Multiphase modelling of precipitation-induced membrane formation

Abstract

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 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 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.dynamically according to equations governing the chemistry. We choose to model the fluid-structure combination as a single multiphase material: one component fluid or solvent and one component structure or precipitate membrane. Such multiphase models have proven useful in a variety of complex-fluid applications, such as bacterial biofilms [13,14], tumor-growth [9,21,37,44], and biological membranes [27]; their formulation is based on averaging momentum and stresses in separated, multi-component fluids [17,18].The multiphase framework developed here builds on previous ones [13,25,52], but with some keys differences that are guided by a combination of physical principles, model simplicity, and the micro-fluidic experiments mentioned above. First, our formulation conserves the total mass of the components solvent, dissolved species, and precipitate membrane throughout evolution. In particular, the model accounts for changes in solute concentrations that result from the formation of new membrane and the associated exclusion of solvent volume. This effect is neglected in previous models that treat reaction chemicals as scalar fields distinct from the multiphase material, but is essential for overall mass conservation. The treatment of reaction chemicals as additional components of a multiphase material has been successfully modeled by many [35, 51, see] but greatly complicates the analysis, interpretation, and simulation of the governing equations. Second, by making certain choices in the averaging procedure for the multicomponent stress, our formulation becomes equivalent to an incompressible Brinkman system with variable permeability. This equivalence is important for a few reasons. First, it guarantees that the model reduces to the Stokes equations in...

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