If A denotes the area of free surface of a liquid and h̅ the depth of an orifice of area of cros

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If A denotes the area of free surface of a liquid and h̅ the...

If $A$ denotes the area of free surface of a liquid and $\bar{h}$ the depth of an orifice of area of cross-section $a$, below the liquid surface, then the velocity $v$ of flow through the orifice is given by (A) $v=\sqrt{(2 g h)}$ (B) $v=\sqrt{(2 g h)} \sqrt{\left(\frac{A^{2}}{A^{2}-a^{2}}\right)}$ (C) $v=\sqrt{2 g h} \sqrt{\left(\frac{A}{A-a}\right)}$ (D) $v=\sqrt{2 g h} \sqrt{\left(\frac{A^{2}-a^{2}}{A^{2}}\right)}$

If $A$ denotes the area of free surface of a liquid and $\bar{h}$ the depth of an orifice of area of cross-section $a$, below the liquid surface, then the velocity $v$ of flow through the orifice is given by
(A) $v=\sqrt{(2 g h)}$
(B) $v=\sqrt{(2 g h)} \sqrt{\left(\frac{A^{2}}{A^{2}-a^{2}}\right)}$
(C) $v=\sqrt{2 g h} \sqrt{\left(\frac{A}{A-a}\right)}$
(D) $v=\sqrt{2 g h} \sqrt{\left(\frac{A^{2}-a^{2}}{A^{2}}\right)}$

Key Concepts

Bernoulli’s Principle

This principle states that for an incompressible, non-viscous fluid, the total mechanical energy along a streamline remains constant. It relates pressures, fluid velocities, and elevation, and forms the foundation for analyzing fluid flow in systems with varying cross-sectional areas.

Torricelli’s Law

Derived from Bernoulli’s Principle, Torricelli’s Law provides the expression for the speed of efflux of a fluid under gravity from an orifice. It predicts that the velocity is proportional to the square root of twice the product of gravitational acceleration and the depth of the orifice relative to the free surface.

Continuity Equation

The continuity equation expresses the conservation of mass in fluid dynamics, stating that the product of the cross-sectional area and the velocity of the fluid is constant along a streamline. This concept is particularly relevant when considering the relationship between the large free surface of the liquid and the small orifice, where assumptions about the velocity of the free surface can impact the analysis.

Adjustments for Non-negligible Free Surface Velocity

When the area of the free surface is not significantly larger than the area of the orifice, the assumption of a stagnant free surface breaks down, and its finite velocity must be accounted for. Such adjustments result in modified expressions for the efflux velocity, incorporating area ratios to correct for the additional kinetic energy at the free surface.

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