The paper establishes the governing equations of the dynamics of a foamy continuum, in the sense of Davini and Podio-Guidugli, which is expected to mimic the behavior of a monodisperse fluid foam. We model this special continuum as a dissipative ordered fluid, following the lead by Sonnet and Virga. Attention to both regular and topological transformations is taken.
PrincenHM. Rheology of foams and highly concentrated emulsions: I. Elastic properties and yield stress of a cylindrical model system. J Colloid Interface Sci1983; 91: 160–175.
2.
PrincenHM. Rheology of foams and highly concentrated emulsions: II. extended experimental study of the yield stress and wall effects for concentrated oil-in-water emulsions. J Colloid Interface Sci1985; 105: 150–171.
3.
KhanSAArmstrongRC. Rheology of foams: I. theory of dry foams. J Non Newton Fluid Mech1986; 22: 1–22.
4.
KraynikAMHansenMG. Foam and emulsion rheology: a quasistatic model for large deformations of spatially-periodic cells. J Rheol1986; 30: 409–439.
5.
KraynikAM. Foam flows. Annu Rev Fluid Mech1988; 20: 325–357.
6.
WeaireDHutzlerS. The physics of foams. Oxford: Oxford University Press, Clarendon Press, 1999.
7.
DaviniC. A continuum model for fluid foams. J Elast2010; 101: 77–99.
8.
DaviniC. A continuum model of fluid foams revisited. J Elast2024; 155: 355–369.
9.
DaviniCPodio-GuidugliP. On the mathematical modelling of 2D fluid foams. special issue for the 100 years of ICTAM, meccanica, 2024
10.
PuglisiGTruskinovskyL. Mechanics of a discrete chain with bi-stable elements. J Mech Phys Solids2000; 48: 1–27.
DaviniC. A proposal for a continuum theory of defective crystals. Arch Rational Mech Anal1986; 96: 295–317.
15.
EricksenJ. Equilibrium theory for x-ray observations of crystals. Arch Rational Mech Anal1997; 139: 181–200.
16.
CardinFFavrettiM. Dynamics of a chain of springs with nonconvex potential energy. Math Mech Solids2003; 8: 651–669.
17.
GrasmanJ. Asymptotic methods for relaxation oscillations and applications. New York: Springer, 1987.
18.
KaperT. An introduction to geometric methods and dynamical systems theory for singular perturbation problems. In: O’MalleyRCroninJ (eds) Analyzing multiscale phenomena using singular perturbation methods, Proceedings of Symposia in Applied Mathematics, vol. 56. Providence, RI: American Mathematical Society, 1999, pp. 85–132.
19.
Podio-GuidugliP. A primer in elasticity. New York: Kluwer, 2000.