheet. By performing many experiments for the same parameter values, we measure the spectrum of smallamplitude perturbations growing on the rim. We show that for a range of parameters in the crown splash regime, the generation of secondary droplets results from a Rayleigh–Plateau instability of the rim, whose shape is almost cylindrical. In our theoretical calculation, we include the time dependence of the base state. The remaining irregularity of the pattern is explained by the finite width of the RayleighPlateau dispersion relation. Alternative mechanisms, such as the Rayleigh–Taylor instability, can be excluded for the experimental parameters of our study.

Axial dispersion in segmented gasliquid flow: Effects of alternating channel curvature
View Description Hide DescriptionThe effects of channel curvature on the axial dispersion in segmented gasliquidflows are studied computationally in a twodimensional setting using a finitevolume/fronttracking method. Passive tracer particles are used to visualize and quantify the axial dispersion. The molecular diffusion is modeled by random walk of tracer particles. It is found that there is significant axial dispersion in serpentine channels even in the absence of molecular diffusion. The lubricating thin liquid layer that persists on the wall of a straight channel is periodically broken in the serpentine channel leading to enhanced axial dispersion. It is also found that the axial dispersion is always larger in the serpentine channel than that in the straight channel but the effects of channel curvature are more pronounced at high Peclet numbers, i.e., . A model is proposed based on the difference between the liquid film thicknesses on the inner and outer side of the bend in the limit as . Good agreement is found between the computational results and the model when the liquid slug is well mixed by the chaotic advection.
 Viscous and NonNewtonian Flows

Mixing of passive tracers in the decay Batchelor regime of a channel flow
View Description Hide DescriptionWe report detailed quantitative studies of passive scalar mixing in a curvilinear channel flow, where elasticturbulence in a dilute polymer solution of high molecular weight polyacrylamide in a high viscosity watersugar solvent was achieved. For quantitative investigation of mixing, a detailed study of the profiles of mean longitudinal and radial components of the velocity in the channel as a function of Wi was carried out. Besides, a maximum of the average value as well as a rms of the longitudinal velocity was used to determine the threshold of the elasticinstability in the channel flow. The rms of the radial derivatives of the longitudinal and radial velocity components was utilized to define the control parameters of the problem, the Weissenberg and the Péclet Pe numbers. The main result of these studies is the quantitative test of the theoretical prediction about the value of the mixing length in the decay Batchelor regime. The experiment shows large quantitative discrepancy, more than 200 times in the value of the coefficient , which appears in the theoretical expression for the mixing length, but with the predicted scaling relation. There are two possible reasons to this discrepancy. First is the assumption made in the theory about the correlated velocity field, which is in odds with the experimental observations. Second, and probably a more relevant suggestion for the significantly increased mixing length and thus reduced mixing efficiency, is the observed jets, the rare, localized, and vigorous ejection of the scalar trapped near the wall, which protrudes into the peripheral region as well as the bulk. They are first found in the recent numerical calculations and then observed in the experiment reported. The jets definitely strongly reduce the mixing efficiency in particular in the peripheral region and so can lead to considerable increase of the mixing length. We hope that this result will initiate further numerical calculations of the mixing length. Finally, we analyze statistical properties of the mixing in the decay Batchelor regime by studying the power spectra, the decay exponents scaling, the structure functions of a tracer and moments of PDF of passive scalar increments, and the temporal and spatialcorrelation functions and find rather satisfactory agreement with theory.

Viscous evolution of point vortex equilibria: The collinear state
View Description Hide DescriptionWe describe the viscous evolution of a collinear threevortex structure that corresponds initially to an inviscid point vortex fixed equilibrium, with the goal of elucidating some of the main transient dynamical features of the flow. Using a multiGaussian “coregrowth” type of model, we show that the system immediately begins to rotate unsteadily, a mechanism we attribute to a “viscously induced” instability. We then examine in detail the qualitative and quantitative evolution of the system as it evolves toward the longtime asymptotic Lamb–Oseen state, showing the sequence of topological bifurcations that occur both in a fixed reference frame and in an appropriately chosen rotating reference frame. The evolution of passive particles in this viscously evolving flow is shown and interpreted in relation to these evolving streamline patterns.

Pair collisions of fluidfilled elastic capsules in shear flow: Effects of membrane properties and polymer additives
View Description Hide DescriptionThe dynamics and pair collisions of fluidfilled elastic capsules during Couette flow in Newtonian fluids and dilute solutions of highmolecular weight (dragreducing) polymers are investigated via direct simulation. Capsule membranes are modeled using either a neoHookean constitutive model or a model introduced by Skalak et al. [“Strain energy function of red bloodcell membranes,” Biophys. J.13, 245 (1973)], which includes an energy penalty for area changes. This model was developed to capture the elastic properties of red blood cells. Polymer molecules are modeled as beadspring trimers with finitely extensible nonlinearly elastic springs; parameters were chosen to loosely approximate 4000 kDa poly(ethylene oxide). Simulations are performed with a novel Stokes flow formulation of the immersed boundary method for the capsules, combined with Brownian dynamics for the polymer molecules. The results for isolated capsules in shear indicate that at the very low concentrations considered here, polymers have a little effect on the capsule shape. In the case of pair collisions, the effect of polymer is strongly dependent on the elastic properties of the capsules’ membranes. For neoHookean capsules or for Skalak capsules with only a small penalty for area change, the net displacement in the gradient direction after collision is virtually unaffected by the polymer. For Skalak capsules with a large penalty for area change, polymers substantially decrease the net displacement when compared to the Newtonian case and the effect is enhanced upon increasing the polymer concentration. The differences between the polymer effects in the various cases are associated with the extensional flow generated in the region between the capsules as they leave the collision. The extension rate is highest when there is a strong resistance to a change in the membrane area and is substantially decreased in the presence of polymer.

Experimental study of dispersion and miscible viscous fingering of initially circular samples in HeleShaw cells
View Description Hide DescriptionExperimental studies are conducted to analyze dispersion and miscibleviscous fingering of initially circular samples of a given solution displaced linearly at constant speed by another solution in horizontal HeleShaw cells (two glass plates separated by a thin gap). In the stable case of a dyed water sample having the same viscosity as that of displacing nondyed water, we analyze the transition between dispersive and advective transport of the passive scalar displaced linearly. At low displacement speed and after a certain time, the length of the sample increases as a square root of time allowing to compute the value of a dispersion coefficient. At larger injection speed, the displacement remains advective for the duration of the experiment, with a length of the sample increasing linearly in time. A parametric study allows to gain insight into the switch from one regime to another as a function of the gap width of the cell. In the unstable case of viscous glycerol samples displaced by dyed water, the rear interface of the sample where less viscous water pushes more viscous glycerol is unstable with regard to viscous fingering. The interface deforms into fingers, the number and size of which depend on the viscosity ratio between the two solutions and on the displacement speed. We study the influence of these viscous fingering phenomena on the increased spreading of the sample for various mobility ratios and injection speeds.