We describe a lattice-Boltzmann model with a multigrid strategy and moving boundaries to solve coupled multiscale flow problems with scalar transport. Whereas we present the details of a two-dimensional model, the method is directly generalizable to 3-D. We apply the methods to our application of nutrient uptake at the epithelium of the small intestine. In this model, the fine grid is embedded within a coarse grid, with an overlap region between the two grids. The transfer of information between the two grids conserves mass and momentum and enforces continuity of stress. The modified moment propagation method is applied to the scalar, allowing higher Schmidt numbers than competing methods. We treat the moving boundaries with second-order boundary conditions that interpolate to the exact wall position. For scalar and scalar flux boundary condition, we formulate the scalar transport from the boundary to a temporary lattice node, and then transfer it back to the node adjacent to the boundary by extrapolation. We demonstrate the application of the method by simulating a macro-scale cavity flow with micro-scale finger-like protuberances in pendular motion on the lower surface as a model of the macro-micro scale interactions in fluid motions, scalar mixing, and scalar uptake at the surface of the villi that line the epithelium of the human intestines.
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