  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.
