# Laboratoire de Mécanique des Fluides et d’Acoustique - UMR 5509

LMFA - UMR 5509
Laboratoire de Mécanique des Fluides et d’Acoustique
Lyon
France

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Article dans Phys. Rev. Fluids (2021)

## Lattice Boltzmann study of miscible viscous fingering for binary and ternary mixtures

Lucien Vienne, Simon Marié

A lattice Boltzmann simulation of a viscous fingering instability with miscible components is performed. The core ingredients of the instability are related to the molecular diffusion and the viscosity discrepancy, or, more specifically for the lattice Boltzmann model (LBM), intermolecular friction forces and partial viscosities. Generally, the instability occurs when a less viscous fluid displaces a more viscous fluid in a porous medium. The viscous fingering is first studied in the case of a binary mixture. At early times in the linear regime, the growth rate of the perturbation is computed for different Péclet numbers. The growth rate, as well as the most dangerous and cutoff numbers, increases with the Péclet number. A good agreement is found with a linear stability analysis in which a quasi-steady-state approximation is used. For intermediate times when strong nonlinear interactions take place, the development of the instability is described globally through the mixing length. A high Péclet number leads to more intense instability. Two regimes are visible : the diffusive regime, when fingers remain small and the growth of the mixing length is dominated by diffusion and proportional to $\sqrt{t^★}$, and the convective regime, when fingers become larger than the diffusive length and the growth of the mixing length is proportional to $t^★$. Finally, it is shown that the LBM can describe complex diffusion specific to the case of three species. For instance, viscous fingering could be induced by reverse diffusion despite having a stable initial flow configuration. At early times, each species exhibits different growth rates and then converges to the same value. Afterward, the same diffusive and convective regimes are also recovered for the three species but are delayed in time due to the complex diffusion mechanisms.