Making use of a mathematical model solving the complete NavierStokes equations for steady flow in coiled rectangular pipes, fully-developed laminar flow in shallow curved channels is analysed physically and mathematically. Transverse convection of momentum by the secondary flow is shown to cause important deformations of the main velocity distribution. The model is also used to investigate simplified computation methods for shallow channels. The usual 'shallow water approximation' is shown to fail here, but a method starting from similarity hypotheses for the main and the secondary flow works well. On the basis of this method, a simplified mathematical model of steady turbulent flow in river bends is developed and verified using the results of laboratory experiments and fully three-dimensional flow computations. This model works well for shallow and mildly curved channels, but it shows important shortcomings if the channel is less shallow or sharplier curved.