Dynamics of a flexible fibre in a sheared two-dimensional foam: Numerical simulations

Abstract

Recently there has been a renewed interest in using foamy suspensions of wood fibres as a carrier fluid in papermaking but there is a lack of fundamental understanding of the dynamics of such a three-phase system. In this article we propose a numerical model for the dynamics of an individual flexible fibre within a flowing foam, based on discrete-element methods. As is observed in a Newtonian shear flow, we observe that the fibre systematically experiences a tumbling instability: the disordered motion of bubbles cannot prevent the pseudo-periodical flip of the fibre. Our simulations show that the tumbling time decreases almost as the inverse of the strain rate. It also decays when the fibre length is increased, though for long enough fibres it reaches a constant value. Similarly the tumbling time is also surprisingly independent of the stiffness of the fibre. Because of their tumbling motion, long and flexible fibres spend most of the time in a coiled geometry. This would imply that using foam as a carrier fluid is not enough to keep fibres aligned with the flow. However, further refinements of the model will need to be considered to arrive at firm conclusions regarding alignment. Dear Editor,We would like to thank the referee for his careful reading and constructive remarks. We tried to follow his recommendations, and to answer his comments and concerns regarding the article. The modifications we brought to the paper are listed below:Reviewer #1 * General questions that seem not to be addressed are: Is there shear-banding or localisation? Does it matter where between the side walls the fibre is placed? (I guess it shouldn't, but I presume that this was checked.)Since no additional wall friction is added (see our previous study of the bubble model in Langlois et al. [2008]), no shear-banding is observed. The average velocity profile is mostly undisturbed by the presence of a unique fibre, and remains linear.The initial y-position of the fibre is of no influence, except in the 'pathological' case where it comes in contact with one of the walls and remains there.* lines 10-20 are not very clear on how the presence of the fibres affects these foam properties; it would be useful to expand this section to give the reader more of a feel for the complexity of this area of research and the way in which a foam interacts with fibres.We have added to the introduction a more thorough description on the effects of fibres on the foam, recently observed experimentally (l.15 to 21).* lines 63 and 96: I find the use of the word "spherical" misleading, since in a 2D experiment bubbles are more discoidal than spherical and they are certainly treated here as discs. Once the description of the bubble model is restricted to 2D, I suggest to use "discs" exclusively.In the simulations the bubbles are indeed treated as disks, though we had initially used the term "spheres" since these disks are meant to mimick a bubble raft made of roughly spherical bubbles seen from above. However we agree that this can lead to some confusion...

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