Thesis Summary:

We consider the coupled Navier-Stokes/advection-diffusion equations, with nonlinear con- ditions across a semi-permeable membrane. This system of partial differential equations arises from reverse osmosis modeling in water desalination processes. To solve the Navier-Stokes equations, we use a conservative hybridizable discontinuous Galerkin (HDG) scheme in primal form. For the advection-diffusion equation, we also employ an HDG method in the primal form. The main advantage of our formulation, in addition to the conservative property, is the fact that the size of the global linear system to solve is smaller compared to non-hybrid schemes for high-order approximations. We show that the scheme is well-posed under smallness assump- tions on the data, obtain error estimates and present numerical experiments illustrating the performance of the scheme.