Question

The Bernoulli equation states that in a steady irrotational flow of an incompressible fluid,

• A. The total potential energy at any point is constant
• B. The total kinetic energy at any point is constant
• C. The total energy at any point is constant
• D. The total pressure energy at any point is constant

Hint:

The Bernoulli equation for irrotational flow states that \( \frac{v^2}{2} + \frac{p}{\rho} + gz \) is constant everywhere in the flow, representing conservation of total mechanical energy.

Answer 1

The Correct Option is C

Step 1: Understand the Bernoulli equation.
The Bernoulli equation applies to the steady, irrotational flow of an incompressible fluid. It is derived from the conservation of energy along a streamline and is often stated for an inviscid fluid (though it can apply with viscosity under certain conditions). The equation is: \[ \frac{v^2}{2} + \frac{p}{\rho} + gz = \text{constant along a streamline}, \] where:
\( \frac{v^2}{2} \): Kinetic energy per unit mass,
\( \frac{p}{\rho} \): Pressure energy per unit mass (where \( p \) is pressure, \( \rho \) is density),
\( gz \): Potential energy per unit mass (where \( g \) is gravitational acceleration, \( z \) is elevation).
The sum of these terms represents the total mechanical energy per unit mass.
Step 2: Analyze the conditions of the problem.
Steady flow: The flow does not change with time.
Irrotational flow: \( \nabla \times \mathbf{v} = 0 \), meaning the flow has no vorticity, which is true for potential flow.
Incompressible fluid: \( \rho \) is constant.
For an irrotational flow, the Bernoulli equation holds not just along a streamline but across the entire flow field (since the velocity can be derived from a potential, \( \mathbf{v} = \nabla \phi \)). The equation states that the total energy (kinetic + pressure + gravitational potential) is constant at any point in the flow.
Step 3: Evaluate the options.
(1) The total potential energy at any point is constant: Incorrect, as the Bernoulli equation involves the sum of kinetic, pressure, and potential energies, not just potential energy. Incorrect.
(2) The total kinetic energy at any point is constant: Incorrect, as kinetic energy varies (e.g., if velocity increases, pressure decreases), but the total energy is constant. Incorrect.
(3) The total energy at any point is constant: Correct, as the Bernoulli equation states that the sum of kinetic, pressure, and gravitational potential energies per unit mass is constant in a steady, irrotational, incompressible flow. Correct.
(4) The total pressure energy at any point is constant: Incorrect, as pressure energy (\( \frac{p}{\rho} \)) varies to balance changes in kinetic and potential energies. Incorrect.
Step 4: Select the correct answer.
The Bernoulli equation states that in a steady irrotational flow of an incompressible fluid, the total energy at any point is constant, matching option (3).