We present a theory for the three-dimensional flow of a Bingham-plastic fluid in a
shallow and wide channel. Focusing attention on slow flows appropriate for gentle
slopes, low discharge rates or the final stage of deposition, we ignore inertia and
apply the long-wave approximation. For steady flows, the velocity distribution, total
discharge, and section-averaged flux are obtained analytically in terms of the fluid
property and the geometry of the channel cross-section. Nonlinear stationary waves,
which connect a uniform depth upstream to another uniform depth downstream,
are then investigated, for both wet and dry beds. A numerical scheme is applied to
calculate the transient flow evolution. The final development of the stationary wave
due to steady discharge upstream is obtained numerically and the relation between
the tongue-like shape of the wave front and the fluid property is discussed. The
phase speed of the stationary wave is also derived analytically. Finally, the transient
spreading of a finite fluid mass released from a reservoir after a dam break is simulated
numerically. The transient development of the front and the final extent of deposition
are examined.