Two-dimensional flow of foam around an obstacle: force measurements

Finite size effects during the penetration of objects in a granular medium

In many industrial or geotechnical applications, objects move through a granular medium and an important issue is the prediction of the force that develops during the motion of the intruder. In this paper, we experimentally study the vertical penetration of intruders into granular media and analyze both the average force and the fluctuations during motion. We investigate configurations where the size of the intruder becomes close to a few grain sizes, a regime that has not been studied before. Finite size effects are observed, showing that both the mean force and the fluctuations significantly increase when decreasing the ratio of the intruder size to the particle size, and scaling laws are identified to characterize this effect. The role of a conical tip in front of the cylinder to facilitate the penetration is also studied, showing that it is more efficient when the aspect ratio between the intruder size and the grain size is low.

Intruder dynamics in a frictional granular fluid: A molecular dynamics study

We study the dynamics of an intruder moving through a fluidized granular medium in three dimensions (d=3). The intruder and grains have both translational and rotational degrees of freedom. The energy-dissipation mechanism is solid friction between all pairs of particles. We keep the granular system fluidized even at rather high densities by randomly perturbing the linear and angular velocities of the grains. We apply a constant external force of magnitude F to the intruder and obtain its steady-state velocity V_{s} in the center-of-mass frame of the grains. The F-V_{s} relation is of great interest in the industrial processing of granular matter and has been the subject of most experiments on this problem. We also obtain the mobility, which is proportional to the inverse viscosity, as a function of the volume fraction ϕ. This is shown to diverge at the jamming volume fraction. For ϕ below the jamming fraction, we find that V_{s}∼F for small F and V_{s}∼F^{1/2} for large F. The intruder shows diffusive motion in the plane perpendicular to the direction of the external force.

Vibration controlled foam yielding

In rheological terms, foams are time independent yield stress fluids, displaying properties of both solid and liquid materials. Here we measure the propagation of a 2D dry foam in a radially symmetric Hele-Shaw cell forcing local yielding. The yield rate is manipulated by mechanical vibration with frequencies from 0 to 150 Hz. The flow speed is then extracted from the video stream and analyzed using digital image correlation software. The data are modeled analytically by a Guzman-Arrhenius type of energy landscape where the local yielding of foam correlates with the number of oscillations, i.e. attempts to cross the energy barrier. The model is confirmed in an auxiliary experiment where the vibrated foam stays in its flowing state at the same small driving pressures, where the flow of the unvibrated foam ceases. We conclude that the yield stress behaviour of foams under an external perturbation can be summarized using a simple energy landscape model. The vibration affects the films causing the stress to occasionally and locally exceed the yield threshold. This, thus, prevents the foam from jamming as in a static configuration even when the global driving is below the yield point of the foam.

A heavy intruder in a locally-shaken granular solid

We experimentally investigate the gravitational-driven motion of a heavy object inside a vertical 2D assembly of identical, plastic cylinders arranged in a regular, triangular lattice. The bottom of the assembly is in contact with a rough plate whose horizontal, sinusoidal motion induces the formation of shear bands in the granular solid, aligned with the edges of the lattice. The intruder sinks when the width of the shear band is as large as its size and halts once the regular configuration of the grains is recovered. The resulting vertical motion of the intruder is random and intermittent, as in disordered granular or colloidal systems near jamming, with alternate flows and blockades. We show, in analogy with earthquakes, that the relation between the size and the duration of the flowing events follows a power-law with an exponent larger than one, and that the statistics of their size is compatible with the Gutenberg-Richter law. We also show that the probability density function of times between flowing events is similar to the Omori law governing the distribution of aftershock sequences following large earthquakes. Finally, the analysis of the velocity fluctuations of the intruder points to a transition from a strong to a weak contact network in the ordered granular assembly, similar to the transition from jammed to fragile states in disordered systems.

X-ray particle tracking velocimetry in liquid foam flow

In this work, we introduce a novel approach to measure the flow velocity of liquid foam by tracking custom-tailored 3D-printed tracers in X-ray radiography. In contrast to optical observations of foam flow in flat cells, the measurement depth equals 100 mm in the X-ray beam direction. Light-weight tracers of millimetric size and tetrapod-inspired shape are additively manufactured from stainless steel powder by selective laser melting. Matching with the foam structure and bubble size, these tracers follow the foam flow. An X-ray beam passes through the radiotransparent foam channel and is detected by an X-ray image intensifier. The X-ray transmission images show the two-dimensional projections of the radiopaque tracers. Utilizing particle tracking velocimetry algorithms, the tracer trajectories are measured with both high spatial (0.2 mm) and temporal (25 fps) resolution. Fine and coarse liquid foam flow of different velocities are studied in a partly curved channel with rectangular cross section. The simultaneous time-resolved measurements of the tracers' translational motion and their intrinsic rotation reveal both the local velocity and vorticity of the foam flow. In the semi-circular curved channel section, the rigid-body-like flow pattern is investigated. Moreover, a relaxation of the foam structure in the transition zone between straight and curved section is observed.

Origin of accelerated and hindered sedimentation of two particles in wet foam

To explore the origin of interactional settling behaviors of multi-particles in wet foam, the sedimentation of two particles placed one above the other as well as placed side by side is studied. According to the average settling velocity in experiment and the average settling drag force of the two particles in numerical simulation, we show that the particles display accelerated sedimentation as placed one above the other while they display hindered sedimentation in the case of the ones positioned side by side. Furthermore, the evolution of structure and force parameters of the bubbles, such as T1 topological events, displacement vector and principal stress fields, shows that the reciprocal action between the foam and the settling particles placed side by side is more significant. The different levels of interplay for these two settling cases also give rise to the diverse changes of bubble pressure response. The bubble pressure component of the average drag force is higher for the particles placed side by side. Especially, for the first time, it reveals that these interactional sedimentation behaviors in the foam are mainly attributed to the changed pressure of bubbles caused by these settling particles at the mesoscopic level. The present results may suggest potential explanations to the cause of the complex accelerated or hindered sedimentation of more particles in wet foam.

Detailed Structural and Mechanical Response of Wet Foam to the Settling Particle

Liquid foam, as a complex fluid, provides an observable prototype for studying a discrete fluid system. In this work, a numerical study on the settling behavior of a round particle in wet polydisperse foam has been conducted on the bubble scale. The local and nonuniform distribution of bubble pressure, as well as the localized plastic events, is presented. It shows a foam region of higher pressure in front of the settling particle due to the extrusion deformation of the bubbles applied by the particle. Additionally, the forces exerted on the particle by the disordered wet foam are measured during the sedimentation. It exhibits in particular a power-law dependence of the drag force caused by the bubble as a function of the foam quality. Moreover, sedimentation experiments are demonstrated to verify this power-law relation. The evolution of the components of drag force is demonstrated when a plastic event occurs in front of the settling particle. The result shows that both the contributions of the pulling force of foam films and the bubble pressure to the drag force decrease in that case. Likewise, the variation of both these contributions to the drag force is illustrated as well when a bubble in the wake detaches from the particle. These results assist in understanding the mesoscopic response of wet foam to a settling particle.

Forces encountered by a sphere during impact into sand

We describe direct measurements of the acceleration of an object impacting on a loosely packed granular bed under various pressures, using an instrumented sphere. The sphere acts as a noninvasive probe that measures and continuously transmits the acceleration as it penetrates into the sand, using a radio signal. The time-resolved acceleration of the sphere reveals the detailed dynamics during the impact that cannot be resolved from the position information alone. Because of the unobstructed penetration, we see a downward acceleration of the sphere at the moment the air cavity collapses. The compressibility of the sand bed is observed through the oscillatory behavior of the acceleration curve for various ambient pressures; it shows the influence of interstitial air on the compaction of the sand as a function of time.

Depth-independent drag force induced by stirring in granular media

The drag force experienced by a horizontal cylinder rotating around the vertical axis in a granular medium under gravity is experimentally investigated. We show that, for deeply buried objects, the drag force dramatically drops after half a turn, as soon as the cylinder crosses its own wake. Whereas the drag during the first half turn increases linearly with the depth, the drag after several rotations appears to be independent of depth, in contradiction with the classical frictional picture stipulating that the drag is proportional to the hydrostatic pressure. We systematically study how the saturated drag force scales with the control parameters and show that this effect may be used to drill deeply in a granular medium without developing high torques.

Elastoplastic flow of a foam around an obstacle

We simulate quasistatic flows of an ideal two-dimensional monodisperse foam around different obstacles, both symmetric and asymmetric, in a channel. We record both pressure and network contributions to the drag and lift forces and study them as a function of obstacle geometry. We show that the drag force increases linearly with the cross section of an obstacles. The lift on an asymmetric aerofoil-like shape is negative and increases with its arc length, mainly due to the pressure contribution.

The continuum theory of shear localization in two-dimensional foam

We review some recent advances in the rheology of two-dimensional liquid foams, which should have implications for three-dimensional foams, as well as other mechanical systems that have a yield stress. We focus primarily on shear localization under steady shear, an effect first highlighted in an experiment by Debrégeas et al. A continuum theory which incorporates wall drag has reproduced the effect. Its further refinements are successful in matching results of more extensive observations and making interesting predictions regarding experiments for low strain rates and non-steady shear. Despite these successes, puzzles remain, particularly in relation to quasistatic simulations. The continuum model is semi-empirical: the meaning of its parameters may be sought in comparison with more detailed simulations and other experiments. The question of the origin of the Herschel-Bulkley relation is particularly interesting.

Journey of an intruder through the fluidization and jamming transitions of a dense granular media

We study experimentally the motion of an intruder dragged into an amorphous monolayer of horizontally vibrated grains at high packing fractions. This motion exhibits two transitions. The first transition separates a continuous motion regime at comparatively low packing fractions and large dragging force from an intermittent motion one at high packing fraction and low dragging force. Associated to these different motions, we observe a transition from a linear rheology to a stiffer response. We thereby call "fluidization" this first transition. A second transition is observed within the intermittent regime when the intruder's motion is made of intermittent bursts separated by long waiting times. We observe a peak in the relative fluctuations of the intruder's displacements and a critical scaling of the burst amplitudes' distributions. This transition occurs at the jamming point phi(J) defined as the point where the static pressure (i.e., the pressure measured in the absence of vibration) vanishes. Investigating the motion of the surrounding grains, we show that below the fluidization transition, there is a permanent wake of free volume behind the intruder. This transition is marked by the evolution of the reorganization patterns around the intruder, which evolve from compact aggregates in the flowing regime to long-range branched shapes in the intermittent regime, suggesting an increasing role of the stress fluctuations. Remarkably, the distributions of the kinetic energy of these reorganization patterns also exhibit a critical scaling at the jamming transition.

Creep motion of an intruder within a granular glass close to jamming

We experimentally study the dynamics of an intruder dragged at a constant force in a horizontally vibrated monolayer of grains. At moderate packing fractions, the intruder moves rapidly as soon as the force is applied. Above some threshold value it has an intermittent creep motion with strong fluctuations reminiscent of "crackling noise". These fluctuations are critical at the jamming transition varphi_{J} unveiled in a previous study. The transition separates a regime with local free volume rearrangements from a regime where the displacement field is strongly heterogeneous and resembles force chain patterns.

Flow in linearly sheared two-dimensional foams: From bubble to bulk scale

We probe the flow of two-dimensional (2D) foams, consisting of a monolayer of bubbles sandwiched between a liquid bath and glass plate, as a function of driving rate, packing fraction, and degree of disorder. First, we find that bidisperse, disordered foams exhibit strongly rate-dependent and inhomogeneous (shear-banded) velocity profiles, while monodisperse ordered foams are also shear banded but essentially rate independent. Second, we adapt a simple model [E. Janiaud, D. Weaire, and S. Hutzler, Phys. Rev. Lett. 97, 038302 (2006)] based on balancing the averaged drag forces between the bubbles and the top plate F[over ]_{bw} and the averaged bubble-bubble drag forces F[over ]_{bb} by assuming that F[over ]_{bw} approximately v;{2/3} and F[over ]_{bb} approximately ( partial differential_{y}v);{beta} , where v and ( partial differential_{y}v) denote average bubble velocities and gradients. This model captures the observed rate-dependent flows for beta approximately 0.36 , and the rate independent flows for beta approximately 0.67 . Third, we perform independent rheological measurements of F[over ]_{bw} and F[over ]_{bb} , both for ordered and disordered systems, and find these to be fully consistent with the forms assumed in the simple model. Disorder thus leads to a modified effective exponent beta . Fourth, we vary the packing fraction phi of the foam over a substantial range and find that the flow profiles become increasingly shear banded when the foam is made wetter. Surprisingly, the model describes flow profiles and rate dependence over the whole range of packing fractions with the same power-law exponents-only a dimensionless number k that measures the ratio of the prefactors of the viscous drag laws is seen to vary with packing fraction. We find that k approximately (phi-phi_{c});{-1} , where phi_{c} approximately 0.84 corresponds to the 2D jamming density, and suggest that this scaling follows from the geometry of the deformed facets between bubbles in contact. Overall, our work shows that the presence of disorder qualitatively changes the effective bubble-bubble drag forces and suggests a route to rationalize aspects of the ubiquitous Herschel-Bulkley (power-law) rheology observed in a wide range of disordered materials.

Simulations of two-dimensional foam rheology: localization in linear Couette flow and the interaction of settling discs

Surface Evolver simulations of flowing two-dimensional foams are described. These are used for two purposes. Firstly, to extract the location of the T 1s, the changes in bubble topology that occur during plastic flow. It is shown that in linear Couette flow the T1 s are localized in space, becoming more so as the polydispersity of the foam decreases. Secondly, the sedimentation of two circular discs through a foam under gravity is studied. If the discs are sufficiently close, they begin to interact and one moves behind the other during their descent.

Discrete rearranging disordered patterns, part II: 2D plasticity, elasticity and flow of a foam

The plastic flow of a foam results from bubble rearrangements. We study their occurrence in experiments where a foam is forced to flow in 2D: around an obstacle; through a narrow hole; or sheared between rotating disks. We describe their orientation and frequency using a topological matrix defined in the companion paper (F. Graner, B. Dollet, C. Raufaste, and P. Marmottant, this issue, 25 (2008) DOI 10.1140/epje/i2007-10298-8), which links them with continuous plasticity at large scale. We then suggest a phenomenological equation to predict the plastic strain rate: its orientation is determined from the foam's local elastic strain; and its rate is determined from the foam's local elongation rate. We obtain a good agreement with statistical measurements. This enables us to describe the foam as a continuous medium with fluid, elastic and plastic properties. We derive its constitutive equation, then test several of its terms and predictions.

A foam film propagating in a confined geometry: analysis via the viscous froth model

A single film (typical of a film in a foam) moving in a confined geometry (i.e. confined between closely spaced top and bottom plates) is analysed via the viscous froth model. In the first instance the film is considered to be straight (as viewed from above the top plate) but is not flat. Instead it is curved (with a circular arc cross-section) in the direction across the confining plates. This curvature leads to a maximal possible steady propagation velocity for the film, which is characterised by the curved film meeting the top and bottom plates tangentially. Next the film is considered to propagate in a channel (i.e. between top and bottom plates and sidewalls, with the sidewall separation exceeding that of the top and bottom plates). The film is now curved along as well as across the top and bottom plates. Curvature along the plates arises from viscous drag forces on the channel sidewall boundaries. The maximum steady propagation velocity is unchanged, but can now also be associated with films meeting channel sidewalls tangentially, a situation which should be readily observable if the film is viewed from above the top plate. Observed from above, however, the film need not appear as an arc of a circle. Instead the film may be relatively straight along much of its length, with curvature pushed into boundary layers at the sidewalls.

Yield drag in a two-dimensional foam flow around a circular obstacle: effect of liquid fraction

We study the two-dimensional flow of foams around a circular obstacle within a long channel. In experiments, we confine the foam between liquid and glass surfaces. In simulations, we use a deterministic software, the Surface Evolver, for bubble details and a stochastic one, the extended Potts model, for statistics. We adopt a coherent definition of liquid fraction for all studied systems. We vary it in both experiments and simulations, and determine the yield drag of the foam, that is, the force exerted on the obstacle by the foam flowing at very low velocity. We find that the yield drag is linear over a large range of the ratio of obstacle to bubble size, and is independent of the channel width over a large range. Decreasing the liquid fraction, however, strongly increases the yield drag; we discuss and interpret this dependence.

Limits of the equivalence of time and ensemble averages in shear flows

In equilibrium systems, time and ensemble averages of physical quantities are equivalent due to ergodic exploration of phase space. In driven systems, it is unknown if a similar equivalence of time and ensemble averages exists. We explore effective limits of such convergence in a sheared bubble raft using averages of the bubble velocities. In independent experiments, averaging over time leads to well-converged velocity profiles. However, the time averages from independent experiments result in distinct velocity averages. Ensemble averages are approximated by randomly selecting bubble velocities from independent experiments. Increasingly better approximations of ensemble averages converge toward a unique velocity profile. Therefore, the experiments establish that in practical realizations of nonequilibrium systems, temporal averaging and ensemble averaging can yield convergent (stationary) but distinct distributions.

Flow of foam past an elliptical obstacle

To investigate the link between discrete small-scale and continuous large scale mechanical properties of a foam, we observe its two-dimensional flow in a channel, around an elliptical obstacle. We measure the drag, lift, and torque acting on the ellipse versus the angle between its major axis and the flow direction. The drag increases with the spanwise dimension, in marked contrast with a square obstacle. The lift passes through a smooth extremum at an angle close to, but smaller than 45 degrees. The torque peaks at a significantly smaller angle 26 degrees. No existing model can reproduce the observed viscous, elastic, plastic behavior. We propose a microscopic visco-elasto-plastic model which agrees qualitatively with the data.

Impact of boundaries on velocity profiles in bubble rafts

Under conditions of sufficiently slow flow, foams, colloids, granular matter, and various pastes have been observed to exhibit shear localization, i.e., regions of flow coexisting with regions of solidlike behavior. The details of such shear localization can vary depending on the system being studied. A number of the systems of interest are confined so as to be quasi two-dimensional, and an important issue in these systems is the role of the confining boundaries. For foams, three basic systems have been studied with very different boundary conditions: Hele-Shaw cells (bubbles confined between two solid plates); bubble rafts (a single layer of bubbles freely floating on a surface of water); and confined bubble rafts (bubbles confined between the surface of water below and a glass plate on top). Often, it is assumed that the impact of the boundaries is not significant in the "quasistatic limit," i.e., when externally imposed rates of strain are sufficiently smaller than internal kinematic relaxation times. In this paper, we directly test this assumption for rates of strain ranging from 10(-3) to 10(-2) s(-1). This corresponds to the quoted rate of strain that had been used in a number of previous experiments. It is found that the top plate dramatically alters both the velocity profile and the distribution of nonlinear rearrangements, even at these slow rates of strain. When a top is present, the flow is localized to a narrow band near the wall, and without a top, there is flow throughout the system.

Viscous instabilities in flowing foams: a Cellular Potts Model approach

The Cellular Potts Model (CPM) successfully simulates drainage and shear in foams. Here we use the CPM to investigate instabilities due to the flow of a single large bubble in a dry, monodisperse two-dimensional flowing foam. As in experiments in a Hele-Shaw cell, above a threshold velocity the large bubble moves faster than the mean flow. Our simulations reproduce analytical and experimental predictions for the velocity threshold and the relative velocity of the large bubble, demonstrating the utility of the CPM in foam rheology studies.

Experimental evidence of flow destabilization in a two-dimensional bidisperse foam

Liquid foam flows in a Hele-Shaw cell were investigated. The plug flow obtained for a monodisperse foam is strongly perturbed in the presence of bubbles whose sizes are larger than the average bubble size by an order of magnitude, at least. Above a velocity threshold, the large bubbles migrate faster than the mean flow. We evidence experimentally this instability and, in the case of a single large bubble, we compare the large bubble velocity to the prediction deduced from scaling arguments. In the case of a bidisperse foam, an attractive interaction between large bubbles induces segregation and the large bubbles organize themselves in columns oriented along the flow.

Anti-inertial lift in foams: a signature of the elasticity of complex fluids

To understand the mechanics of a complex fluid such as a foam we propose a model experiment (a bidimensional flow around an obstacle) for which an external sollicitation is applied, and a local response is measured, simultaneously. We observe that an asymmetric obstacle (cambered airfoil profile) experiences a downwards lift, opposite to the lift usually known (in a different context) in aerodynamics. Correlations of velocity, deformations, and pressure fields yield a clear explanation of this inverse lift, involving the elasticity of the foam. We argue that such an inverse lift is likely common to complex fluids with elasticity.