Volume 23, Issue 1, January 2011

Using a stereoscopic vision method, we have experimentally investigated the time evolution of a free thin disk motion with six degrees of freedom for the first time. It is found that, as the dimensionless moment of inertia decreases, the trajectory of the disk transits from planar to nonplanar. New types of free falling motions were identified for small values, including the spiral state and the transitional state. An extended phase diagram corresponding to different flow regimes was given. The underlying physics associated with the transition is found to be connected to the interactions between the moving object and induced vortices.
 ANNOUNCEMENTS


Announcement: The 2010 François Naftali Frenkiel Award for Fluid Mechanics
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 EDITORIALS


Referee acknowledgment for 2010
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 AWARD AND INVITED PAPERS


Simple models of turbulent flows^{a)}
View Description Hide DescriptionStochastic Lagrangian models provide a simple and direct way to modelturbulent flows and the processes that occur within them. This paper provides an introduction to this approach, aimed at the nonspecialist, and providing some historical perspective. Basic models for the Lagrangian velocity (i.e., the Langevin equation) and composition are described and applied to the simple but revealing case of dispersion from a line source in grid turbulence. With simple extensions, these models are applied to inhomogeneous turbulent reactive flows, where they form the core of probability density function (PDF) methods. The use of PDF methods is illustrated for the case of a lifted turbulent jet flame.Lagrangian time series are now accessible both from experiments and from direct numerical simulations, and this information is used to scrutinize and improve stochastic Lagrangian models. In particular, we describe refinements to account for the observed strong Reynoldsnumber effects including intermittency. It is emphasized that all models of turbulence are necessarily approximate and incomplete, and that simple models are valuable in many applications in spite of their limitations.
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 LETTERS


Theoretical model for spectra in dispersed multiphase flows
View Description Hide DescriptionThe spectrum of a signal consisting of a sum of localized random bursts can exhibit, under certain conditions, an intermediate subrange evolving as the power −3 of the wavenumber . These bursts should have a smooth and regular pattern, their strength and size should be statistically independent, and their size should be uniformly distributed between two finite wavelengths. This is probably the explanation for the subrange that is commonly observed in the velocity spectra of bubbleinducedturbulence, which results from the interaction of the localized velocity disturbances caused by the bubbles.

Experimental study of freely falling thin disks: Transition from planar zigzag to spiral
View Description Hide DescriptionUsing a stereoscopic vision method, we have experimentally investigated the time evolution of a free thin disk motion with six degrees of freedom for the first time. It is found that, as the dimensionless moment of inertia decreases, the trajectory of the disk transits from planar to nonplanar. New types of free falling motions were identified for small values, including the spiral state and the transitional state. An extended phase diagram corresponding to different flow regimes was given. The underlying physics associated with the transition is found to be connected to the interactions between the moving object and induced vortices.

Visualizing the verylargescale motions in turbulent pipe flow
View Description Hide DescriptionTimeresolved stereoscopic particle imagevelocimetry is used to investigate the structure of the verylargescale motions (VLSMs) in fully developed turbulent pipe flow. The motions are visualized using snapshot proper orthogonal decomposition. It is shown that the structures can be reconstructed using a small number of the most energetic modes. The results strongly suggest a possible connection between the origin of the VLSM and linear stability analysis. The structures are seen to be highly threedimensional, meandering azimuthally and radially. At this Reynolds number, they occasionally extend from the nearwall region to the wake region of the pipe.

The effect of particle inertia on triaxial ellipsoids in creeping shear: From drift toward chaos to a single periodic solution
View Description Hide DescriptionThe motion of inertial, triaxial ellipsoids in creeping shear flow is explained for a wide range of aspect ratios and Stokes numbers (St, quantifying particle inertia). Particle inertia induces a drift toward rotation around the shortest axis, with this axis aligned with the vorticity axis of the flow. For aspect ratio combinations in a certain region, this periodic state is unstable for low St and the particle moves in a chaotic manner. At higher St, the instability is stabilized and one single final periodic motion is well defined also for (in the limit of ) unstable aspect ratios.
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 ARTICLES

 Biofluid Mechanics

Instability regimes in flowing suspensions of swimming microorganisms
View Description Hide DescriptionThe effects of an external shear flow on the dynamics and pattern formation in a dilute suspension of swimming microorganisms are investigated using a linear stability analysis and threedimensional numerical simulations, based on the kinetic model previously developed by [D. Saintillan and M. J. Shelley, Phys. Fluids20, 123304 (2008)]. The external shear flow is found to damp the instabilities that occur in these suspensions by controlling the orientation of the particles. We demonstrate in our simulations that the rate of damping is directiondependent: it is fastest in the flow direction, but slowest in the direction perpendicular to the shear plane. As a result, transitions from three to two to onedimensional instabilities are observed to occur as shear rate increases, and above a certain shear rate the instabilities altogether disappear. The density patterns and complex flows that arise at long time in the suspensions are also analyzed from the numerical simulations using standard techniques from the literature on turbulent flows. The imposed shear flow is found to have an effect on both density patterns and flow structures, which typically align with the extensional axis of the external flow. The disturbance flows in the simulations are shown to exhibit similarities with turbulent flows, and in particular two of the seemingly universal characteristics of turbulent flows also occur, namely: (i) the bias of plots toward the second and fourth quadrants, corresponding to stable focus/stretching and unstable node/saddle/saddle flow topologies, respectively, and (ii) the alignment of the vorticity vector with the intermediate strainrate eigenvector. However, the flows described herein also significantly differ from turbulent flows owing to the strong predominance of large scales, as exemplified by the very rapid decay of the kinetic energy spectrum, an effect further enhanced after the transitions to two and onedimensional instabilities.

Passive hydrodynamic synchronization of twodimensional swimming cells
View Description Hide DescriptionSpermatozoa flagella are known to synchronize when swimming in close proximity. We use a model consisting of twodimensional sheets propagating transverse waves of displacement to demonstrate that fluid forces lead to such synchronization passively. Using two distinct asymptotic descriptions (small amplitude and long wavelength), we derive the synchronizing dynamics analytically for arbitrarily shaped waveforms in Newtonian fluids, and show that phaselocking will always occur for sufficiently asymmetric shapes. We characterize the effect of the geometry of the waveforms and the separation between the swimmers on the synchronizing dynamics, the final stable conformations, and the energy dissipated by the cells. For two closely swimming cells, synchronization always occurs at the inphase or oppositephase conformation, depending solely on the geometry of the cells. In contrast, the work done by the swimmers is always minimized at the inphase conformation. As the swimmers get further apart, additional fixed points arise at intermediate values of the relative phase. In addition, computations for large amplitude waves using the boundary integral method reveal that the two asymptotic limits capture all the relevant physics of the problem. Our results provide a theoretical framework to address other hydrodynamic interactions phenomena relevant to populations of selfpropelled organisms.
 Micro and Nanofluid Mechanics

Modeling drag reduction and meniscus stability of superhydrophobic surfaces comprised of random roughness
View Description Hide DescriptionPrevious studies dedicated to modeling drag reduction and stability of the airwater interface on superhydrophobicsurfaces were conducted for microfabricated coatings produced by placing hydrophobic microposts/microridges arranged on a flat surface in aligned or staggered configurations. In this paper, we model the performance of superhydrophobicsurfaces comprised of randomly distributed roughness (e.g., particles or microposts) that resembles natural superhydrophobicsurfaces, or those produced via random deposition of hydrophobic particles. Such fabrication method is far less expensive than microfabrication, making the technology more practical for large submerged bodies such as submarines and ships. The present numerical simulations are aimed at improving our understanding of the drag reduction effect and the stability of the airwater interface in terms of the microstructure parameters. For comparison and validation, we have also simulated the flow over superhydrophobicsurfaces made up of aligned or staggered microposts for channel flows as well as streamwise or spanwise ridges configurations for pipe flows. The present results are compared with theoretical and experimental studies reported in the literature. In particular, our simulation results are compared with work of Sbragaglia and Prosperetti, and good agreement has been observed for gas fractions up to about 0.9. The numerical simulations indicate that the random distribution of surface roughness has a favorable effect on drag reduction, as long as the gas fraction is kept the same. This effect peaks at about 30% as the gas fraction increases to 0.98. The stability of the meniscus, however, is strongly influenced by the average spacing between the roughness peaks, which needs to be carefully examined before a surface can be recommended for fabrication. It was found that at a given maximum allowable pressure, surfaces with random post distribution produce less drag reduction than those made up of staggered posts.

A full analytical solution for the forcedriven compressible Poiseuille gas flow based on a nonlinear coupled constitutive relation
View Description Hide DescriptionThe compressible Poiseuille gas flow driven by a uniform force is analytically investigated using a phenomenological nonlinear coupled constitutive relation model. A new fully analytical solution in compact tangent (or hyperbolic tangent in the case of diatomic gases) functional form explains the origin behind the central temperature minimum and a heat transfer from the cold region to the hot region. The solution is not only proven to satisfy the conservation laws exactly but also welldefined for all physical conditions (the Knudsen number and a forcerelated dimensionless parameter). It is also shown that the nonFourier law associated with the coupling of force and viscous shear stress in the constitutive relation is responsible for the existence of the central temperature minimum, while a kinematic constraint on viscous shear and normal stresses identified in the velocity shear flow is the main source of the nonuniform pressure distribution. In addition, the convex pressure profile with a maximum at the center is theoretically predicted for diatomic gases. Finally, the existence of the Knudsen minimum in the mass flow rate is demonstrated by developing an exact analytical formula for the average temperature of the bulk flow.

Twocell circulation in a liquid meniscus driven by a swirling gas jet
View Description Hide DescriptionA liquid issuing from a capillary needle adopts a conejet structure if the liquid is further driven by a coflowing gas jet. In the present work, flow patterns appearing in this conejet structure are studied by the volumeoffluid numerical method. Axisymmetric motions of the liquid and gas, both treated as viscous incompressible fluids, are simulated. As the gas/liquid mass ratio increases, the meridional circulation develops in the meniscus region of the liquidflow. As the ratio exceeds a threshold, the flow becomes time periodic and droplet generating. Swirl, added in the gas jet, affects the liquidflow in two ways. First, the threshold value increases with swirl. Second, the circulation region transforms from the bubblelike into ringlike pattern and then becomes twocellular. As swirl further increases, the cells separate, one cell disappears, and a new cell emerges being attached to the needle wall. The predicted metamorphoses of the flowtopology might be important for atomization of a liquid fuel.

Terraced spreading of nanofilms under a nonmonotonic disjoining pressure
View Description Hide DescriptionA thin film of a viscous, essentially nonvolatile liquid spreads over a substrate controlled by the disjoining pressure exerted by the two interfaces on one another. Such films are commonly used as hard disk lubricants in the magnetic recording industry. Macroscopic nonuniformities in the film caused by a perturbation of the uniformly spread state flow away and the film is “healed” in a time frame governed by the appropriate hydrodynamic equations.Lubrication theory may be used to derive a diffusionequation for the local film height as a function of position and time which shows that an effective heightdependent diffusion coefficient controls the spreading dynamics, where is the bulk liquidviscosity and is a function accounting for any variation of local viscosity near the substrate due to molecularity of the liquid. Such an approach is possible due to the very small ratio of the film height to the inplane length scale of the disturbance. Provided the disjoining pressure is positive and monotonically decreasing with film thickness, the motion of the film is unexceptional, exhibiting the usual smooth profiles associated with diffusive flow with time. However, for nonmonotonic disjoining pressures, the film is experimentally observed to exhibit vertical terraces. These abrupt jumps in height do not disappear with time and they move slowly or are stationary. This phenomenon is investigated here. We demonstrate how a physically consistent “weak” solution of the diffusionequation can be constructed, where only positive values of the diffusion coefficient are sampled. The film heights at the jump discontinuity are determined by an equal area rule for the disjoining pressure. Numerical simulations for a realistic nonmonotonic disjoining pressure exhibit finite termination on the lowside and vertical terraces, thereby matching the behavior observed in experimental systems.
 Interfacial Flows

Numerical simulation of a liquid bridge in a coaxial gas flow
View Description Hide DescriptionThe dynamical response of an isothermal liquid bridge to a coaxial gas stream is examined from axisymmetric numerical simulations of the Navier–Stokes equations. The simulation method is previously validated by calculating the temporal evolution of the first oscillation mode in both cylindrical and axisymmetric liquid bridges. The comparison with other theoretical approaches and experiments shows good agreement in most cases, although significant discrepancies are found between the simulation and the experimental values of the damping rate for hexadecane. The simulation of a liquid bridge in a coaxial gas stream shows that a recirculation cell always appears in the liquid driven by the gas viscous stress on the free surface. The recirculation cell speed depends quasilinearly on the gas velocity for the range of gas flow rates considered. If the gas stream and gravity have the same direction, then the speed of the recirculation cell increases considerably due to the free surface deformation of the liquid bridge at equilibrium. This effect does not occur when gravity has the opposite direction because viscous dissipation in the liquid increases in this case. If the gas stream and gravity point downward, the liquid bridge shrinks at the upper part and bulges at the lower owing to the accumulation of momentum there. The same occurs for zero gravity, but noncylindrical liquid bridges deform more than cylindrical shapes with the same slenderness. If one inverts the direction of the gravity force, the interface deformation caused by the gas stream is the opposite, and its magnitude is smaller. The magnitude of the free surface deformation depends almost linearly on the gas stream velocity for both zero and normal gravity conditions.

Parametric stability and dynamic buckling of an encapsulated microbubble subject to acoustic disturbances
View Description Hide DescriptionStability analysis of the radial pulsations of a gas microbubble that is encapsulated by a thin viscoelastic shell and surrounded by an ideal incompressible liquid is carried out. Small axisymmetric disturbances in the microbubble shape are imposed and their long and short term stability is examined depending on the initial bubble radius, the shell properties, and the parameters, i.e., frequency and amplitude, of the external acoustic excitation. Owing to the anisotropy of the membrane that is forming the encapsulating shell, two different types of elastic energy are accounted for, namely, the membrane and bending energy per unit of initial area. They are used to describe the tensions that develop on the shell due to shell stretching and bending, respectively. In addition, two different constitutive laws are used in order to relate the tensions that develop on the membrane as a result of stretching, i.e., the Mooney–Rivlin law describing materials that soften as deformation increases and the Skalak law describing materials that harden as deformation increases. The limit for static buckling is obtained when the external overpressure exerted upon the membrane surpasses a critical value that depends on the membrane bending resistance. The stability equations describing the evolution of axisymmetric disturbances, in the presence of an external acoustic field, reveal that static buckling becomes relevant when the forcing frequency is much smaller than the resonance frequency of the microbubble, corresponding to the case of slow compression. The resonance frequencies for shape oscillations of the microbubble are also obtained as a function of the shell parameters. Floquet analysis shows that parametric instability, similar to the case of an oscillating free bubble, is possible for the case of a pulsating encapsulated microbubble leading to shape oscillations as a result of subharmonic or harmonic resonance. These effects take place for acoustic amplitude values that lie above a certain threshold but below those required for static buckling to occur. They are quite useful in providing estimates for the shell elasticity and bending resistance based on a frequency/amplitude sweep that monitors the onset of shape oscillations when the forcing frequency resonates with the radial pulsation, , or with a certain shape mode, . An acceleration based instability, identified herein as dynamic buckling, is observed during the compression phase of the pulsation, evolving over a small number of periods of the forcing, when the amplitude of the acoustic excitation is further increased. It corresponds to the Rayleigh–Taylor instability observed for free bubbles, and has been observed with contrast agents as well, e.g., BR14. Finally, phase diagrams for contrast agent BR14 are constructed and juxtaposed with available experimental data, illustrating the relevance and range of the above instabilities.

Nonlinear development of oscillatory instability in a threelayer system under the joint action of buoyancy and thermocapillary effect
View Description Hide DescriptionThe nonlinear development of oscillatory instability under the joint action of buoyant and thermocapillary effects in multilayer system is investigated. The nonlinear convective regimes are studied by the finite difference method. The calculations have been performed for twodimensional flows. The interfaces are assumed to be nondeforming. Rigid heatinsulated lateral walls are considered. Transitions between the flows with different spatial structures are studied. Specific types of nonlinear flows—symmetric and asymmetric oscillations—have been found. It is shown that the oscillatory flow takes place in an interval of Grashof number values bounded both from below by the quiescent mechanical equilibrium, and from above by a convecting steady state. Cavities with different lengths are considered.

Droplet charging regimes for ultrasonic atomization of a liquid electrolyte in an external electric field
View Description Hide DescriptionDistinct regimes of dropletcharging, determined by the dominant charge transport process, are identified for an ultrasonicdroplet ejector using electrohydrodynamic computational simulations, a fundamental scale analysis, and experimental measurements. The regimes of dropletcharging are determined by the relative magnitudes of the dimensionless Strouhal and electric Reynolds numbers, which are a function of the process (pressure forcing), advection, and chargerelaxation time scales for charge transport. Optimal (net maximum) dropletcharging has been identified to exist for conditions in which the electric Reynolds number is of the order of the inverse Strouhal number, i.e., the chargerelaxation time is on the order of the pressure forcing (droplet formation) time scale. The conditions necessary for optimal dropletcharging have been identified as a function of the dimensionless Debye number (i.e., liquid conductivity), external electric field (magnitude and duration), and atomization drive signal (frequency and amplitude). The specific regime of dropletcharging also determines the functional relationship between dropletcharge and chargingelectric field strength. The commonly expected linear relationship between dropletcharge and external electric field strength is only found when either the inverse of the Strouhal number is less than the electric Reynolds number, i.e., the charge relaxation is slower than both the advection and external pressure forcing, or in the electrostatic limit, i.e., when charge relaxation is much faster than all other processes. The analysis provides a basic understanding of the dominant physics of dropletcharging with implications to many important applications, such as electrospray mass spectrometry, ink jet printing, and dropondemand manufacturing.

Analysis of timedependent nonlinear dynamics of the axisymmetric liquid film on a vertical circular cylinder: Energy integral model
View Description Hide DescriptionThe nonlinear dynamics of an axisymmetric liquid film falling on the outer surface of a vertical cylinder is investigated in the framework of the set of two coupled evolution equations derived recently using the energy integral method (EIM). This set of EIM evolution equations is solved numerically and its solutions are compared with the traveling wave solutions derived from it using AUTO. We find that traveling wave solutions of EIM equations can bifurcate either supercritically or subcritically from the base state. The type of bifurcation depends on the parameter set of the problem. The set of EIM equations studied here admits both traveling wave and nonstationary wave flows. We demonstrate that in the case of subcritical primary bifurcation the film dynamics is sensitive to the choice of the initial condition and coexistence of up to five different flows is possible for the same parameter set in the domain of a given periodicity. The case of supercritical primary bifurcation exhibits much lesser dependence on the initial condition, though coexistence of two different flows for the same parameter set is possible. The synergetic approach based on both direct numerical solution of the governing evolution equations and search of traveling wave solutions using AUTO facilitate a discovery of a large variety of flows and help to conclude about stability of the traveling wave flows found using AUTO.

Wall energy relaxation in the Cahn–Hilliard model for moving contact lines
View Description Hide DescriptionThe Cahn–Hilliard model uses diffusion between fluid components to regularize the stress singularity at a moving contact line. In addition, it represents the dynamics of the nearwall layer by the relaxation of a wall energy. The first part of the paper elucidates the role of the wall relaxation in a flowing system, with two main results. First, we show that wall energy relaxation produces a dynamic contact angle that deviates from the static one, and derive an analytical formula for the deviation. Second, we demonstrate that wall relaxation competes with Cahn–Hilliard diffusion in defining the apparent contact angle, the former tending to “rotate” the interface at the contact line while the latter to “bend” it in the bulk. Thus, varying the two in coordination may compensate each other to produce the same macroscopic solution that is insensitive to the microscopic dynamics of the contact line. The second part of the paper exploits this competition to develop a computational strategy for simulating realistic flows with microscopic slip length at a reduced cost. This consists in computing a moving contact line with a diffusion length larger than the real slip length, but using the wall relaxation to correct the solution to that corresponding to the small slip length. We derive an analytical criterion for the required amount of wall relaxation, and validate it by numerical results on dynamicwetting in capillary tubes and drop spreading.

Dynamics of nearly unstable axisymmetric liquid bridges
View Description Hide DescriptionThe dynamics of a noncylindrical, axisymmetric, marginally unstable liquid bridge between two equal disks is analyzed in the inviscid limit. The resulting model allows for the weakly nonlinear description of both the (first stage of) breakage for unstable configurations and the (slow) dynamics for stable configurations. The analysis is made for both slender and short liquid brides. In the former range, the dynamics breaks reflection symmetry on the midplane between the supporting disks and can be described by a standard Duffing equation, while for short bridges reflection symmetry is preserved and the equation is still Duffinglike but exhibiting a quadratic nonlinearity. The asymptotic results compare well with existing experiments.