Porous two-dimensional crystals offer many promises for water desalination applications. For computer simulation to play a predictive role in this area, however, one needs to have reliable methods for simulating an atomistic system with hydrodynamic currents and interpretative tools to relate microscopic interactions to emergent macroscopic dynamical quantities, such as friction, slip length, and permeability. In this article, we use Gaussian dynamics, a nonequilibrium molecular dynamics method that provides microscopic insights into the interactions that control the flows of both simple liquids and liquid water through atomically small channels. In simulations of aqueous transport, we mimic the effect of changing the membrane chemical composition by adjusting the attractive strength of the van der Waals interactions between the membrane atoms and water. We find that the wetting contact angle, a common measure of a membranes hydrophobicity, does not predict the permeability of a membrane. Instead, the hydrophobic effect is subtle, with both static and dynamic effects that can both help and hinder water transport through these materials. The competition between the static and dynamical hydrophobicity balances an atomic membranes tendency to wet against hydrodynamic friction, and determines an optimal contact angle for water passage through nonpolar membranes. To a reasonable approximation, the optimal contact angle depends only on the aspect ratio of the pore. We also find that water molecules pass through the most hydrophobic membranes in a punctuated series of bursts that are separated by long pauses. A continuous-time Markov model of these data provides evidence of a molecular analogue to the clogging transition, a phenomenon observed in driven granular flows.