Slow dow of a Bingham fluid in a shallow channel of finite width

We present a theory for the three-dimensional flow of a Bingham-plastic flu id in a shallow and wide channel. Focusing attention on slow flows appropri ate 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 t he channel cross-section. Nonlinear stationary waves, which connect a unifo rm depth upstream to another uniform depth downstream, are then investigate d, for both wet and dry beds. A numerical scheme is applied to calculate th e transient flow evolution. The final development of the stationary wave du e to steady discharge upstream is obtained numerically and the relation bet ween the tongue-like shape of the wave front and the fluid property is disc ussed. The phase speed of the stationary wave is also derived analytically. Finally, the transient spreading of a finite fluid mass released from a re servoir after a dam break is simulated numerically. The transient developme nt of the front and the final extent of deposition are examined.