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Abstract

We investigate a lattice Boltzmann model which can be applied for the simulation of foams as a multiphase system. In order to decrease the surface tension and increase the lifetime of bubbles, we use the multirange pseudopotential scheme proposed in [1]. Here, the interactions of a given site with single-range neighbors x_i + c_kΔt, as well as with the double-range ones x_i +2c_kΔt are taken into account with coefficients G¹_k and G²_k, respectively. Such scheme allows one to vary independently both the pressure from the equation of state which is determined by G¹ +2G², and the surface tension proportional to G¹ +8G² . Particularly, setting G¹ +8G² = 0 significantly increases the decay time of the foam. Unlike the scheme of [2], the method used preserves the isotropy. To simulate arbitrary equation of state, we apply the method proposed in [3].

This scheme stabilizes the stationary foam. Under shear, however, the bubbles begin to coalesce, and the foam decays rather fast. In order to overcome this effect, we introduce into the LB equation a second component which acts like a surfactant. It is attracted to the interfaces (regions where the gradient of the density of the main components is large). We introduce an interaction force which is proportional to ∇|∇ρ|² and also to the density of surfactant. The relative motion of the surfactant and the main component produces a friction force proportional to ρρ_s(u − u_s). We use van der Waals equation of state for the main component, and the ideal gas one for the surfactant. Introduction of the surfactant leads to a significant increase of the foam lifetime for a certain range of parameters.

In order to further stabilize the foam (and to get closer to the reality), we introduce one more component which represents the gas filling the bubbles. To prevent the dissolution of the gas, a repulsion force between it and the main component is used, which hinders the collapse of bubbles.

[1] M. Sbragaglia, R. Benzi, L. Biferale, S. Succi, K. Sugiyama, and F. Toschi. Generalized lattice Boltzmann method with multirange pseudopotential. Phys. Rev. E 75(2):026702 (2007).
[2] G. Falcucci, S. Chibbaro, S. Succi, X. Shan, and H. Chen. Lattice Boltzmann spray-like fluids. Europhys. Lett. 82:24005 (2008).
[3] A. L. Kupershtokh, D. A. Medvedev, D. I. Karpov. On equations of state in a lattice Boltzmann method. Computers & Mathematics with Applications 58(5):965 (2009)