Question

The assumptions made in derivation of Bernoulli’s theorem are:
A. The fluid is ideal i.e viscosity is zero
B. The flow is steady
C. The flow is incompressible
D. The flow is irrotational
E. The flow is compressible
Choose the correct answer from the options given below:

• A. B and C only
• B. A, B, D and E only
• C. A, B, C and D only
• D. A, B and E only

Hint:

To remember Bernoulli's assumptions, think of the acronym S-I-I-N:
- Steady flow
- Incompressible fluid
- Irrotational flow
- Non-viscous (ideal) fluid

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Bernoulli's theorem is a fundamental principle in fluid mechanics that describes the conservation of energy along a streamline in a flowing fluid.
It relates pressure, velocity, and elevation in a moving fluid under specific idealized conditions.

Step 2: Detailed Explanation:
Bernoulli's equation is derived by integrating Euler's equation of motion along a streamline.
To perform this integration and simplify the energy conservation equation, several key assumptions must be made about the fluid and the nature of its flow:
- A. The fluid is ideal (inviscid): The viscosity of the fluid is assumed to be zero.
This means there are no frictional shear forces between fluid layers, and no energy is dissipated as heat due to viscous resistance. This matches statement A.
- B. The flow is steady: The fluid properties (such as velocity, pressure, and density) at any point in the flow field do not change with respect to time (\( \frac{\partial \mathbf{v}}{\partial t} = 0 \)). This matches statement B.
- C. The flow is incompressible: The density of the fluid remains constant throughout the flow (\( \rho = \text{constant} \)).
This assumption is generally valid for liquids and for gases moving at low Mach numbers. This matches statement C.
- D. The flow is irrotational: The fluid particles do not rotate about their own axes as they move along a streamline.
This allows the velocity field to be represented by a scalar potential. This matches statement D.
Since the flow is assumed to be incompressible (C), it cannot be compressible (E).
Therefore, statements A, B, C, and D are the correct assumptions.

Step 3: Final Answer:
The assumptions made are A, B, C, and D only.