# Velocity Profiles in Slowly Sheared Bubble Rafts

Velocity Profiles in Slowly Sheared Bubble Rafts, John Lauridsen, Gregory Chanan, and Michael Dennin, PRL vol.93 018303 (2004) [1]

### Keywords

Foam, Yield Stress, Jamming, Couette Viscometer

### Original Abstract from Paper

"Measurements of average velocity profiles in a bubble raft subjected to slow, steady shear demonstrate the coexistence between a flowing state and a jammed state similar to that observed for three-dimensional foams and emulsions [P. Coussot , Phys. Rev. Lett. 88, 218301 (2002)]. For sufficiently slow shear, the flow is generated by nonlinear topological rearrangements. We report on the connection between this short-time motion of the bubbles and the long-time averages. We find that velocity profiles for individual rearrangement events fluctuate, but a smooth, average velocity is reached after averaging over only a relatively few events."

### Soft Matter

This study looks at the jamming of a flow of bubble rafts in a Couette viscometer. The experiment consists of first forming a single layer of bubbles at a fluid-air interface by flowing nitrogen through a solution of 44% glycerine, 28% each of water and "miracle bubble" (presumable containing some surfactant). The single layer of bubbles permits the authors to speak of a bubble "raft" and treat this as a two dimensional problem. In the paper, the units of stress are appropriatly modified for 2D (ie. <math>[ \sigma ] = [ N/m ]</math>). The bubble rafts are located between the concentric cylinders that define the Couette cylinder. The outer cylinder is driven at a constant rotation rate, while the torque on the inner cylinder is measured via a torsion wire (this cylinder is passive - not driven). For a rigid body rotation, one expects an angular velocity profile <math>v_{\theta} = \Omega r</math> from simple classical mechanics. Thus, the authors non-dimensionalize the velocity of the bubbles by <math>v(r) = v_{\theta} / \Omega r</math>, noting that jamming can be identified by <math>v(r_c)=1</math> where <math>r_c</math> is a critical jamming radius. Closer to the inner cylinder than <math>r_c</math>, the stress imparted on the rafts is below the yield stress and hence the raft does not flow: it is jammed. The first row of bubbles adjoining the cylinder surfaces were always observed to stick, and so the definition of the radial distance was modified to reflect this effect, although the offset is not important with respect to the paper's observations.

The rotation rate of the outer cylinder was varied in the range of <math>10^{-3} - 10^{-4} rad/s</math> and is thus very slow. The movement of the bubbles is recorded via video tape and is analysed offline. From the video data, the positions and velocity of each bubble in the raft may be tabulated. The angular velocity profiles are seen to follow an existing theoretical model called the Herschel-Bulkley model, up until a certain distance from the outer cylinder at which point it becomes jammed. The onset of this jamming is observed to have discontinuous behaviour, that is the jamming does not appear in a continuous fashion; the transition is in a sense discrete. Figure 1 shows the normalized velocity profile inside the viscometer, with corresponding solid line plots which represents fits from the Harschel-Bulkley model (a continuous model). Figure 2 depicts actual bubble positions within the viscometer during periods known as stress drops. These at points in at time at which the torque on the inner cylinder exhibits a sharp and sudden decrease. The black circles and grey circles represent bubbles moving in opposite directions (the grey ones moving in the direction of the applied shear). The bubbles moving the in the opposite sense are likely undergoing a packing rearrangement and are responsible for the stress drop. This observation is complemented by the velocity profiles plotted in Figure 2. The boundary between black and grey circles follows the same trend as the jamming radius - that is when the critical jamming stress is achieved closer to the inner wall, so too is the position at which these structural rearrangements take place.

This experiment has looked at the jamming dynamics in a quasi two-dimensional foam. The paper is largely qualitative owing to the extremely complex nature of the jamming phenomena, not too mention the complications which arise when the particles are soft and multiphased.