Hydraulic jump in a rectangular channel — governing equations used When deriving the relationship between conjugate depths and the initial Froude number for a hydraulic jump in a horizontal rectangular channel, which fundamental equations are used?

Introduction / Context:
A hydraulic jump is a rapidly varied flow phenomenon where supercritical flow transitions to subcritical flow with significant energy dissipation. The classic sequent (conjugate) depth formula in a rectangular channel ties the downstream depth to the upstream Froude number. Understanding which conservation laws drive this formula is critical for correct analysis and design (stilling basins, spillways, energy dissipators).

Given Data / Assumptions:

• Control volume spans the jump in a horizontal, prismatic rectangular channel.
• Steady flow; hydrostatic pressure distribution outside the roller.
• Large, non-negligible energy loss across the jump.

Concept / Approach:
Because the jump dissipates energy, the specific energy is not conserved. Therefore the energy equation alone cannot give the conjugate depth relation. Instead, apply continuity (discharge equality) and the momentum equation (force–momentum balance) between upstream and downstream sections; eliminating discharge yields the standard sequent-depth relation as a function of the upstream Froude number.

Step-by-Step Solution:
Apply continuity: q = by1v1 = by2v2 (rectangular, width b).Apply momentum: M1 = M2 for the control volume (external forces from channel bed/walls cancel in the simplified form).Combine to eliminate velocities and solve for y2/y1 in terms of Fr1.Conclude that energy is not conserved; the energy loss is obtained after y2 is known.

Verification / Alternative check:
Textbook formula for sequent depths derives strictly from momentum + continuity; subsequent energy loss is ΔE = E1 − E2 > 0.

Why Other Options Are Wrong:
Energy equation (alone or with zero loss) is invalid across a jump due to large dissipation.
Including all three is unnecessary and misleading; momentum + continuity suffice.
GVF equation applies to gradually varied flow; a jump is rapidly varied.

Common Pitfalls:

• Forgetting that hydrostatic pressure assumption at sections must be outside the roller.
• Using specific-energy minima logic (critical depth) incorrectly to relate conjugate depths.

Final Answer:
Continuity and momentum equations