Complete governing equations of fluid movement

The movement of water in an open channel (1D unsteady flow) is described by the Saint-Venant equations, a pair of equations from conservation of mass and momentum.

Continuity (conservation of mass):

$$\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=q$$

Momentum (conservation of momentum):

$$\underbrace{\frac{\partial Q}{\partial t}}_{\text{local acc.}}+\underbrace{\frac{\partial}{\partial x}\!\left(\frac{Q^2}{A}\right)}_{\text{convective acc.}}+\underbrace{gA\frac{\partial y}{\partial x}}_{\text{pressure force}}-\underbrace{gAS_0}_{\text{gravity}}+\underbrace{gAS_f}_{\text{friction}}=0$$

where $A$ = flow area, $Q$ = discharge, $y$ = flow depth, $q$ = lateral inflow per unit length, $S_0$ = bed slope, $S_f=\dfrac{n^2Q|Q|}{A^2R^{4/3}}$ = friction slope (Manning), $g$ = gravity. These are the Saint-Venant equations (1D unsteady open-channel flow).

In terms of velocity $V$ and depth $y$ for a wide rectangular channel:

$$\frac{\partial y}{\partial t}+V\frac{\partial y}{\partial x}+y\frac{\partial V}{\partial x}=0,\qquad \frac{\partial V}{\partial t}+V\frac{\partial V}{\partial x}+g\frac{\partial y}{\partial x}=g(S_0-S_f)$$

Assumptions: one-dimensional flow, hydrostatic pressure, small bed slope, uniform velocity over the section, and friction given by the steady-flow (Manning) formula. The two equations have two unknowns, $Q$ (or $V$) and $y$ (or $A$), as functions of $x$ and $t$.