Multiphase modeling of precipitation-induced membrane formation

We formulate a model for the dynamic growth of a membrane developing in a flow as the result of a precipitation reaction, a situation inspired by recent microfluidic experiments. The precipitating solid introduces additional forces on the fluid and eventually forms a membrane that is fixed in the flow due to adhesion with a substrate. A key challenge is that the location of the immobile membrane is unknown $\textit{a priori}$. To model this situation, we use a multiphase framework with fluid and membrane phases; the aqueous chemicals exist as scalar fields that react within the fluid to induce phase change. To verify that the model exhibits desired fluid-structure behaviors, we make a few simplifying assumptions to obtain a reduced form of the equations that is amenable to exact solution. This analysis demonstrates no-slip behavior on the developing membrane without $\textit{a priori}$ assumptions on its location. The model has applications towards precipitate reactions where the precipitate greatly affects the surrounding flow, a situation appearing in many laboratory and geophysical contexts including the hydrothermal vent theory for the origin of life. More generally, this model can be used to address fluid-structure interaction problems that feature the dynamic generation of structures.