Volume 23, Issue 7, July 2011

The ability to generate complete, or almost complete, chaotic mixing is of great interest in numerous applications, particularly for microfluidics. For this purpose, we propose a strategy that allows us to quickly target the parameter values at which complete mixing occurs. The technique is applied to a time periodic, twodimensional electroosmotic flow with spatially and temporally varying HelmholtzSmoluchowski slip boundary conditions. The strategy consists of following the linear stability of some key periodic pathlines in parameter space (i.e., amplitude and frequency of the forcing), particularly through the bifurcation points at which such pathlines become unstable.
 LETTERS


The bipolar behavior of the Richtmyer–Meshkov instability
View Description Hide DescriptionA numerical study of the evolution of the multimode planar RichtmyerMeshkov instability(RMI) in a lightheavy (airSF_{6}, Atwood number A = 0.67) configuration involving a Mach number Ma = 1.5 shock is carried out. Our results demonstrate that the initial material interface morphology controls the evolution characteristics of RMI (for fixed A, Ma), and provide a significant basis to develop metrics for transition to turbulence. Depending on initial rms slope of the interface, RMI evolves into linear or nonlinear regimes, with distinctly different flow features and growth rates, turbulence statistics, and materialmixing rates. We have called this the bipolar behavior of RMI. Some of our findings are not consistent with heuristic notions of mixing in equilibrium turbulence: more turbulent flow—as measured by spectral bandwidth, can be associated with higher materialmixing but, paradoxically, to lower integral measures of turbulent kinetic energy and mixing layer width.
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 ARTICLES

 Biofluid Mechanics

Modeling hydrodynamic selfpropulsion with Stokesian Dynamics. Or teaching Stokesian Dynamics to swim
View Description Hide DescriptionWe develop a general framework for modeling the hydrodynamic selfpropulsion (i.e., swimming) of bodies (e.g., microorganisms) at low Reynolds number via Stokesian Dynamics simulations. The swimming body is composed of many spherical particles constrained to form an assembly that deforms via relative motion of its constituent particles. The resistance tensor describing the hydrodynamic interactions among the individual particles maps directly onto that for the assembly. Specifying a particular swimming gait and imposing the condition that the swimming body is force and torquefree determine the propulsive speed. The body’s translational and rotational velocities computed via this methodology are identical in form to that from the classical theory for the swimming of arbitrary bodies at low Reynolds number. We illustrate the generality of the method through simulations of a wide array of swimming bodies: pushers and pullers, spinners, the Taylor/Purcell swimming toroid, Taylor’s helical swimmer, Purcell’s threelink swimmer, and an amoebalike body undergoing largescale deformation. An open source code is a part of the supplementary material and can be used to simulate the swimming of a body with arbitrary geometry and swimming gait.
 Micro and Nanofluid Mechanics

Frictional slip lengths for unidirectional superhydrophobic grooved surfaces
View Description Hide DescriptionThe exact solutions due to Philip [ZAMP 23, 353 (1972)] for Stokes shear flow over a periodic array of noshear slots embedded in a noslip surface are generalized to account for an arbitrary pattern of noshear slots in each period window. The slots, or grooves, in each period window run parallel to each other, and are of infinite length, but their widths and separations can be specified arbitrarily. Explicit solutions are found both for longitudinal and transverse flows over the composite grooved surface. Analytical expressions for the transverse and longitudinal slip lengths associated with the microstructured surfaces are then found as functions of the geometrical parameters characterizing the surface. The formulae are relevant to a wide class of flow geometries and are expected to provide a useful tool in the design, analysis, and optimization of the friction properties of grooved microstructured superhydrophobicsurfaces. The results are used to show that introducing even a very small wetted region in a noshear slot can have a significant influence on the effective slip length.

Complete chaotic mixing in an electroosmotic flow by destabilization of key periodic pathlines
View Description Hide DescriptionThe ability to generate complete, or almost complete, chaotic mixing is of great interest in numerous applications, particularly for microfluidics. For this purpose, we propose a strategy that allows us to quickly target the parameter values at which complete mixing occurs. The technique is applied to a time periodic, twodimensional electroosmotic flow with spatially and temporally varying HelmholtzSmoluchowski slip boundary conditions. The strategy consists of following the linear stability of some key periodic pathlines in parameter space (i.e., amplitude and frequency of the forcing), particularly through the bifurcation points at which such pathlines become unstable.

Concentration polarization and secondkind electrokinetic instability at an ionselective surface admitting normal flow
View Description Hide DescriptionThe passage of ionic current across a chargeselective surface has been studied for over a century and is relevant to wellestablished processes such as electrodialysis, electrodeposition, and electrochromatography. Recent years have witnessed a resurgence of interest in this subject, motivated by experiments demonstrating chargeselective transport of ions and solutes in nanofluidic devices. In this paper, we revisit and build upon the prototypical problem of onedimensional ion transport across a flat ideally ionselective surface, by examining the influence of imposed fluid flows on concentration polarization, overlimiting current, and secondkind (nonequilibrium) electroosmoticinstability at the surface. Specifically, we consider a simple model system of a cationselective surface or membrane that admits a uniform fluid flow across itself. The membrane resides against a binary symmetric electrolyte, whose concentration is uniform in a “wellmixed” region at a prescribed distance from the membrane. A potential difference across the system drives an ionic current, leading to concentration polarization in the “unstirred layer” between the membrane and wellmixed bulk. The concentration polarization profile reflects a balance between advection of ions with the imposed “normal flow” and diffusion. The relative importance of these effects is parameterized by a Pećlet number Pe; notably, Pe is a signed quantity as the flow can be imposed toward or away from the membrane. An asymptotic analysis in the thinDebyelayer limit reveals a significant impact of normal flow on concentration polarization and the advectiondiffusion limiting current across the membrane. In particular, there exists a nonlinear concentration profile in the unstirred layer for nonzero Pe, in contrast to the familiar linear (diffusive) concentration polarization at Pe = 0. Next, we use matched asymptotic expansions to explore the structure of the unstirred layer at overlimiting currents, wherein a nonequilibrium spacecharge layer develops near the membrane surface. A key step in this process is the derivation of a “generalized master equation” for the electric field across the unstirred layer. Finally, we examine the instability of the quiescent concentration polarization resulting from secondkind electroosmotic slip in the spacecharge layer. A linear stability analysis shows that normal flow can either enhance or retard the instability, depending on the flow direction.

Generation of small monodisperse bubbles in axisymmetric Tjunction: The role of swirl
View Description Hide DescriptionThe dynamics of microbubble formation in an axisymmetric Tjunction for a gasliquid system is analyzed. The approach adopted involves the creation of a tapering gasliquidmeniscus from which a steady gas ligament issues by the introduction of a coaxial swirl in the liquid stream. A simple and easy geometry (an axisymmetric Tjunction) suffices to introduce the swirl and to stabilize the meniscus, leading to the formation of small monodisperse bubbles. Full threedimensional simulations (3D) have also been conducted to show that, even when the liquid injection is not perfectly axisymmetric, the bubbles generated under conditions of some focusing swirl are distinctively smaller than bubbles created in the absence of swirl. In such cases, the bubbles, produced at the trail of the vortex axis, become a serendipitous tool to visualize the nonaxisymmetrical behavior of the vortex core, as shown by the simulations.

Physical origins of apparently enhanced viscosity of interfacial fluids in electrokinetic transport
View Description Hide DescriptionA key concept in classical electrokinetic theories is that the viscosity of interfacial fluids is much higher than that of bulk fluids, and this concept is indirectly supported by experimental evidence and molecular dynamics simulations. However, a universal mechanism that encompasses the breadth of experimental evidence is still lacking. Here we show, using molecular dynamics simulations, that the “apparent” thickening of interfacial fluids in electrokinetic transport near molecularly smooth surface originates mainly from the fact that ionwall interactions are not accounted for in the hydrodynamic model of classical electrokinetic theories. Specifically, strong ionwall interactions cause intermittent adsorption of ions on charged walls, and this in turn leads to loss of driving force for flow and screening of fluid flow by the adsorbed ions. Although not considered in the classical electrokinetic theories, these effects can significantly suppress electrokinetic transport. Consequently, when the classical theories are used to interpret the electrokinetic data, the viscosity of interfacial fluids appears to be greatly enhanced even if their material viscosity is similar to that of the bulk fluids. This mechanism for the apparent thickening of interfacial fluids is applicable to electrokinetic transport near any type of charged surface.

Elastic deformation due to tangential capillary forces
View Description Hide DescriptionA sessile liquiddrop can deform the substrate on which it rests if the solid is sufficiently “soft.” In this paper we compute the detailed spatial structure of the capillary forces exerted by the drop on the solid substrate using a model based on Density Functional Theory. We show that, in addition to the normal forces, the drop exerts a previously unaccounted tangential force. The resultant effect on the solid is a pulling force near the contact line directed towards the interior of the drop, i.e., not along the interface. The resulting elastic deformations of the solid are worked out and illustrate the importance of the tangential forces.

A theoretical study of inducedcharge dipolophoresis of ideally polarizable asymmetrically slipping Janus particles
View Description Hide DescriptionWe consider the non linear electrophoretic transport of uncharged, ideally polarizable hydrodynamic Janus spheres, the inhomogeneity of which is produced by a variable Navier slip condition at the particle surface. A general, three dimensional formulation enabling calculation of the electrophoreticmobility of any patchy particle, with an arbitrary tensorial slip boundary condition is provided. The solution avoids the common assumption of an infinitely thin electric double layer (λ) and Navier slip coefficient () and is thereby valid for finite values of these parameters, which is of particular importance at the nanoscale. The specific case of a Janus sphere, consisting of two equal hemispheres, each with a different but constant slip boundary condition is solved semianalytically and numerically. In the instance where the slip coefficients at each hemisphere are equal, induced charge electroosmotic flow is evident at an increased rate as compared to a homogeneous sphere with no slip. If the slip coefficients differ from each other, the particle is found to selfalign with the electric field and travel with the slip surface facing forward. The increased pumping rates and mobility found in the cases of the homogeneous and Janus spheres respectively, occur as a function of the ratio and are most significant for the combination of a thin electric double layer(EDL) and large slip length. However, it is also illustrated that the size of the EDL independently dominates the effects of slip.

Numerical study of the formation process of ferrofluid droplets
View Description Hide DescriptionThis paper numerically investigates the influence of a uniform magnetic field on the dropletformation process at a microfluidicflow focusing configuration. The mathematical model was formulated by considering the balance of forces such as interfacial tension, magnetic force, and viscous stress across the liquid/liquid interface. A linearly magnetizable fluid was assumed. The magnetic force acts as a body force where the magnetic permeability jumps across the interface. The governing equations were solved with finite volume method on a Cartesian fixed staggered grid. The evolution of the interface was captured by the particle level set method. The code was validated with the equilibrium steady state of a ferrofluiddroplet exposed to a uniform magnetic field. The evolution of the dropletformation in a flow focusing configuration was discussed. The paper mainly analyzes the effects of magnetic Bond number and the susceptibility on the velocity field and the droplet size. The droplet size increased with increasing magnetic strength and susceptibility.
 Interfacial Flows

Minimising wave drag for free surface flow past a twodimensional stern
View Description Hide DescriptionFree surface flow past a twodimensional semiinfinite curved plate is considered, with emphasis given to solving the shape of the resulting wave train that appears downstream on the surface of the fluid. This flow configuration can be interpreted as applying near the stern of a wide blunt ship. For steady flow in a fluid of finite depth, we apply the WienerHopf technique to solve a linearised problem, valid for small perturbations of the uniform stream. Weakly nonlinear results obtained by a forced KdV equation are also presented, as are numerical solutions of the fully nonlinear problem, computed using a conformal mapping and a boundary integral technique. By considering different families of shapes for the semiinfinite plate, it is shown how the amplitude of the waves can be minimised. For plates that increase in height as a function of the direction of flow, reach a local maximum, and then point slightly downwards at the point at which the free surface detaches, it appears the downstream wavetrain can be eliminated entirely

Novel pattern forming states for Marangoni convection in volatile binary liquids
View Description Hide DescriptionWe describe experiments on Marangoni convection in thin evaporating liquid films. The films are binary mixtures of ethanol and water exposed to the ambient room air during all experimental runs. These experiments exhibit a variety of different, often novel, patterns, depending on the concentration (weight fraction) c of ethanol. Among these are mobile circular convective patterns, which have not been previously observed, to our knowledge. The convective patterns evolve due to the evaporation of both the solvent and the solute, and their size increases substantially with the initial concentration c. The patterns reported here differ from those found in binary mixtures of NaCl and water, where only water evaporates.

Contact line dynamics on heterogeneous surfaces
View Description Hide DescriptionContact line dynamics on rough or chemically heterogeneous surfaces has been a subject of great interest. Most previous work focused on the issue of contact angle hysteresis in the static limit. This paper is devoted to the study of contact line dynamics on a chemically patternedsurface over a wide range of contact line speed. Numerical simulations are carried out for two immiscible fluids confined in a channel and driven by either the shear motion of the two confining walls or an external force. It is found that in the lowspeed regime when the averaged contact line speed , with being the surface tension of the fluid interface and the friction coefficient at the contact line, the behavior of the contact line dynamics is very similar to that of the static limit, namely it undergoes a stickslip motion and the contact angle exhibits hysteretic behavior. At finite speed, the stickslip behavior gradually diminishes, and the contact line motion becomes more smooth. The effect of these microscale dynamics on the averaged force between the fluid and the solid is investigated. It is found that while the friction force increases linearly with the averaged contact line speed, the force at the contact line due to the defect decreases with U. It is nonzero in the static limit and this is the cause of the contact angle hysteresis. As a result, the total force at the contact line may become nonmonotone as a function of the contact line speed. This gives rise to an unstable regime for the contact line dynamics, which is indeed observed in the simulation when the dynamics is driven by an external force.

Sinking of small sphere at low Reynolds number through interface
View Description Hide DescriptionA dense solid sphere gently released on an airliquid interface slowly sinks into liquid due to gravity, while the motion is resisted by viscous and capillary forces. Here, we predict the sinking velocity of the interfacestraddling sphere by a simplified model and experimentally corroborate the results. The viscous drag on the sphere is determined by integrating the surface stress, which is the solution of the Stokes equation, over the wetted area that changes with time. To compute the interfacial tension force that depends on the meniscus profile, we solve the dynamic boundary condition for the normal and tangential stresses at the airliquid interface. The predicted sinking velocity, a function of the sphere density and radius, liquid density, viscosity and surface tension, and the dynamic contact angle, is in good agreement with the experimental measurements except for the late stages when meniscus snapping occurs. We also construct a scaling law for the steady velocity of a sinking sphere, which gives the characteristic sinking time.

Forced instability of coreannular flow in capillary constrictions
View Description Hide DescriptionInstability of fluid cylinders and jets, a highly nonlinear hydrodynamic phenomenon, has fascinated researchers for nearly 150 years. A subset of the phenomenon is the coreannular flow, in which a nonwetting core fluid and a surrounding wallwetting annulus flow through a solid channel. The model, for example, represents the flow of oil in petroleum reservoirs. The flow may be forced to break up when passing through a channel’s constriction. Although it has long been observed that the breakup occurs near the neck of the constriction, the exact conditions for the occurrence of the forced breakup and its dynamic theory have not been understood. Here, we test a simple geometric conjecture that the fluid will always break in the constrictions of all channels with sufficiently long wavelengths, regardless of the fluid properties. We also test a theory of the phenomenon. Four constricted glass tubes were fabricated above and below the critical wavelength required for the fluid disintegration. In a direct laboratory experiment, the breakup occurred according to the conjecture: the fluids were continuous in the shorter tubes but disintegrated in the longer tubes. The evolution of the interface to its pinchoff was recorded using highspeed digital photography. The experimentally observed coreannulus interface profiles agreed well with the theory, although the total durations of the process agreed less satisfactorily. Nonetheless, as the theory predicts, the ratio between the experimental and theoretical times of the breakup process tends to one with decreasing capillary number. The breakup condition and the dynamic theory of fluid disintegration in constricted channels can serve as quantitative models of this important natural and technical phenomenon.

Particle accumulation on periodic orbits by repeated free surface collisions
View Description Hide DescriptionThe motion of small particles suspended in cylindrical thermocapillary liquid bridges is investigated numerically in order to explain the experimentally observed particle accumulation structures (PAS) in steady two and timedependent threedimensional flows. Particles moving in this flow are modeled as perfect tracers in the bulk, which can undergo collisions with the free surface. By way of freesurface collisions the particles are transferred among different streamlines which represents the particle trajectories in the bulk. The interstreamline transferprocess near the free surface together with the passive transport through the bulk is used to construct an iterative map that can describe the accumulation process as an attraction to a stable fixed point which represents PAS. The flowtopology of the underlying azimuthally traveling hydrothermal wave turns out to be of key importance for the existence of PAS. In a frame of reference exactly rotating with the hydrothermal wave the threedimensional flow is steady and exhibits coexisting regular and chaotic streamlines. We find that particles are attracted to accumulation structures if a closed regular streamline exists in the rotating frame of reference which closely approaches the free surface locally. Depending on the closed streamline and the particle radius PAS can arise as a specific trajectory which winds about the closed regular streamline or as the surface of a particular stream tube containing the closed streamline.
 Viscous and NonNewtonian Flows

The influence of pressure relaxation on the structure of an axial vortex
View Description Hide DescriptionGoverning equations including the effects of pressure relaxation have been utilized to study an incompressible, steadystate viscous axial vortex with specified farfield circulation. When sound generation is attributed to a velocity gradient tensorpressure gradient product, the modified conservation of momentum equations that result yield an exact solution for a steady, incompressible axial vortex. The vortexvelocity profile has been shown to closely approximate experimental vortexmeasurements in air and water over a wide range of circulationbased Reynolds numbers. The influence of temperature and humidity on the pressure relaxation coefficient in air has been examined using theoretical and empirical approaches, and published axial vortex experiments have been employed to estimate the pressure relaxation coefficient in water. Nonequilibrium pressure gradient forces have been shown to balance the viscous stresses in the vortex core region, and the predicted pressure deficits that result from this nonequilibrium balance can be substantially larger than the pressure deficits predicted using a Bernoulliequation approach. Previously reported pressure deficit distributions for dust devils and tornados have been employed to validate the nonequilibrium pressure deficit predictions.

Electrohydrodynamic quenching in polymer melt electrospinning
View Description Hide DescriptionInfrared thermal measurements on polymer melt jets in electrospinning have revealed rapid quenching by ambient air, an order of magnitude faster than predicted by the classical Kase and Matsuo correlation. This drastic heat transfer enhancement can be linked to electrohydrodynamic (EHD) effects. Analysis of EHDdriven air flow was performed and included into a comprehensive model for polymer meltelectrospinning. The analysis was validated by excellent agreement of both predicted jet radius and temperature profiles with experimental results for electrospinning of Nylon6 (N6), polypropylene (PP), and polylactic acid (PLA) melts. Based on this analysis, several methods that can be used to inhibit or enhance the quenching are described.

On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media
View Description Hide DescriptionWe report on the controversial dependence of the inertial correction to Darcy’s law upon the filtration velocity (or Reynolds number) for onephase Newtonian incompressible flow in modelporous media. Our analysis is performed on the basis of an upscaled form of the NavierStokes equation requiring the solution of both the microscale flow and the associated closure problem. It is carried out with a special focus on the different regimes of inertia (weak and strong inertia) and the crossover between these regimes versus flow orientation and structural parameters, namely porosity and disorder. For ordered structures, it is shown that (i) the tensor involved in the expression of the correction is generally not symmetric, despite the isotropic feature of the permeability tensor. This is in accordance with the fact that the extra force due to inertia exerted on the structure is not pure drag in the general case; (ii) the Forchheimer type of correction (which strictly depends on the square of the filtration velocity) is an approximation that does not hold at all for particular orientations of the pressure gradient with respect to the axes of the structure; and (iii) the weak inertia regime always exists as predicted by theoretical developments. When structural disorder is introduced, this work shows that (i) the quadratic dependence of the correction upon the filtration velocity is very robust over a wide range of the Reynolds number in the strong inertia regime; (ii) the Reynolds number interval corresponding to weak inertia, that is always present, is strongly reduced in comparison to ordered structures. In conjunction with its relatively small magnitude, it explains why this weak inertia regime is most of the time overlooked during experiments on natural media. In all cases, the Forchheimer correction implies that the permeability is different from the intrinsic one.

Controllable adhesion using fieldactivated fluids
View Description Hide DescriptionWe demonstrate that fieldresponsive magnetorheological fluids can be used for variablestrength controllable adhesion. The adhesive performance is measured experimentally in tensile tests (a.k.a. probetack experiments) in which the magnetic field is provided by a cylindrical permanent magnet. Increasing the magnetic field strength induces higher peak adhesive forces. We hypothesize that the adhesion mechanism arises from the shear resistance of a yield stressfluid in a thin gap. This hypothesis is supported by comparing the experimentally measuredadhesive performance to the response predicted by a lubrication model for a nonNewtonian fluid with a fielddependent yield stress. The model predictions are in agreement with experimental data up to moderate field strengths. Above a critical magnetic field strength the model overpredicts the experimentally measured values indicating nonideal conditions such as local fluid dewetting from the surface.