Volume 26, Issue 8, August 2014
Index of content:

We solve for the first time the classical linear CauchyPoisson problem—the time evolution of an initial surface disturbance—when a shear current of uniform vorticity is present beneath the surface. The solution is general, including the effects of gravity, surface tension, and constant finite depth. The particular case of an initially Gaussian disturbance of width b is studied for different values of three system parameters: a “shear Froude number” (S is the vorticity), the Bond number and the depth relative to the initial perturbation width. Different phase and group velocity in different directions yield very different wave patterns in different parameter regimes when the shear is strong, and the wellknown pattern of diverging ring waves in the absence of shear can take on very different qualitative behaviours. For a given shear Froude number, both finite depth and nonzero capillary effects are found to weaken the influence of the shear on the resulting wave pattern. The various patterns are analysed and explained in light of the shearmodified dispersion relation.
 LETTERS


Analytic solutions for three dimensional swirling strength in compressible and incompressible flows
View Description Hide DescriptionEigenvalues of the 3D critical point equation (∇u)ν = λν are normally computed numerically. In the letter, we present analytic solutions for 3D swirling strength in both compressible and incompressible flows. The solutions expose functional dependencies that cannot be seen in numerical solutions. To illustrate, we study the difference between using fluctuating and total velocity gradient tensors for vortex identification. Results show that mean shear influences vortex detection and that distortion can occur, depending on the strength of mean shear relative to the vorticity at the vortex center.

 ARTICLES


Biofluid Mechanics

Hydrodynamic tracer diffusion in suspensions of swimming bacteria
View Description Hide DescriptionWe present theoretical predictions, simulations, and experimental measurements of the diffusion of passive, Brownian tracer particles in the bulk of threedimensional suspensions of swimming bacteria performing runtumble random walks. In the theory, we derive an explicit expression for the “hydrodynamic” tracer diffusivity that results from the fluid disturbances created by a slenderbody model of bacteria by ensemble averaging the mass conservation equation of the tracer over the space of tracerbacterium interactions which are assumed to be binary. The theory assumes that the orientations of the bacterium before and after a tumble are uncorrelated and the fluid velocity disturbance created by the bacterium is small compared to its swimming speed. The dependence of the nondimensional hydrodynamic diffusivity obtained by scaling the dimensional hydrodynamic diffusivity by nL ^{3} U s L on the persistence in bacterial swimming and the Brownian diffusivity of the tracer are studied in detail through two nondimensional parameters—a Peclet number Pe = U s L/D which is the ratio of the time scale of bacterial swimming to the tracer diffusion time scale and a nondimensional persistence time τ^{*} = U s τ/L obtained by scaling the dimensional bacterial persistence time by the time that a bacterium takes to swim over a distance equal to its length. Here, n, U s , τ, and L are the concentration, swimming speed, tumbling time, and the overall length of the bacteria, respectively, and D is the Brownian diffusivity of the tracer. is found to be a monotonically increasing function of τ^{*} and a nonmonotonic function of Pe with a Pe ^{1/2} scaling in the Pe ≪ 1 limit, an intermediate peak and a constant value in the Pe ≫ 1 limit for the typical case of wildtype bacteria with τ^{*} = O(1). In the simulation study we compute the bacterial contribution to the tracer diffusivity from explicit numerical simulations of binary tracerbacterium interactions to examine the validity of the weak disturbance assumption made in the theory, and to investigate the effects of correlations in the pre and posttumble bacterium orientations and the excluded volume (steric) interactions between the bacterium and the tracer. It is found that the weak disturbance assumption does not have a statistically significant effect on and correlations among pre and posttumble bacterium orientations and bacteriumtracer excluded volume interactions are found to enhance the tracer diffusivity by modest but statistically significant factors. Finally, we measure the effective diffusion coefficient of 1.01 μm diameter colloidal tracer particles in the bulk of a suspension of wildtype E. Coli cells and compare the experimental measurements with the predictions made by the theory and simulations.

Locomotion in complex fluids: Integral theorems
View Description Hide DescriptionThe biological fluids encountered by selfpropelled cells display complex microstructures and rheology. We consider here the general problem of lowReynolds number locomotion in a complex fluid. Building on classical work on the transport of particles in viscoelastic fluids, we demonstrate how to mathematically derive three integral theorems relating the arbitrary motion of an isolated organism to its swimming kinematics in a nonNewtonian fluid. These theorems correspond to three situations of interest, namely, (1) squirming motion in a linear viscoelastic fluid, (2) arbitrary surface deformation in a weakly nonNewtonian fluid, and (3) smallamplitude deformation in an arbitrarily nonNewtonian fluid. Our final results, valid for a wideclass of swimmer geometry, surface kinematics, and constitutive models, at most require mathematical knowledge of a series of Newtonian flow problems, and will be useful to quantity the locomotion of biological and synthetic swimmers in complex environments.

Micro and Nanofluid Mechanics

Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies
View Description Hide DescriptionRecent experiments showed that standing acoustic waves could be exploited to induce selfpropulsion of rigid metallic particles in the direction perpendicular to the acoustic wave. We propose in this paper a physical mechanism for these observations based on the interplay between inertial forces in the fluid and the geometrical asymmetry of the particle shape. We consider an axisymmetric rigid nearsphere oscillating in a quiescent fluid along a direction perpendicular to its symmetry axis. The kinematics of oscillations can be either prescribed or can result dynamically from the presence of an external oscillating velocity field. Steady streaming in the fluid, the inertial rectification of the timeperiodic oscillating flow, generates steady stresses on the particle which, in general, do not average to zero, resulting in a finite propulsion speed along the axis of the symmetry of the particle and perpendicular to the oscillation direction. Our derivation of the propulsion speed is obtained at leading order in the Reynolds number and the deviation of the shape from that of a sphere. The results of our model are consistent with the experimental measurements, and more generally explains how time periodic forcing from an acoustic field can be harnessed to generate autonomous motion.

Particle dynamics and rapid trapping in electroosmotic flow around a sharp microchannel corner
View Description Hide DescriptionWe study here the curious particle dynamics resulting from electroosmotic flow around a microchannel junction corner whose dielectric walls are weakly polarizable. The hydrodynamic velocity field is obtained via superposition of a linear irrotational term associated with the equilibrium zeta potentials of both the microchannel and particle surfaces and the nonlinear inducedcharge electroosmotic flow which originates from the interaction of the externally applied electric field on the charge cloud it induces at the solidliquid interface. The particle dynamics are analyzed by considering dielectrophoretic forces via the addition of a mobility term to the flow field in the limit of Stokes drag law. The former, nondivergence free term is responsible for migration of particles towards the sharp microchannel junction corner, where they can potentially accumulate. Experimental observations of particle trapping for various applied electric fields and microparticle size are rationalized in terms of the growing relative importance of the dielectrophoretic force and inducedcharge contributions to the global velocity field with increasing intensity of the externally applied electric field.

Multiscale liquid drop impact on wettable and textured surfaces
View Description Hide DescriptionThe impact of microscopic liquid drops on solids with a variety of surface characteristics is studied using numerical simulations. The focus is on relatively low impact velocities leading to bouncing or spreading drops, and the effects of wettability. Molecular dynamics and lattice Boltzmann simulation methods are used for nanometersized and continuum drops, respectively, and the results of the two methods are compared in terms of scaled variables. We consider surfaces which are flat, curved or pillared, with either homogeneous interactions or crossshaped patterns of wettability. In most situations we observe similar drop behavior at both length scales; the two methods agree best at low impact velocities on wettable surfaces while discrepancies are most pronounced for strongly hydrophobic surfaces and for higher velocities.

Evidence of slippage breakdown for a superhydrophobic microchannel
View Description Hide DescriptionA full characterization of the water flow past a silicon superhydrophobic surface with longitudinal microgrooves enclosed in a microfluidic device is presented. Fluorescence microscopy images of the flow seeded with fluorescent passive tracers were digitally processed to measure both the velocity field and the position and shape of the liquidair interfaces at the superhydrophobic surface. The simultaneous access to the meniscus and velocity profiles allows us to put under a strict test the noshear boundary condition at the liquidair interface. Surprisingly, our measurements show that air pockets in the surface cavities can sustain nonzero interfacial shear stresses, thereby hampering the friction reduction capabilities of the surface. The effects of the meniscus position and shape as well as of the liquidair interfacial friction on the surface performances are separately assessed and quantified.

Interfacial Flows

Pilotwave hydrodynamics in a rotating frame: Exotic orbits
View Description Hide DescriptionWe present the results of a numerical investigation of droplets walking on a rotating vibrating fluid bath. The drop's trajectory is described by an integrodifferential equation, which is simulated numerically in various parameter regimes. As the forcing acceleration is progressively increased, stable circular orbits give way to wobbling orbits, which are succeeded in turn by instabilities of the orbital center characterized by steady drifting then discrete leaping. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but its statistical behavior reflects the influence of the unstable orbital solutions. The study results in a complete regime diagram that summarizes the dependence of the walker's behavior on the system parameters. Our predictions compare favorably to the experimental observations of Harris and Bush [“Droplets walking in a rotating frame: from quantized orbits to multimodal statistics,” J. Fluid Mech.739, 444–464 (2014)].

Marangoni waves in a twolayer film under the action of an inclined temperature gradient
View Description Hide DescriptionThe development of the longwave deformational instability of a thermocapillary flow in a twolayer film under the action of an inclined temperature gradient is studied in the framework of the lubrication approximation. The stability boundaries with respect to different oscillatory modes are calculated by means of the linear stability theory. In a contradistinction to the case of a vertical temperature gradient, these boundaries strongly depend on the direction of the wave propagation. Numerical simulations of spatially periodic nonlinear regimes are fulfilled. It is shown that because of the anisotropy of the problem, the most typical kind of patterns is a traveling wave. For small inclination of the temperature gradient, temporally quasiperiodic waves are observed. A number of new threedimensional traveling wave planforms is revealed.

Effects of hierarchical features on longevity of submerged superhydrophobic surfaces with parallel grooves
View Description Hide DescriptionWhile the air–water interface over superhydrophobic surfaces decorated with hierarchical micro or nanosized geometrical features have shown improved stability under elevated pressures, their underwater longevity—the time that it takes for the surface to transition to the Wenzel state—has not been studied. The current work is devised to study the effects of such hierarchical features on the longevity of superhydrophobic surfaces. For the sake of simplicity, our study is limited to superhydrophobic surfaces composed of parallel grooves with side fins. The effects of fins on the critical pressure—the pressure at which the surface starts transitioning to the Wenzel state—and longevity are predicted using a mathematical approach based on the balance of forces across the air–water interface. Our results quantitatively demonstrate that the addition of hierarchical fins significantly improves the mechanical stability of the air–water interface, due to the high advancing contact angles that can be achieved when an interface comes in contact with the fins sharp corners. For longevity on the contrary, the hierarchical fins were only effective at hydrostatic pressures below the critical pressure of the original smoothwalled groove. Our results indicate that increasing the length of the fins decreases the critical pressure of a submerged superhydrophobic groove but increases its longevity. Increasing the thickness of the fins can improve both the critical pressure and longevity of a submerged groove. The mathematical framework presented in this paper can be used to customdesign superhydrophobic surfaces for different applications.

Initial surface disturbance on a shear current: The CauchyPoisson problem with a twist
View Description Hide DescriptionWe solve for the first time the classical linear CauchyPoisson problem—the time evolution of an initial surface disturbance—when a shear current of uniform vorticity is present beneath the surface. The solution is general, including the effects of gravity, surface tension, and constant finite depth. The particular case of an initially Gaussian disturbance of width b is studied for different values of three system parameters: a “shear Froude number” (S is the vorticity), the Bond number and the depth relative to the initial perturbation width. Different phase and group velocity in different directions yield very different wave patterns in different parameter regimes when the shear is strong, and the wellknown pattern of diverging ring waves in the absence of shear can take on very different qualitative behaviours. For a given shear Froude number, both finite depth and nonzero capillary effects are found to weaken the influence of the shear on the resulting wave pattern. The various patterns are analysed and explained in light of the shearmodified dispersion relation.

Line tension and reduction of apparent contact angle associated with electric double layers
View Description Hide DescriptionThe line tension of an electrolyte wetting a nonpolar substrate is computed analytically and numerically. The results show that, depending on the value of the apparent contact angle, positive or negative line tension values may be obtained. Furthermore, a significant difference between Young's contact angle and the apparent contact angle measured several Debye lengths remote from the threephase contact line occurs. When applying the results to water wetting highly charged surfaces, line tension values of the same order of magnitude as found in recent experiments can be achieved. Therefore, the theory presented may contribute to the understanding of line tension measurements and points to the importance of the electrostatic line tension. Being strongly dependent on the interfacial charge density, electrostatic line tension is found to be tunable via the pH value of the involved electrolyte. As a practical consequence, the stability of nanoparticles adsorbed at fluidfluid interfaces is predicted to be dependent on the pH value. The theory is suited for future incorporation of effects due to surfactants where even larger line tension values can be expected.

Selfsimilar solution for oblique impact of a water column with sharp front on a wall and its zero inner angle steady limit
View Description Hide DescriptionThe hydrodynamic problem of a threedimensional (3D) water column impacting on a solid wall is investigated. The focus is on cases of selfsimilar flow and its related implications in physics. It is demonstrated that limiting process of the selfsimilar flow can become the steady flow. The problem involves a free surface whose shape is unknown and on which the boundary conditions are nonlinear. It is solved through boundary element method (BEM) with quadrilateral elements within an iterative procedure. The application of the BEM to such a problem has some major challenges. During the process, mesh used in the BEM is regularly regenerated to follow the deformation of the free surface and the data in the old mesh are transferred to the new one. Coordinate rotation technique is used to resolve the difficulties caused by the multivalued free surface, together with the technique for the thin liquid film over the wall surface. Results are provided for the free surface shapes and pressure distributions for perpendicular and oblique impact cases, including both selfsimilar flow and its limiting steady flow. Their physics about jet root, the peak pressure and its location, as well as the limiting process to the steady flow are discussed.

Subharmonic resonant wave interactions in the presence of a linear interfacial instability
View Description Hide DescriptionThis work considers the role of nonlinear subharmonic resonant wave interactions in the development of interfacial waves, which may be under the influence of a linear interfacial instability, in an inviscid twofluid stratified flow through a horizontal channel. We begin by examining the case of resonant interactions between one linearly unstable mode and its subharmonic that may be linearly stable or unstable. Using the method of multiple scales, we derive the nonlinear interaction equations governing the time evolution of interacting wave amplitudes. These nonlinear equations account for the combined effects of both the nonlinear resonant interaction and the linear instability. We show that through this nonlinear coupling, the linearly stable subharmonic mode can achieve faster than exponential growth. It is found that such a mechanism is capable of generating largeamplitude long waves that are stable by linear stability analysis. The analytical predictions are cross validated by comparisons with direct nonlinear numerical simulations based on a more general perturbation based spectral method. Good agreement between the analytical and numerical solutions is observed. The more complicated case where a single mode is simultaneously involved in multiple (subharmonic and triad) resonances is also investigated numerically. The results demonstrate that chains of resonances can permit the energy generated by the linear instability, among high wavenumber components, to be passed across the spectrum to the longest wave components creating an efficient mechanism for the generation of largeamplitude long waves from unstable short waves.

Creating localizeddroplet train by traveling thermal waves
View Description Hide DescriptionWe investigate the nonlinear dynamics of a twolayer system consisting of a thin liquid film and an overlying gas layer driven by the Marangoni instability induced by thermal waves propagating along the solid substrate. In the case of a stationary thermal wave with sufficiently large amplitude and Marangoni number, liquid film rupture takes place with a flattish wide trough. For sufficiently small but not too small frequencies of the thermal wave, a periodic structure consisting of localized drops interconnected by thin liquid bridges emerges. This train of drops travels unidirectionally along the heated substrate following the thermal wave. For larger thermal wave frequencies, the thickness of the bridges increases enabling fluid flow between the neighboring drops. The droptrain regimes may be utilized in microfluidic applications for directed transport of liquid content enclosed in drops formed by thermocapillary forces.

Dynamics of stepemulsification: From a single to a collection of emulsion droplet generators
View Description Hide DescriptionMicrofluidics has proven to be an efficient tool for making fine and calibrated emulsion droplets. The parallelization of drop makers is required for producing large amounts. Here, we investigate the generation of emulsion drops along a series of shallow microchannels emerging in a deep one, in other words the stepemulsification process. The dynamics of a single drop maker is first characterized as a function of interfacial tension and viscosities of both phases. The characteristic time scale of drop formation, namely, the necking time that finally sets drop size, is shown to be principally governed by the outer phase viscosity to interfacial tension ratio with a minor correction linked to the viscosity ratio. The step emulsification process experiences a transition of fragmentation regime where both the necking time and drop size suddenly raise. This transition, that corresponds to a critical period of drop formation and thus defines a maximum production rate of small droplets, is observed to be ruled by the viscosity ratio of the two phases. When drops are produced along an array of microchannels with a cross flow of the continuous phase, a configuration comparable to a onedimensional membrane having rectangular pores, a drop boundary layer develops along the drop generators. In the small drop regime, the local dynamics of drop formation is shown to be independent on the emulsion cross flow. Moreover, we note that the development of the drop boundary layer is even beneficial to homogenize drop size along the production line. On the other hand, in the large drop regime, drop collision can trigger fragmentation and thus lead to size polydispersity.

Wettability model for varioussized droplets on solid surfaces
View Description Hide DescriptionThe wetting phenomenon is crucial for the formation of stable liquid films on solid surfaces. The wettability of a liquid on a solid surface is characterized by the Young equation, which represents an equilibrium condition of a droplet at the three phase contact line. In general, the surface force in the vertical direction on a solid surface is ignored because of the resistance of the solid surface. However, considering the adhesion energy of the droplet rather than the force balance at the contact line, the vertical component of the surface force can be expected to be an important factor during wetting. Based on this concept, an analytical model is developed herein by considering the energy balance including adhesion forces acting not only in the horizontal but also in the vertical direction, in addition to the effect of gravity on the droplet. The validity of the developed model is then evaluated by experimental observation of the wetting phenomena of droplets on low and highsurfaceenergy solids. Existing data are also used for evaluation of our model. The developed model describes the wetting phenomena of droplets with sizes ranging from nano to millimeters under all experimental conditions and exhibits universality. In addition, on the basis of our model, the line tension is discussed. The results indicate that the line tension approach may be considered as a method to explain wetting phenomena by considering gravitational potential and other macroscopic parameters as a single parameter (i.e., line tension).

Viscous and NonNewtonian Flows

Freeboundary models of a meltwater conduit
View Description Hide DescriptionWe analyse the crosssectional evolution of an englacial meltwater conduit that contracts due to inward creep of the surrounding ice and expands due to melting. Making use of theoretical methods from freeboundary problems in Stokes flow and Hele–Shaw squeeze flow we construct an exact solution to the coupled problem of external viscous creep and internal heating, in which we adopt a Newtonian approximation for ice flow and an idealized uniform heat source in the conduit. This problem provides an interesting variant on standard freeboundary problems, coupling different internal and external problems through the kinematic condition at the interface. The boundary in the exact solution takes the form of an ellipse that may contract or expand (depending on the magnitudes of effective pressure and heating rate) around fixed focal points. Linear stability analysis reveals that without the melting this solution is unstable to perturbations in the shape. Melting can stabilize the interface unless the aspect ratio is too small; in that case, instabilities grow largest at the thin ends of the ellipse. The predictions are corroborated with numerical solutions using boundary integral techniques. Finally, a number of extensions to the idealized model are considered, showing that a contracting circular conduit is unstable to all modes of perturbation if melting occurs at a uniform rate around the boundary, or if the ice is modelled as a shearthinning fluid.

Particulate, Multiphase, and Granular Flows

Free falling and rising of spherical and angular particles
View Description Hide DescriptionDirect numerical simulations of freely falling and rising particles in an infinitely long domain, with periodic lateral boundary conditions, are performed. The focus is on characterizing the free motion of cubical and tetrahedral particles for different Reynolds numbers, as an extension to the wellstudied behaviour of freely falling and rising spherical bodies. The vortical structure of the wake, dynamics of particle movement, and the interaction of the particle with its wake are studied. The results reveal mechanisms of path instabilities for angular particles, which are different from those for spherical ones. The rotation of the particle plays a more significant role in the transition to chaos for angular particles. Following a framework similar to that of Mougin and Magnaudet [“Wakeinduced forces and torques on a zigzagging/spiralling bubble,” J. Fluid Mech.567, 185–194 (2006)], the balance of forces and torques acting on particles is discussed to gain more insight into the path instabilities of angular particles.

Laminar Flows

Pitchingmotionactivated flapping foil near solid walls for power extraction: A numerical investigation
View Description Hide DescriptionA numerical investigation on the power extraction of a pitchingmotionactivated flapping foil near solid walls is performed by using an immersed boundarylattice Boltzmann method in this study. The flapping motions of the foil include a forced pitching component and an induced plunging component. The foil is placed either near a solid wall or between two parallel plane walls. Compared to previous work on the flapping foil for power extraction, the effect of the walls is first considered in this work. At a Reynolds number of 1100 and with the position of the foil pitching axis at third chord, the influences of the mechanical parameters (such as damping coefficient and spring constant) of the foil, the amplitude and frequency of the pitching motion and the clearance between the foil pitching axis and the wall on the power extraction performance of the flapping foil are systematically evaluated. Compared to the situation of free stream, the power extraction performance of the foil near the wall is improved. For given amplitude and frequency, as the clearance decreases the net power extraction efficiency improves. Moreover, as the foil is placed near one wall, there is a transverse shift to the plunging motion that consequently weakens the improvement of net power extraction efficiency. In contrast, the shift can be significantly eliminated as the foil is placed between two walls, which can further improve the net power extraction efficiency. In addition, it is found that the efficiency improvement is essentially from the increased power extraction, which is due to the generation of high lift force.
