In this study, the well-known Multiphase Mixture Model (MMM) was improved with a new single domain approach which was used to model the flow behaviour and performance of a Flowing Electrolyte - Direct Methanol Fuel Cell (FE-DMFC). Emphasis was placed on the methanol and water crossover fluxes. Unlike the existing methods, the proposed method only requires the mixture variables, thereby decoupling the requirement for information about the gaseous state. This new approach was demonstrated by describing the physics of the fuel cell under base line operating conditions, as well as different cathode humidifications. The observed trends were found to be consistent with experimental and modeling results in DMFC literature. The model suggests that the flowing electrolyte channel (FEC) effectively washes unreacted methanol out of the system, yielding a negligible methanol crossover flux at the cathode catalyst layer. Furthermore, it was determined that the convection caused by the back-pressure within the FEC is an important factor to consider; since this effect is sufficiently strong to cause water to flow from the FEC to the anode. The model also suggests that decreasing the cathode humidification increases both the methanol and water crossover fluxes as well as the ohmic resistance within the membranes and FEC. This result suggests that some liquid water within the cathode could be beneficial for the fuel cell, to decrease these losses.