Gradually varied flow (GVF) in open channels:\nHow is the flow classified when depth varies slowly along the channel but does not change with time?

Introduction / Context:
Open-channel flows are classified by temporal variation (steady vs unsteady) and spatial variation (uniform vs non-uniform). Gradually varied flow (GVF) is ubiquitous—occurring upstream of dams, along mild slopes, and near control structures. Correct classification underpins the use of the GVF differential equation and profile analysis (M1, M2, S1, etc.).

Given Data / Assumptions:

• Depth and velocity change with distance but slowly (gradually), so hydrostatic pressure distribution holds.
• No change with time at a fixed section (steady).
• Prismatic channel with slowly varying boundary conditions.

Concept / Approach:
“Steady” means ∂()/∂t = 0; “uniform” means properties are constant along the channel (d()/dx = 0). GVF has d()/dx ≠ 0 but small, and ∂()/∂t = 0. Hence it is steady and non-uniform. The gradual assumption allows using hydrostatic pressure and the energy (or momentum) approach with small-slope approximations.

Step-by-Step Solution:

Verification / Alternative check:
GVF equation dy/dx = (S_0 − S_f)/(1 − Fr^2) is a spatial differential relation, appropriate only when the flow is steady and depth varies slowly with x.

Why Other Options Are Wrong:

• Steady uniform: Would require constant depth along x (normal depth everywhere).
• Unsteady uniform / unsteady non-uniform: These imply time variation, not given here.

Common Pitfalls:
Confusing GVF with rapidly varied flow (RVF) such as hydraulic jumps; assuming “gradually varied” means time-varying rather than space-varying.

Final Answer:
steady non-uniform flow