Lattice Bhatnagar-Gross-Krook models for miscible fluids

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Abstract
Lattice Bhatnagar-Gross-Krook models for miscible fluid flow in two (2D) and three (3D) dimensions are introduced. The convection-diffusion (CD) equation and the Navier-Stokes (NS) equation describing the macroscopic behavior of the models are derived using the Chapman-Enskog expansion technique. Corrections to the CD equation of higher order in the flow velocity are obtained, and it is shown how the present models are linked with the existing Boltzmann model. It is also shown how the Navier-Stokes dynamics is explicitly decoupled from the diffusive behavior of the model. The results obtained from both 2D and 3D simulations are observed to be in excellent agrement with the analytic predictions. In particular it is shown that the models are well described by theory for high-Péclet-number flows. We also present simulation results confirming the anomalous (but small) velocity dependence of the CD equation, and we investigate the models’ sensitivity to large gradients in the concentration profile.

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