Volume 25, Issue 6, June 2013
Index of content:

The velocity of a twodimensional aqueous foam has been measured as it flows through two parallel channels, at a constant overall volumetric flow rate. The flux distribution between the two channels is studied as a function of the ratio of their widths. A peculiar dependence of the velocity ratio on the width ratio is observed when the foam structure in the narrower channel is either single staircase or bamboo. In particular, discontinuities in the velocity ratios are observed at the transitions between double and single staircase and between single staircase and bamboo. A theoretical model accounting for the viscous dissipation at the solid wall and the capillary pressure across a film pinned at the channel outlet predicts the observed nonmonotonic evolution of the velocity ratio as a function of the width ratio. It also predicts quantitatively the intermittent temporal evolution of the velocity in the narrower channel when it is so narrow that film pinning at its outlet repeatedly brings the flow to a near stop.
 LETTERS


Spontaneous autophoretic motion of isotropic particles
View Description Hide DescriptionSuspended colloidal particles interacting chemically with a solute can selfpropel by autophoretic motion when they are asymmetrically patterned (Janus colloids). Here we demonstrate theoretically that such anisotropy is not necessary for locomotion and that the nonlinear interplay between surface osmotic flows and solute advection can produce spontaneous and selfsustained motion of isotropic particles. Solving the classical autophoretic framework for isotropic particles, we show that, for given material properties, there exists a critical particle size (or Péclet number) above which spontaneous symmetrybreaking and autophoretic motion occur. A hierarchy of instabilities is further identified for quantized critical Péclet numbers.

 ARTICLES

 Biofluid Mechanics

Coarsegrained theory to predict the concentration distribution of red blood cells in wallbounded Couette flow at zero Reynolds number
View Description Hide DescriptionWe develop a coarsegrained theory to predict the concentration distribution of a suspension of vesicles or red blood cells in a wallbound Couette flow. This model balances the wallinduced hydrodynamic lift on deformable particles with the flux due to binary collisions, which we represent via a secondorder kinetic master equation. Our theory predicts a depletion of particles near the channel wall (i.e., the FahraeusLindqvist effect), followed by a nearwall formation of particle layers. We quantify the effect of channel height, viscosity ratio, and shearrate on the cellfree layer thickness (i.e., the FahraeusLindqvist effect). The results agree with in vitro experiments as well as boundary integral simulations of suspension flows. Lastly, we examine a new type of collective particle motion for red blood cells induced by hydrodynamic interactions near the wall. These “swapping trajectories,” coined by ZuritaGotor et al. [J. Fluid Mech.592, 447–469 (Year: 2007)10.1017/S0022112007008701], could explain the origin of particle layering near the wall. The theory we describe represents a significant improvement in terms of time savings and predictive power over current largescale numerical simulations of suspension flows.

Helical swimming in Stokes flow using a novel boundaryelement method
View Description Hide DescriptionWe apply the boundaryelement method to Stokes flows with helical symmetry, such as the flow driven by an immersed rotating helical flagellum. We show that the twodimensional boundary integral method can be reduced to one dimension using the helical symmetry. The computational cost is thus much reduced while spatial resolution is maintained. We review the robustness of this method by comparing the simulation results with the experimental measurement of the motility of model helical flagella of various ratios of pitch to radius, along with predictions from resistiveforce theory and slenderbody theory. We also show that the modified boundary integral method provides reliable convergence if the singularities in the kernel of the integral are treated appropriately.
 Micro and Nanofluid Mechanics

Analysis of the trajectory of a sphere moving through a geometric constriction
View Description Hide DescriptionWe present a numerical study of the effect that fluid and particle inertia have on the motion of suspended spherical particles through a geometric constriction to understand analogous microfluidic settings, such as pinched flow fractionation devices. The particles are driven by a constant force in a quiescent fluid, and the constriction (the pinching gap) corresponds to the space between a plane wall and a second, fixed sphere of the same size (the obstacle). The results show that, due to inertia and/or the presence of a geometric constriction, the particles attain smaller separations to the obstacle. We then relate the minimum surfacetosurface separation to the effect that shortrange, repulsive nonhydrodynamic interactions (such as solidsolid contact due to surface roughness, electrostatic double layer repulsion, etc.) would have on the particle trajectories. In particular, using a simple hardcore repulsive potential model for such interactions, we infer that the particles would experience larger lateral displacements moving through the pinching gap as inertia increases and/or the aperture of the constriction decreases. Thus, separation of particles based on differences in density is in principle possible, owing to the differences in inertia associated with them. We also discuss the case of significant inertia in which the presence of a small constriction may hinder separation by reducing inertia effects.

A Fokker–Planck based kinetic model for diatomic rarefied gas flows
View Description Hide DescriptionA Fokker–Planck based kinetic model is presented here, which also accounts for internal energy modes characteristic for diatomic gas molecules. The model is based on a Fokker–Planck approximation of the Boltzmann equation for monatomic molecules, whereas phenomenological principles were employed for the derivation. It is shown that the model honors the equipartition theorem in equilibrium and fulfills the Landau–Teller relaxation equations for internal degrees of freedom. The objective behind this approximate kinetic model is accuracy at reasonably low computational cost. This can be achieved due to the fact that the resulting stochastic differential equations are continuous in time; therefore, no collisions between the simulated particles have to be calculated. Besides, because of the devised energy conserving time integration scheme, it is not required to resolve the collisional scales, i.e., the mean collision time and the mean free path of molecules. This, of course, gives rise to much more efficient simulations with respect to other particle methods, especially the conventional direct simulation Monte Carlo (DSMC), for small and moderate Knudsen numbers. To examine the new approach, first the computational cost of the model was compared with respect to DSMC, where significant speed up could be obtained for small Knudsen numbers. Second, the structure of a high Mach shock (in nitrogen) was studied, and the good performance of the model for such out of equilibrium conditions could be demonstrated. At last, a hypersonic flow of nitrogen over a wedge was studied, where good agreement with respect to DSMC (with level to level transition model) for vibrational and translational temperatures is shown.
 Interfacial Flows

Energy transfer between the shape and volume modes of a nonspherical gas bubble
View Description Hide DescriptionA model of a nonspherical gas bubble is developed in which the RayleighPlesset equation is augmented with second order terms that backcouple the volume mode to a single shape mode. These additional terms in the RayleighPlesset equation permit oscillation energy to be transferred back and forth between the shape and volume modes. The parametric stability of the shape mode is analyzed for a driven bubble, and it is seen that the bidirectional coupling yields an enhanced, albeit minor, stabilizing effect on the shape mode when compared with a model where the shapevolume coupling is unidirectional. It is also demonstrated how a pure shape distortion can excite significant volume pulsations when the volume mode is in 2:1 internal resonance with the shape mode.

Linear oscillations of a supported bubble or drop
View Description Hide DescriptionShape oscillations of a spherical bubble or drop, for which part of its interface is fixed due to contact with a solid support, are studied analytically using variational methods. Linear oscillations and irrotational flow are assumed. The present analysis is parallel to those of Strani and Sabetta [“Free vibrations of a drop in partial contact with a solid support,” J. Fluid Mech.141, 233–247 (Year: 1984)]10.1017/S0022112084000811; and Bostwick and Steen [“Capillary oscillations of a constrained liquid drop,” Phys. Fluids21, 032108 (Year: 2009)]10.1063/1.3103344 but is also able to determine the response of bubbles or drops to movements imposed on their supports or to variations of their volumes. The analysis leads to equations of motion with a simple structure, from which the eigenmodes and frequency response to periodic forcing are easily determined.

Surface tension effects on the motion of a freefalling liquid sheet
View Description Hide DescriptionThe stationary motion of a liquid curtain falling under the effects of inertia, gravity, and surface tension is analyzed. An original equation governing the streamwise distribution of thickness and velocity is derived by means of a Taylor expansion in the lateral distance from the mean line of the sheet. Approximate solutions are obtained by means of perturbation approaches involving the two parameters governing the problem, namely, the slenderness ratio ɛ and the Weber number We. The numerical procedure employed in order to integrate the nonlinear equation is discussed and a parametric study is presented, together with a comparison with the approximate asymptotic solutions valid for small ɛ and We.

Volume oscillations of a constrained bubble
View Description Hide DescriptionThe behavior of a single acoustically driven bubble tethered to a wire ring is considered. The method of restraining the bubble against rising by attaching it to a wire is a common procedure in conducting precision acoustic measurements. The dynamics of the tethered bubble differs from those of free bubble due to variation in inertial (or added) mass. The objective of this study is to obtain a closedform, leading order solution for the volume oscillations, assuming smallness of the bubble radius R 0 in comparison with the acoustic wavelength λ. It was shown, by using the invariance of the Laplace equation to conformal transformations and the geometry of the problem, that the toroidal coordinates provide separation of variables and are most suitable for analysis of the oscillations of the tethered bubble. Thus, the dynamics of the bubble restraining by a wire loop in toroidal coordinates can be investigated by using analytical approach and by analogy to the dynamics of a free spherical bubble.

Biaxial extensional motion of an inertially driven radially expanding liquid sheet
View Description Hide DescriptionWe consider the inertially driven, timedependent biaxial extensional motion of inviscid and viscous thinning liquid sheets. We present an analytic solution describing the base flow and examine its linear stability to varicose (symmetric) perturbations within the framework of a longwave model where transient growth and longtime asymptotic stability are considered. The stability of the system is characterized in terms of the perturbation wavenumber, Weber number, and Reynolds number. We find that the isotropic nature of the base flow yields stability results that are identical for axisymmetric and general twodimensional perturbations. Transient growth of shortwave perturbations at early to moderate times can have significant and lasting influence on the longtime sheet thickness. For finite Reynolds numbers, a radially expanding sheet is weakly unstable with bounded growth of all perturbations, whereas in the inviscid and Stokes flow limits sheets are unstable to perturbations in the shortwave limit.

Multiple states of finger propagation in partially occluded tubes
View Description Hide DescriptionRecent experiments by Pailha et al. [Phys. Fluids24, 021702 (Year: 2012)10.1063/1.3682772] uncovered a rich array of propagation modes when air displaces oil from axially uniform tubes that have local variations in flow resistance within their crosssections. The behaviour is particularly surprising because only a single, symmetric mode has been observed in tubes of regular crosssection, e.g., circular, elliptical, rectangular, and polygonal. In this paper, we present experimental results describing a new mode, an asymmetric localised air finger, that persists in the limit of zero propagation speed. We show that the experimental observations are consistent with a model based on capillary static calculations within the tube's crosssection, and the observed bistability is a consequence of the existence of multiple solutions to the Young–Laplace equations. The model also provides an upper bound for the previously reported symmetrybreaking bifurcation[A. de Lózar, A. Heap, F. Box, A. L. Hazel, and A. Juel, Phys. Fluids21, 101702 (Year: 2009)10.1063/1.3247879].

Thermocapillary instability of irradiated transparent liquid films on absorbing solid substrates
View Description Hide DescriptionThe thermocapillary instability of irradiated transparent liquid films on absorbing solid substrates is investigated by means of linear stability analysis. Under such circumstances, incident light passes through a film and is absorbed by the substrate, and the film is then heated by the heat influx across the interface with the substrate. The optical absorption in the substrate is affected by optical reflection. The energy reflectance varies periodically with the film thickness due to optical interference between light waves reflected from the gasliquid and liquidsolid interfaces. The periodic variation of the reflectance strongly affects the film stability, which also varies periodically with the film thickness. Characteristic scales of the instability are also affected by the substrate thickness and incident light intensity. While qualitative aspects of the stability can be easily obtained from the analysis based on a simplified model that is derived under the thinsubstrate assumption, the quantitative evaluation for the case of substrates of moderate to large thickness should be based on a more generalized model that allows for substrates of arbitrary thickness.

Predicting longevity of submerged superhydrophobic surfaces with parallel grooves
View Description Hide DescriptionA mathematical framework is developed to predict the longevity of a submerged superhydrophobic surface made up of parallel grooves. Timedependent integrodifferential equations predicting the instantaneous behavior of the air–water interface are derived by applying the balance of forces across the air–water interface, while accounting for the dissolution of the air in water over time. The calculations start by producing a differential equation for the initial steadystate shape and equilibrium position of the air–water interface at t = 0. Analytical and/or numerical solutions are then developed to solve the timedependent equations and to compute the volume of the trapped air in the grooves over time until a Wenzel state is reached as the interface touches the groove's bottom. For demonstration, a superhydrophobic surface made of parallel grooves is considered, and the influence of the groove's dimensions on the longevity of the surface under different hydrostatic pressures is studied. It was found that for grooves with higher widthtodepth ratios, the critical pressure (pressure at which departure from the Cassie state starts) is higher due to stronger resistance to deflection of the air–water interface from the air trapped in such grooves. However, grooves with higher widthtodepth ratios reach the Wenzel state faster because of their greater air–water interface areas.

Stability of viscous long liquid filaments
View Description Hide DescriptionWe study the collapse of an axisymmetric liquid filament both analytically and by means of a numerical model. The liquid filament, also known as ligament, may either collapse stably into a single droplet or break up into multiple droplets. The dynamics of the filament are governed by the viscosity and the aspect ratio, and the initial perturbations of its surface. We find that the instability of long viscous filaments can be completely explained by the RayleighPlateau instability, whereas a low viscous filament can also break up due to end pinching. We analytically derive the transition between stable collapse and breakup in the Ohnesorge number versus aspect ratio phase space. Our result is confirmed by numerical simulations based on the slender jet approximation and explains recent experimental findings by CastréjonPita et al. [Phys. Rev. Lett.108, 074506 (Year: 2012)]10.1103/PhysRevLett.108.074506.
 Viscous and NonNewtonian Flows

Structuredependent mobility of a dry aqueous foam flowing along two parallel channels
View Description Hide DescriptionThe velocity of a twodimensional aqueous foam has been measured as it flows through two parallel channels, at a constant overall volumetric flow rate. The flux distribution between the two channels is studied as a function of the ratio of their widths. A peculiar dependence of the velocity ratio on the width ratio is observed when the foam structure in the narrower channel is either single staircase or bamboo. In particular, discontinuities in the velocity ratios are observed at the transitions between double and single staircase and between single staircase and bamboo. A theoretical model accounting for the viscous dissipation at the solid wall and the capillary pressure across a film pinned at the channel outlet predicts the observed nonmonotonic evolution of the velocity ratio as a function of the width ratio. It also predicts quantitatively the intermittent temporal evolution of the velocity in the narrower channel when it is so narrow that film pinning at its outlet repeatedly brings the flow to a near stop.

An experimental study of highly transient squeezefilm flows
View Description Hide DescriptionThe aim of this work was to extend a previous investigation of the flow between two parallel disks (one of which was stationary) that have been subjected to a constant energy impact arising from a falling mass onto the upper disk assembly. Whereas the previous work considered the measurement of centreline pressures and distance between the plates only, for a single case, the current work in addition entailed monitoring of pressures at 45% and 90% of disk radius, under 28 combinations of drop height (100 to 1000 mm), drop mass (10 to 55 kg), and initial disk separation (3 to 10 mm), each with 5 repeat tests. Over the duration of the phenomenon (about 3.5 to 10 ms), four basic features were identified: (1) during initial impact under the dominance of temporal inertia, a preliminary pressure spike with peak pressures occurring at a displacement change of less than 0.25 mm from the initial disk separation; (2) an intermediate region with lower pressures; (3) pressure changes arising from a succession of elastic momentum exchanges (bounces) between the colliding masses; and (4) the final largest pressure spike towards the end of the phenomenon, where viscous effects dominate. Regions (1) and (4) became merged for smaller values of initial disk separation, with region (2) being obscured. A previously developed quasisteady linear (QSL) model conformed satisfactorily with pressures measured at the centre of the lower disk; however, substantial deviations from radially parabolic pressure distributions were encountered over a range of operating parameters during the preliminary pressure phenomenon, unexpected because they implicitly conflict with the generally accepted concept of parallel flows and radially selfsimilar velocity profiles in such systems. Measurements of maximum pressures encountered during the preliminary and final pressure events agreed satisfactorily, both with the QSL model and with a simple but effective scaling analysis.
 Particulate, Multiphase, and Granular Flows

Selfdiffusion of wet particles in rotating drums
View Description Hide DescriptionAxial mixing of wet particles in rotating drums was investigated by the discrete element method with the capillary force explicitly considered. Different flow regimes were observed by varying the surface tension of liquid and keeping other conditions unchanged. The analysis of the concentration and mean square displacement of particles indicated that the axial motion of wet particles was a diffusive process characterised by Fick's law. Particle diffusivity decreased with increasing interparticle cohesion and drum filling level but increased with increasing drum rotation speed. Two competing mechanisms were proposed to explain these effects. A theoretical model based on the relation between local diffusivity and shear rate was developed to predict particle diffusivity as a function of drum operation conditions. It was also observed that despite the high inhomogeneity of particle flow in rotating drums, the mean diffusivity of flow exhibited a strong correlation with granular temperature, defined as the mean square fluctuating velocity of particles.

Wall forces on a sphere in a rotating liquidfilled cylinder
View Description Hide DescriptionWe experimentally study the behavior of a particle slightly denser than the surrounding liquid in solid body rotating flow. Earlier work revealed that a heavy particle has an unstable equilibrium point in unbounded rotating flows[G. O. Roberts, D. M Kornfeld, and W. W Fowlis, J. Fluid Mech.229, 555–567 (Year: 1991)10.1017/S0022112091003166]. In the confinement of the rotational flow by a cylindrical wall a heavy sphere with density 1.05 g/cm3 describes an orbital motion in our experiments. This is due to the effect of the wall near the sphere, i.e., a repulsive force (F W ). We model F W on the sphere as a function of the distance from the wall (L): F W ∝L −4 as proposed by Takemura et al. [J. Fluid Mech.495, 235–253 (Year: 2003)10.1017/S0022112003006232]. Remarkably, the path evaluated from the model including F W reproduces the experimentally measured trajectory. In addition during an orbital motion the particle does not spin around its axis, and we provide a possible explanation for this phenomenon.

Velocity profile variations in granular flows with changing boundary conditions: insights from experiments
View Description Hide DescriptionWe present results of detailed velocity profile measurements in a large series of granular flow experiments in a dambreak setup. The inclination angle, bead size, and roughness of the running surface were varied. In all experiments, the downstream velocity profiles changed continuously from the head to the tail of the avalanches. On rough running surfaces, an inflection point developed in the velocity profiles. These velocity profiles cannot be modeled by the large class of constitutive laws which relate the shear stress to a power law of the strain rate. The velocity profile shape factor increased from the head to the tail of the avalanches. Its maximum value grew with increasing roughness of the running surface. We conclude that flow features such as velocity profiles are strongly influenced by the boundary condition at the running surface, which depends on the ratio of bead size to the typical roughness length of the surface. Furthermore, we show that varying velocity profile shape factors inside gravitationally driven finitemass flows give rise to an additional term in the depthaveraged momentum equation, which is normally solved in the simulation software of hazardous geophysical flows. We therefore encourage time dependent velocity profile measurements inside hazardous geophysical flows, to learn about the importance of this “new” term in the mathematical modeling of these flows.

Granular shear flows of flat disks and elongated rods without and with friction
View Description Hide DescriptionGranular shear flows of flat disks and elongated rods are simulated using the Discrete Element Method. The effects of particle shape, interparticle friction, coefficient of restitution, and Young's modulus on the flow behavior and solid phase stresses have been investigated. Without friction, the stresses decrease as the particles become flatter or more elongated due to the effect of particle shape on the motion and interaction of particles. In dense flows, the particles tend to have their largest dimension aligned in the flow direction and their smallest dimension aligned in the velocity gradient direction, such that the contacts between the particles are reduced. The particle alignment is more significant for flatter disks and more elongated rods. The interparticle friction has a crucial impact on the flow pattern, particle alignment, and stress. Unlike in the smooth layer flows with frictionless particles, frictional particles are entangled into large masses which rotate like solid bodies under shear. In dense flows with friction, a sharp stress increase is observed with a small increase in the solid volume fraction, and a spacespanning network of force chains is rapidly formed with the increase in stress. The stress surge can occur at a lower solid volume fraction for the flatter and more elongated particles. The particle Young's modulus has a negligible effect on dilute and moderately dense flows. However, in dense flows where the spacespanning network of force chains is formed, the stress depends strongly on the particle Young's modulus. In shear flows of nonspherical particles, the stress tensor is found to be symmetric, but anisotropic with the normal component in the flow direction greater than the other two normal components. The granular temperature for the nonspherical particle systems consists of translational and rotational temperatures. The translational temperature is not equally partitioned in the three directions with the component in the flow direction greater than the other two. The rotational temperature is less than the translational temperature at low solid volume fractions, but may become greater than the translational temperature at high solid volume fractions.