Which governing equation set is applicable to unsteady (time-varying) flow in open channels? Choose the most appropriate theoretical framework used in open-channel hydraulics.

Introduction / Context:
Open-channel flows such as river floods, dam-break waves, and surges are often unsteady and nonuniform. Their transient behaviour is described by depth-averaged equations that capture time variation and spatial gradients of depth and discharge.

Given Data / Assumptions:

• Free-surface flow dominated by gravity (shallow water assumption).
• One-dimensional depth-averaged modelling is acceptable over channel reaches.
• Time dependence must be represented explicitly.

Concept / Approach:
The Saint-Venant (shallow-water) equations comprise continuity and momentum partial differential equations with source terms for bed slope, friction, and external inputs. They generalize gradually varied flow to unsteady conditions and are the standard basis of flood routing and surge simulation.

Step-by-Step Solution:
Continuity: ∂A/∂t + ∂Q/∂x = q_l (lateral inflow per unit length).Momentum: ∂Q/∂t + ∂(βQ^2/A)/∂x + gA ∂y/∂x = gA(S0 − Sf) + external forces.These equations allow wave celerity, backwater effects, and reflections to be modelled.Numerical schemes (e.g., finite volume) integrate them for hydrographs and stage predictions.

Verification / Alternative check:
Special cases recover steady GVF (when ∂/∂t = 0) or kinematic waves (when inertia is neglected). Hence they correctly subsume many open-channel formulae as limits.

Why Other Options Are Wrong:
Chezy/Manning: empirical steady uniform relations; do not model time dependence or spatial storage.Bernoulli: steady inviscid along a streamline; not a full unsteady channel model.Darcy–Weisbach: pipe head loss, not open-channel unsteady theory.

Common Pitfalls:

• Applying steady uniform formulas to floods or surges.
• Neglecting friction/source terms in the momentum equation.

Final Answer:
Saint-Venant equations (unsteady continuity and momentum for shallow water)