Question

Correct Bernoulli’s equation is (symbols have their usual meaning):

• A. \( P + mgh + \frac{1}{2} mv^2 = \text{constant} \)
• B. \( P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant} \)
• C. \( P + \rho gh + \rho v^2 = \text{constant} \)
• D. \( P + \frac{1}{2} \rho gh + \frac{1}{2} \rho v^2 = \text{constant} \)

Hint:

Bernoulli’s theorem is fundamental for ideal fluid systems. Always ensure that terms represent energy per unit volume (pressure, potential energy, and kinetic energy) and use consistent units throughout.

Answer 1

The Correct Option is B

Bernoulli’s theorem elucidates the conservation of energy within a flowing fluid. This principle is derived by examining the work performed by pressure forces, gravitational effects, and variations in velocity along a streamline. The mathematical formulation is expressed as:\[P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant},\]wherein: \( P \): Represents pressure energy per unit volume. \( \rho \): Denotes the fluid's density. \( g \): Signifies gravitational acceleration. \( h \): Indicates the height relative to a reference level. \( v \): Represents the fluid's velocity. Analysis of the provided options: Option (a): Substituting \( m \) (mass) for \( \rho \) (density) is erroneous, as Bernoulli’s equation quantifies energy per unit volume, not mass. Option (c): Altering the potential energy term to \( mg \) instead of \( \rho gh \) results in dimensional inconsistency. Option (d): Modifying the kinetic energy term from \( \frac{1}{2} \rho v^2 \) to \( \frac{1}{2} m v^2 \) is inappropriate for fluid dynamics, as it deviates from the energy-per-unit-volume convention. Only option (b) accurately maintains the energy balance per unit volume:\[P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}.\] Final Answer:\[\boxed{P + \rho gh + \frac{1}{2} \rho v^2 = \text{constant}}\]