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We use a lattice gas cellular automata model in the presence of randomdynamic scattering sites and quenched disorder in the two-phase immisciblemodel with the aim of producing an interface dynamics similar to that observedin Hele-Shaw cells. The dynamics of the interface is studied as one fluiddisplaces the other in a clean lattice and in a lattice with quenched disorder.For the clean system, if the fluid with a lower viscosity displaces the other,we show that the model exhibits the Saffman-Taylor instability phenomenon,whose features are in very good agreement with those observed in real (viscous)fluids. In the system with quenched disorder, we obtain estimates for thegrowth and roughening exponents of the interface width in two cases:viscosity-matched fluids and the case of unstable interface. The first case isshown to be in the same universality class of the random deposition model withsurface relaxation. Moreover, while the early-time dynamics of the interfacebehaves similarly, viscous fingers develop in the second case with thesubsequent production of bubbles in the context of a complex dynamics. We alsoidentify the Hurst exponent of the subdiffusive fractional Brownian motionassociated with the interface, from which we derive its fractal dimension andthe universality classes related to a percolation process.

physical review. e, statistical, nonlinear and soft matter physics

Interface dynamics of immiscible two-phase lattice-gas cellular  automata: A model with random dynamic scatterers and quenched disorder in two  dimensions