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Abstract We derive a class of exact solutions for Stokes flow in infinite and semi‐infinite channel geometries with permeable walls. These simple, explicit, series expressions for both pressure and Stokes flow are valid for all permeability values. At the channel walls, we impose a no‐slip condition for the tangential fluid velocity and a condition based on Darcy's law for the normal fluid velocity. Fluid flow across the channel boundaries is driven by the pressure drop between the channel interior and exterior; we assume the exterior pressure to be constant. We show how the ground state is an exact solution in the infinite channel case. For the semi‐infinite channel domain, the ground‐state solutions approximate well the full exact solution in the bulk and we derive a method to improve their accuracy at the transverse wall. This study is motivated by the need to quantitatively understand the detailed fluid dynamics applicable in a variety of engineering applications including membrane‐based water purification, heat and mass transfer, and fuel cells. 

Microfiltration is a widely used engineering technology for fresh water production and water treatment. The major concern in many applications is the formation of a biological fouling layer leading to increased hydraulic resistance and flux decline during membrane operations. The growth of bacteria constituting such a biological layer implicates the formation of a multispecies biofilm and the consequent increase of operational costs for reactor management and cleaning procedures. To predict the biofouling evolution, a mono-dimensional continuous free boundary model describing biofilm dynamics and EPS production in different operational phases of microfiltration systems has been well studied. The biofouling growth is governed by a system of hyperbolic PDEs. Substrate dynamics are modeled through parabolic equations accounting for diffusive and advective fluxes generated during the filtration process. The free boundary evolution depends on both microbial growth and detachment processes. What is not addressed is the interplay between biofilm dynamics, filtration, and water recovery. In this study, filtration and biofilm growth modeling principles have been coupled for the definition of an original mathematical model able to reproduce biofouling evolution in membrane systems. The model has been solved numerically to simulate biologically relevant conditions, and to investigate the hydraulic behavior of the membrane. It has been calibrated and validated using lab-scale data. Numerical results accurately predicted the pressure drop occurring in the microfiltration system. A calibrated model can give information for optimization protocols as well as fouling prevention strategies.