Upscaling of a Cahn–Hilliard Navier–Stokes model with precipitation and dissolution in a thin strip

We consider a phase-field model for the incompressible flow of two immiscible fluids. This extends widespread models fluid phases by including third, solid phase, which can evolve due to e.g. precipitation and dissolution. simple, two-dimensional geometry thin strip, still be seen as representation single pore throat in porous medium. Under moderate assumptions on Peclet number capillary number, we investigate limit case when ratio between width length strip is going zero. In this way employing transversal averaging, derive an upscaled model. The result multi-scale consisting equations total flux ion transport, while equation has solved cell-problems at scale determine position interfaces. also sharp-interface model, parameter approaches 0. resulting consists only Darcy-scale equations, explicitly. Notably find asymptotic consistency, that upscaling process commute. use numerical results validity discontinuities are formed