We consider a mathematical model for addressing coupled fluid flow problems arising from reverse osmosis modeling in water desalination processes. It consists of the coupled Navier-Stokes/transport equations, with nonlinear conditions across a semipermeable membrane. To solve these set of partial differential equations, we present two different numerical methods. First, we employ a mixed finite element method able to capture several variables of interest such as the salt concentration level, pressure drop and fluid velocity. Secondly, a conservative hybridizable discontinuous Galerkin method is also employed. Through diverse numerical simulations and a variety of configurations, we illustrate the capability of both methods to accurately capture the behavior of saline water when passing through membrane-based reverse osmosis desalination channels.