In fluid mechanics, for a totally submerged orifice discharging between two reservoirs, the discharge Q is directly proportional to which head measure? (Assume steady flow, sharp-edged orifice, and fully submerged upstream and downstream openings.)

Introduction / Context:
An orifice is an opening in a wall or plate through which fluid flows. When both the upstream and downstream openings are submerged, it is a totally submerged (drowned) orifice. Understanding how discharge depends on head is a core hydrostatics–hydrodynamics concept used in tanks, gates, and culverts.

Given Data / Assumptions:

• Steady, incompressible flow of water.
• Sharp-edged orifice with coefficient of discharge Cd approximately constant in the considered range.
• Both sides submerged; head driving flow is the difference in water surface elevations (ΔH).

Concept / Approach:

From Bernoulli and continuity, the theoretical discharge through an orifice is Q_th = A * √(2 * g * ΔH), where A is the orifice area, g is gravitational acceleration, and ΔH is the difference in piezometric head. Real discharge is Q = Cd * A * √(2 * g * ΔH). Hence, for fixed A and Cd, Q varies with the square root of ΔH, not linearly with ΔH.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: √(g * ΔH) has dimensions of velocity. Multiplying by area A gives volume flow per unit time—consistent with discharge units.

Why Other Options Are Wrong:

(a) Linear proportionality to ΔH is incorrect; the dependence is on the square root. (c) “Square root of the opening” is dimensionally inconsistent. (d) Reciprocal area contradicts continuity; Q increases with area. (e) is unnecessary since (b) is correct.

Common Pitfalls:

Confusing free orifice head (to atmosphere) with submerged head difference; assuming linear dependence on head without checking Bernoulli-based velocity relation.

Final Answer:

Square root of the difference in elevation of water surfaces