There are at least two ways that hydrodynamic processes can limit continuous flow electrophoresis. One arises from the sensitivity of the flow to small temperature gradients, especially at low flow rates and power levels. This sensitivity can be suppressed, at least in principle, by providing a carefully tailored, stabilizing temperature gradient in the cooling system that surrounds the flow channel. At higher power levels another limitation arises due to a restructuring of the main flow. This restructuring is caused by buoyancy, which is in turn affected by the electro-osmotic crossflow. Approximate solutions to appropriate partial differential equations have been computed by finite difference methods. One set of results is described here to illustrate the strong coupling between the structure of the main (axial) flow and the electro-osmotic flow.