Hydraulic equations are derived for a stratified (two-layer) flow in which the horizontal velocity varies continuously in the vertical. Viscosity is included in the governing equations, and the effect of friction in hydraulically controlled flows is examined. The analysis yields Froude numbers which depend upon the integrated inverse square of velocity but reduce to the original layered Froude numbers when velocity is constant with depth. The Froude numbers reveal a critical condition for hydraulic control, which equates to the arrest of internal gravity waves.
Solutions are presented for the case of unidirectional flow through a lateral constriction, both with and without bottom drag. In the free-slip lower boundary case, viscosity transports momentum from the faster to the slower layer, thereby shifting the control point downstream and reducing the flux through the constriction. However, while the velocity shear at the interface between the two layers is reduced, the top-to-bottom velocity difference of the controlled solution is increased for larger values of viscosity. This counter-intuitive result is due to the restrictions placed on the flow at the hydraulic control point. When bottom drag is included in the model, the total flux may increase, in some cases exceeding that of the inviscid solution.