We discuss mixing and transport in three-dimensional, steady, Navier–Stokes flows with the no-slip condition at the boundaries. The advective flux is related to the dynamics of the Navier–Stokes equations and a prediction is made of the scaling of the advective flux with the Reynolds number: the flux is expected to decay as the Reynolds number goes to infinity. This prediction is made via a Melnikov-type calculation together with boundary layer concepts through which the flow is split into an integrable and a small non-integrable part. The rate of decay is related to the details of viscous flow in boundary layers. The Melnikov function is related to the Bernoulli integral of the underlying Euler flow. The effects of molecular diffusivity are discussed and the effective axial diffusivity scaling predicted as a function of Reynolds and Péclet numbers. Using these ideas, we study the mass transport in the wavy vortex flow in the Taylor–Couette apparatus as a particular example. We propose an explanation of the observed non-monotonic behaviour of flux with increasing Reynolds number that was not captured in any of the previous models. It is shown that there is a Reynolds number at which the axial flux in the wavy vortex flow is maximized. At the low range of Reynolds numbers for which the wavy vortex flow is stable the flux increases, while for large Reynolds numbers it decreases. We compare these predictions with the available experimental and numerical data on the wavy vortex flow.