Complex isochoric flows in a domain of space that is long compared to its width are studied for a viscoplastic and perfectly rigid Herschel–Bulkley model. It is argued here that no continuous yield surface can exist along the flow direction in these either confined or open channel flows. A similarity analysis is performed that shows that normal stresses cannot be neglected. For open channel flows the influence of normal stresses can be estimated through comparison of the yield stress value to the hydrostatic pressure value at the channel bed. Generalized Barré de Saint Venant one‐dimensional equations are obtained. The influence of the yield stress value on wave velocity and on gradually varied flows and critical depth has been deduced.