Volume 25, Issue 2, February 2013
Index of content:

We introduce a modified tomographic PIV technique using four highspeed video cameras and a scanning pulsed laservolume. By rapidly illuminating adjacent subvolumes onto separate video frames, we can resolve a larger total volume of velocity vectors, while retaining good spatial resolution. We demonstrate this technique by performing timeresolved measurements of the turbulent structure of a round jet, using up to 9 adjacent volume slices. In essence this technique resolves more velocity planes in the depth direction by maintaining optimal particle image density and limiting the number of ghost particles. The total measurement volumes contain between 1 ×106 and 3 ×106 velocity vectors calculated from up to 1500 reconstructed depthwise image planes, showing timeresolved evolution of the largescale vortical structures for a turbulent jet of Re up to 10 000.
 LETTERS


A simple criterion for filament breakup in dropondemand inkjet printing
View Description Hide DescriptionJets from dropondemand inkjet printheads consist of a main drop with a trailing filament, which either condenses into the main drop, or breaks up into satellite drops. Filament behaviour is quantitatively similar to that of larger, free symmetrical filaments, and can be predicted from the aspect ratio and Ohnesorge number. Symmetrical filaments generated from inkjet printheads show the same behaviour. A simple model, based on competition between the processes of axial shortening and radial necking, predicts the critical aspect ratio below which the jet condenses into a single drop. The success of this simple criterion supports the underlying physical model.

Formulation for volume flowrate induced by vortex rings, through circular orifice of arbitrary section profile
View Description Hide DescriptionA new theoretical formulation that relates the volume flowrate to the vorticity distribution for the circular orifice of an arbitrary section profile is presented. The formulation is based on axisymmetric potential theory involving Green's function of Stokes’ streamfunction. The results showed that the volume flowrate through the orifice opening can be separated into potential and vortical portions, and the kernel of the vortical portion is the homogeneous streamfunction of the orifice. The formulation for the thin circular orifice was verified by comparing analytical solutions.

Some exact properties of the effective slip over surfaces with hydrophobic patternings
View Description Hide DescriptionShear flows of viscous fluid layers over nonuniformly hydrophobic surfaces are characterized in the farfield by an effective slip velocity, which relates to the applied stress through some mobility tensor characterizing the surface. Here, we identify two methods to determine the mobility tensor for flat surfaces with arbitrary sliplength variations. A family of “Cross Flow Identities” is then analyzed, which equate mobility components of different unidirectional patternings. We also calculate an analytical mobility solution for a family of continuously varying patterns. We validate the results numerically and discuss implications in various limits.

Probing highReynoldsnumber effects in numerical boundary layers
View Description Hide DescriptionWe study the highReynoldsnumber behavior of a turbulent boundary layer in the low supersonic regime through verylargescale direct numerical simulation (DNS). For the first time a Reynolds number is attained in DNS ( , where δ is the boundary layer thickness and δ v is the viscous length scale) at which theoretical predictions and experiments suggest the occurrence of phenomena pertaining to the asymptotic Reynolds number regime. From comparison with previous DNS data at lower Reynolds number we find evidence of a continuing trend toward a stronger imprint of the outerlayer structures onto the nearwall region. This effect is clearly manifested both in flow visualizations, and in energy spectra. More than a decade of nearlylogarithmic variation is observed in the mean velocity profiles, with loglaw constants k ≈ 0.394, C ≈ 4.84, and a trend similar to experiments. We find some supporting evidence for the debated existence of a k −1 region in the power spectrum of streamwise velocity fluctuations, which extends up to y + ≈ 150, and of a k −5/3 spectral range in the outer layer.

 ARTICLES

 Biofluid Mechanics

Stability of passive locomotion in inviscid wakes
View Description Hide DescriptionWe consider the passive locomotion of rigid bodies in inviscid pointvortex wakes. This work is motivated by a common belief that live and inanimate objects may extract energy from unsteady flows for locomotory advantages. Studies on energy extraction from unsteady flows focus primarily on energy efficiency. Besides efficiency, a fundamental aspect of energy extraction for locomotion purposes is stability of motion. Here, we propose idealized wake models using periodically generated point vortices to emulate shedding of vortices from an unmodeled moving or stationary object. We assess the stability of these pointvortex wakes and find that they are stable for a range of periods, unlike the von Kármán street model which is mainly unstable. We then investigate the dynamics of a rigid body submerged in such wakes. In particular, we calculate periodic trajectories where the rigid body “swims” passively against the flow by extracting energy from the ambient vortices. All the periodic trajectories we find are unstable. The largest instabilities reported are for elliptic bodies where rotational effects play a role in destabilizing their motion. Within the context of this model, we conclude that passive locomotion of rigid bodies in inviscid wakes is unstable. Questions as to whether passive stability can be achieved when accounting for fluid viscosity and body elasticity remain open.

Aerodynamic forces and vortical structures in flapping butterfly's forward flight
View Description Hide DescriptionForward flights of a bilaterally symmetrically flapping butterfly modeled as a fourlink rigidbody system consisting of a thorax, an abdomen, and left and right wings are numerically simulated. The joint motions of the butterflies are adopted from experimental observations. Three kinds of the simulations, distinguished by ways to determine the position and attitude of the thorax, are carried out: a tethered simulation, a prescribed simulation, and freeflight simulations. The upward and streamwise forces as well as the wake structures in the tethered simulation, where the thorax of the butterfly is fixed, reasonably agree with those in the corresponding tethered experiment. In the prescribed simulation, where the thoracic trajectories as well as the joint angles are given by those observed in a freeflight experiment, it is confirmed that the butterfly can produce enough forces to achieve the flapping flights. Moreover, coherent vortical structures in the wake and those on the wings are identified. The generation of the aerodynamic forces due to the vortical structures are also clarified. In the freeflight simulation, where only the joint angles are given as periodic functions of time, it is found that the free flight is longitudinally unstable because the butterfly cannot maintain the attitude in a proper range. Focusing on the abdominal mass, which largely varies owing to feeding and metabolizing, we have shown that the abdominal motion plays an important role in periodic flights. The necessity of control of the thoracic attitude for periodic flights and maneuverability is also discussed.
 Micro and Nanofluid Mechanics

Sample dispersion in isotachophoresis with Poiseuille counterflow
View Description Hide DescriptionA particular mode of isotachophoresis (ITP) employs a pressuredriven flow opposite to the sample electromigration direction in order to anchor a sample zone at a specific position along a channel or capillary. We investigate this situation using a twodimensional finitevolume model based on the NernstPlanck equation. The imposed Poiseuille flow profile leads to a significant dispersion of the sample zone. This effect is detrimental for the resolution in analytical applications of ITP. We investigate the impact of convective dispersion, characterized by the areaaveraged width of a sample zone, for various values of the sample Pécletnumber, as well as the relative mobilities of the sample and the adjacent electrolytes. A onedimensional model for the areaaveraged concentrations based on a TaylorAristype effective axial diffusivity is shown to yield good agreement with the finitevolume calculations. This justifies the use of such simple models and opens the door for the rapid simulation of ITP protocols with Poiseuille counterflow.

Frequency dependence and frequency control of microbubble streaming flows
View Description Hide DescriptionSteady streaming from oscillating microbubbles is a powerful actuating mechanism in microfluidics, enjoying increased use due to its simplicity of manufacture, ease of integration, low heat generation, and unprecedented control over the flow field and particle transport. As the streaming flow patterns are caused by oscillations of microbubbles in contact with walls of the setup, an understanding of the bubble dynamics is crucial. Here we experimentally characterize the oscillation modes and the frequency response spectrum of such cylindrical bubbles, driven by a pressure variation resulting from ultrasound in the range of 1 kHz 100 kHz. We find that (i) the appearance of 2D streaming flow patterns is governed by the relative amplitudes of bubble azimuthal surface modes (normalized by the volume response), (ii) distinct, robust resonance patterns occur independent of details of the setup, and (iii) the position and width of the resonance peaks can be understood using an asymptotic theory approach. This theory describes, for the first time, the shape oscillations of a pinned cylindrical bubble at a wall and gives insight into necessary mode couplings that shape the response spectrum. Having thus correlated relative mode strengths and observed flow patterns, we demonstrate that the performance of a bubble micromixer can be optimized by making use of such flow variations when modulating the driving frequency.

Nanodrop impact on solid surfaces
View Description Hide DescriptionThe impact of nanometer sized drops on solid surfaces is studied using molecular dynamics simulations. Equilibrated floating drops consisting of short chains of LennardJones liquids with adjustable volatility are directed normally onto an atomistic solid surface where they are observed to bounce, stick, splash, or disintegrate, depending on the initial velocity and the nature of the materials involved. Drops impacting at low velocity bounce from nonwetting surfaces but stick and subsequently spread slowly on wetting surfaces. Higher velocity impacts produce an prompt splash followed by disintegration of the drop, while at still higher velocity, drops disintegrate immediately. The disintegration can be understood as either a loss of coherence of the liquid or as the result of a local temperature exceeding the liquidvapor coexistence value. In contrast to macroscopic drops, the presence of vapor outside the drop does not effect the behavior in any significant way. Nonetheless, the transition between the splashing and bouncing/sticking regimes occurs at Reynolds and Weber numbers similar to those found for larger drops.
 Interfacial Flows

Thin film flow down a porous substrate in the presence of an insoluble surfactant: Stability analysis
View Description Hide DescriptionThe stability of a gravitydriven film flow on a porous inclined substrate is considered, when the film is contaminated by an insoluble surfactant, in the frame work of OrrSommerfeld analysis. The classical longwave asymptotic expansion for small wave numbers reveals the occurrence of two modes, the Yih mode and the Marangoni mode for a clean/a contaminated film over a porous substrate and this is confirmed by the numerical solution of the OrrSommerfeld system using the spectralTau collocation method. The results show that the Marangoni mode is always stable and dominates the Yih mode for small Reynolds numbers; as the Reynolds number increases, the growth rate of the Yih mode increases, until, an exchange of stability occurs, and after that the Yih mode dominates. The role of the surfactant is to increase the critical Reynolds number, indicating its stabilizing effect. The growth rate increases with an increase in permeability, in the region where the Yih mode dominates the Marangoni mode. Also, the growth rate is more for a film (both clean and contaminated) over a thicker porous layer than over a thinner one. From the neutral stability maps, it is observed that the critical Reynolds number decreases with an increase in permeability in the case of a thicker porous layer, both for a clean and a contaminated film over it. Further, the range of unstable wave number increases with an increase in the thickness of the porous layer. The film flow system is more unstable for a film over a thicker porous layer than over a thinner one. However, for small wave numbers, it is possible to find the range of values of the parameters characterizing the porous medium for which the film flow can be stabilized for both a clean film/a contaminated film as compared to such a film over an impermeable substrate; further, it is possible to enhance the instability of such a film flow system outside of this stability window, for appropriate choices of the porous substrate characteristics.

Shape of a large drop on a rough hydrophobic surface
View Description Hide DescriptionLarge drops on solid surfaces tend to flatten due to gravitational effect. Their shapes can be predicted by solving the YoungLaplace equation when their apparent contact angles are precisely given. However, for large drops sitting on rough surfaces, the apparent contact angles are often unavailable a priori and hard to define. Here we develop a model to predict the shape of a given volume of large drop placed on a rough hydrophobic surface using an overlapping geometry of double spheroids and the free energy minimization principle. The drop shape depends on the wetting state, thus our model can be used not only to predict the shape of a drop but also to infer the wetting state of a large drop through the comparison of theory and experiment. The experimental measurements of the shape of large water drops on various micropillar arrays agree well with the model predictions. Our theoretical model is particularly useful in predicting and controlling shapes of large drops on surfaces artificially patterned in microscopic scales, which are frequently used in microfluidics and labonachip technology.

Longitudinal instability of a liquid rim
View Description Hide DescriptionWe study the transverse instability of a retracting liquid rim using a long wavelength approximation model and full numerical simulations. We observe that the instability of the rim is driven both by the RayleighTaylor mechanism because of the initial rim acceleration, and by the RayleighPlateau one. The coupling between the rim and the sheet stabilizes the rim at long wavelength. Full numerical simulations are in good agreement with the model and the subsequent breakup of droplets is observed in the numerical simulations when the instability is strong enough.

The effect of viscoelasticity on the dynamics of gas bubbles near free surfaces
View Description Hide DescriptionThe dynamics of bubbles immersed in a viscoelastic fluid directly beneath an initially plane free surface is modelled using the boundary integral method. The model predicts a range of dynamics that is dependent on the Deborah number, the Reynolds number and the proximity of the bubble to the free surface. The motion of the free surface jet caused by the collapse of a bubble in a viscoelastic fluid can be significantly retarded compared with the Newtonian case. The axial jet predicted in many instances in the Newtonian case is not observed when the inertial forces are sufficiently small. In this case an annular jet forms that can penetrate the bubble. At high Deborah numbers, there is a return to Newtonianlike dynamics since the effects of viscosity are abated by elasticity to such an extent that inertia is the prevailing influence on bubble dynamics.

Film drainage of viscous liquid on top of bare bubble: Influence of the Bond number
View Description Hide DescriptionWe present experimental results of film drainage on top of gas bubbles pushed by gravity towards the free surface of highly viscous Newtonian liquid with a uniform interface tension. The temporal evolution of the thickness of the film between a single bubble and the air/liquid interface is investigated via interference method. Experiments under various physical conditions (range of viscosities and surface tension of the liquid, and bubble sizes) evidence the influence of the deformation of the thin film on the thinning rate and confirm the slow down of film drainage with Bond number as previously reported by numerical work of Pigeonneau and Sellier [Phys. Fluids23, 092102 (Year: 2011)]10.1063/1.3629815. Considering the liquid flow in the cap squeezed by buoyancy force of the bubble, we provide an approximation of thinning rate as a function of Bond number that agrees with experimental and numerical data. Qualitatively, the smaller the area of the thin film compare to the surface of the bubble, the faster the drainage.
 Viscous and NonNewtonian Flows

RayleighBénard convection for viscoplastic fluids
View Description Hide DescriptionThe influence of rheological and interfacial properties of yield stress fluids is investigated on the onset of the RayleighBénard convection. Different Carbopol® (B.F. Goodrich) gels are used in a circular cell for RayleighBénard experimental setup. The influence of the boundary conditions is also investigated by controlling either slip or noslip conditions. The onset of thermoconvection is shown by measuring temperature differences and also by using shadowgraph flow visualization. Experimental results show that convection occurs in the range of our experiments. Considering Carbopol gels as elastoplastic materials with a yield stress τ y , a generalized Rayleigh number is obtained: Ra g = Y −1, with Y the yield number, which represents the balance between the yield stress of the gel and the buoyancy effects. The results show that the Rayleigh number is proportional to d, the height of the setup, and that the control parameter is the yield number at the onset of convection. Critical values of Y −1 have been determined for slip conditions as well as for noslip conditions . It highlights that the change in surface conditions affect significantly the critical conditions.
 Particulate, Multiphase, and Granular Flows

Longitudinal and transverse disturbances in unbounded, gasfluidized beds
View Description Hide DescriptionA temporal linear stability analysis is performed on the continuumaveraged equations of motion for the fluid and dispersed particle phases which describe a uniform, boundless fluidized bed. The present analysis considers disturbances ranging from purely transverse to purely longitudinal, with respect to the flow direction of the basestate fluid velocity. It is found that disturbances over the entire horizontalvertical wavenumber spectrum may be unstable, although the dominant disturbance is either the classically recognized longitudinal disturbance travelling in the vertical direction or a nonoscillating, purely transverse disturbance. The nature of the preferred mode depends upon the dimensionless parameters of the system. For all parameter cases the growth constants of the two modes remain within an order of magnitude of each other, thereby highlighting the importance of both modes in stability considerations of this system. Dispersion relations for each of the separated horizontal and vertical modes are derived and longwavelength analyses are performed on the corresponding relations. It is shown that the unstable transverse mode shares similar mechanisms with the longitudinal mode, and that the dominant stabilizing mechanism for each mode is the same for closelypacked beds.

Unconfined slumping of a granular mass on a slope
View Description Hide DescriptionThis study investigates the gravitationally driven dynamics of dense granular materials, released from rest and allowed to flow down a slope until they stop moving. Laboratory experiments were performed in which a measured volume of material was released from rest in a cylindrical tube and spread across an unconfined rigid plane inclined at angles less than the angle of repose. Upon release, the particles initially spread outward radially. However, upslope motion is rapidly suppressed while downslope motion is promoted, which leads to an approximately ellipsoidally shaped deposit once the flow has been fully arrested. The flows were modeled under the shallow layer approximation and integrated numerically to capture the motion from initiation to final arrest. In modeling, two types of Coulombtype friction models were employed. One had a constant friction coefficient, and another had a friction coefficient that depends upon the dimensionless inertial number of the motion. When the initial aspect ratio of a granular mass is small and the slope angle is low (<5°), the model with a constant friction coefficient can capture the shape of the deposit. However, when the slope angle is increased, the friction model that is dependent on inertial number becomes more important. For granular columns of initially high aspect ratios, the shallow water model fails to reproduce some aspects of the experimental observations. Finally, the dependence of the shape and depth of the deposit upon dimensionless parameters that characterize the system is examined under the constant friction coefficient model, demonstrating that the deduced scaling arguments are borne out by the numerical simulations and laboratory data.
 Laminar Flows

Unsteady separated stagnationpoint flow of an incompressible viscous fluid on the surface of a moving porous plate
View Description Hide DescriptionUsing grouptheoretic method, an analysis is presented for a similarity solution of boundary layer equations which represents an unsteady twodimensional separated stagnationpoint (USSP) flow of an incompressible fluid over a porous plate moving in its own plane with speed u 0(t). It is observed that the solution to the governing nonlinear ordinary differential equation for the USSP flow admits of two solutions (in contrast with the corresponding steady flow where the solution is unique): one is the attached flow solution (AFS) and the other is the reverse flow solution (RFS). A novel result of the analysis is that in the case of stationary plate (u 0(t) = 0), after a certain value of the magnitude of the blowing d (<0) at the plate, only the AFS exists and the solution becomes unique. For a stationary plate (u 0(t) = 0), the USSP flow is found to be separated for all values of d in both the cases of AFS and RFS. It is also observed that when u 0(t) = 0, in the RFS flow with wall suction d (>0), there are two stagnationpoints in the flow but in the presence of blowing d (<0), there is only one stagnationpoint in the flow which moves further and further up with increase in d. Suction is shown to increase the wall shear stress while blowing has an opposite effect. Streamlines for an USSP flow when u 0(t) ≠ 0 are also plotted. It is found that in this case, the USSP flow is not in general separated.

Deformation of vortex patches by boundaries
View Description Hide DescriptionThe deformation of twodimensional vortex patches in the vicinity of fluid boundaries is investigated. The presence of a boundary causes an initially circular patch of uniform vorticity to deform. Sufficiently far away from the boundary, the deformed shape is well approximated by an ellipse. This leading order elliptical deformation is investigated via the elliptic moment model of Melander, Zabusky, and Styczek [J. Fluid Mech.167, 95 (Year: 1986)10.1017/S0022112086002744]. When the boundary is straight, the centre of the elliptic patch remains at a constant distance from the boundary, and the motion is integrable. Furthermore, since the straining flow acting on the patch is constant in time, the problem is that of an elliptic vortex patch in constant strain, which was analysed by Kida [J. Phys. Soc. Jpn.50, 3517 (Year: 1981)10.1143/JPSJ.50.3517]. For more complicated boundary shapes, such as a square corner, the motion is no longer integrable. Instead, there is an adiabatic invariant for the motion. This adiabatic invariant arises due to the separation in times scales between the relatively rapid time scale associated with the rotation of the patch and the slower time scale associated with the selfadvection of the patch along the boundary. The interaction of a vortex patch with a circular island is also considered. Without a background flow, the conservation of angular impulse implies that the motion is again integrable. The addition of an irrotational flow past the island can drive the patch towards the boundary, leading to the possibility of large deformations and breakup.

Numerical simulation of vortexinduced vibration of a square cylinder at a low Reynolds number
View Description Hide DescriptionVortexinduced vibrations (VIV) of a square cylinder at a Reynolds number of 100 and a low mass ratio of 3 are studied numerically by solving the NavierStokes equations using the finite element method. The equation of motion of the square cylinder is solved to simulate the vibration and the Arbitrary Lagrangian Eulerian scheme is employed to model the interaction between the vibrating cylinder and the fluid flow. The numerical model is validated against the published results of flow past a stationary square cylinder and the results of VIV of a circular cylinder at low Reynolds numbers. The effect of flow approaching angle (α) on the response of the square cylinder is investigated. It is found that α affects not only the vibration amplitude but also the lockin regime. Among the three values of α (α = 0°, 45°, and 22.5°) that are studied, the smallest vibration amplitude and the narrowest lockin regime occur at α = 0°. It is discovered that the vibration locks in with the natural frequency in two regimes of reduced velocity for α = 22.5°. Single loop vibration trajectories are observed in the lockin regime at α = 22.5° and 45°, which is distinctively different from VIV of a circular cylinder. As a result, the vibration frequency in the inline direction is the same as that in the crossflow direction.