Dissipative Rheology of Foams
- Simon Cox (Aberystwyth University, United Kingdom)
- Isabelle Cantat (Université Rennes I, France)
- Reinhard Hohler (Université Paris-Est, Marne-la-Vallée, France)
- Stefan Hutzler (Trinity College Dublin, Ireland)
A groundswell of opinion in the community of researchers interested in foams and emulsions suggests that while the "quasistatic" flow of these materials is understood in broad terms, and can often be predicted with current numerical methods, the origin of the dissipation observed at finite shear rate is not so clear. A good deal of current theoretical and experimental activity aims to understand and predict the response of disordered foams and emulsions at finite shear-rate and, in particular, to include dissipative processes due to stretching and generation of new films and bubble rearrangements at a mesoscopic level. It is this coupling between different length-scales (mesoscopic bubble vs film) that provides a major barrier to progress.
Soft materials such as foams and emulsions are complex fluids which show both solid and liquid properties [1,2]. Although consisting mostly of air, their macroscopic response can be elastic, plastic or viscous, depending on the applied stress and composition. This behaviour is due to coupled processes on three length scales, set by the surfactant-laden liquid interfaces, the bubbles and clusters of bubbles undergoing intermittent rearrangements, often subsumed into a mean field or continuum approach.
Many sources of dissipation within a flowing foam or emulsion have been described . They include viscous effects from the bulk  and surfaces, interfacial rheology and the motion of surfactant molecules [5, 6, 8,9]. Our goal is to develop computationally-efficient multi-scale methods that accurately represent these processes. This is an interdisciplinary effort, involving applied mathematicians, physicists and chemical engineers, among others, and requiring the use of techniques from hydrodynamics, physical chemistry and numerical analysis. We recognise that advances in neighbouring fields, for example soft glassy materials , amorphous solids  and suspensions , may therefore contribute to this effort.
A number of recent continuum models [5,13,14,15,16] have shear-rate as a variable, but here the difficulty is in recognising the effect of structural disorder and microscopic processes on the resulting behaviour.
Current effort is also directed at developing bubble-scale models, pioneered by Princen , to predict a foam's rheological response, but all fail in certain limits, for example fast disordered flows. Durian's bubble model , which is based upon particles that interact via a potential, is appropriate only to the wet limit; Surface Evolver [19,20] and vertex models  work well in the dry, quasi-static regime; the Viscous Froth model [22,23] describes dry 2D flows at finite shear-rate well; and hydrodynamical models [24,25] include the fluid flow in the interstices between the bubbles but are limited to small, ordered systems.
 R. Hohler and S. Cohen-Addad. Rheology of Liquid Foam. J. Phys.: Condens. Matter, 17:R1041R1069, 2005.
 D. Weaire and S. Hutzler. The Physics of Foams. Clarendon Press, Oxford, 1999.
 D.M.A. Buzza, C.-Y. D. Lu, and M.E. Cates. Linear Shear Rheology of Incompressible Foams. J. Phys. II France, 5:3752, 1995.
 A.M. Kraynik and M.G. Hansen. Foam rheology: A model of viscous phenomena. J. Rheol., 31:175205, 1987.
 N.D. Denkov, S. Tcholakova, K. Golemanov, K.P. Ananthapadmanabhan, and A. Lips. Viscous friction in foams and concentrated emulsions under steady shear. Phys. Rev. Lett., 100:138301, 2008.
 M. Durand and H. Stone. Relaxation time of the topological T1 process in a two-dimensional foam. Phys. Rev. Lett., 97:226101, 2006.
 K. Krishan, A. Helal, R. Höhler and S. Cohen-Addad. Fast relaxations in foam: Linking interfacial and bulk rheology, Physical Review E, in press, 2010.
 S.Besson and G. Debrégeas. Statics and dynamics of adhesion between two soap bubbles. Euro. Phys. J. E, 24, 109-117, 2007.
 P. Sollich. Rheological constitutive equation for a model of soft glassy materials. Phys. Rev. E, 58:738-759, 1998.
 S. Karmakar, A. Lemaître, E. Lerner, and I. Procaccia. Predicting Plastic Flow Events in Athermal Shear-Strained Amorphous Solids. Phys. Rev. Lett. 104: 215502, 2010.
 J.F. Brady and J.F. Morris. Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fl. Mech., 348,103-139.
 S. Benito, C.-H. Bruneau, T. Colin, C. Gay, and F. Molino. An elasto-visco-plastic model for immortal foams or emulsions. Eur. Phys. J. E, 25:225251, 2008.
 I. Cheddadi, P. Saramito, C. Raufaste, P. Marmottant, and F. Graner. Numerical modelling of foam couette flows. Euro. Phys. J. E, 27:123133, 2008.
 E. Janiaud, D. Weaire, and S. Hutzler. Two dimensional foam rheology with viscous drag. Phys. Rev. Lett., 97:038302, 2006.
 G. Katgert, M.E. Mobius, and M. van Hecke. Rate dependence and role of disorder in linearly sheared two-dimensional foams. Phys. Rev. Lett., 101:058301, 2008.
 H.M. Princen. Rheology of Foams and Highly Concentrated Emulsions. I. Elastic Properties and Yield Stress of a Cylindrical Model System, J. Coll. Int. Sci., 91, 160-174, 1983.
 D.J. Durian. Foam Mechanics at the Bubble Scale. Phys. Rev. Lett., 75:47804783, 1995.
 K. Brakke. The Surface Evolver. Exp. Math., 1:141165, 1992.
 A. Wyn, I.T. Davies, and S.J. Cox. Simulations of two-dimensional foam rheology: localization in linear couette flow and the interaction of settling discs. Euro. Phys. J. E, 26:8189, 2008.
 I. Cantat and R. Delannay. Dissipative flows of 2D foams. Euro. Phys. J. E, 18:5567, 2005.
 T.E. Green, P. Grassia, L. Lue, and B. Embley. Viscous froth model for a bubble staircase structure under rapid applied shear: An analysis of fast flowing foam. Coll. Surf. A, 348:4958, 2009.
 N. Kern, D. Weaire, A. Martin, S. Hutzler, and S.J. Cox. Two-dimensional viscous froth model for foam dynamics. Phys. Rev. E, 70:041411, 2004.
 J. Higdon. See http://appliedmath.math.uiuc.edu/faculty_profile_higdon.html
 X. Li, H. Zhou, and C. Pozrikidis. A numerical study of the shearing motion of emulsions and foams. J. Fluid Mech., 286:379404, 1995.