Question

A flow has a velocity potential given by \( \phi = Ax^3 \) where ‘A’ is a non-zero constant. Which of the following statement(s) is/are true about the flow?

• A. The flow is incompressible.
• B. The flow is irrotational.
• C. The flow has local acceleration.
• D. The flow has convective acceleration.

Hint:

For velocity potentials, check the derivatives to determine whether the flow is irrotational, incompressible, or has acceleration. Local acceleration occurs when the velocity changes with time, while convective acceleration is caused by the change in velocity as a fluid particle moves.

Answer 1

The Correct Option is B, D

We are given that the velocity potential \( \phi = Ax^3 \), where \( A \) is a non-zero constant. We need to analyze the flow based on this potential function and determine which statements are true.
Step 1: Check for incompressibility.
A flow is incompressible if the divergence of the velocity is zero. The velocity components can be derived from the velocity potential function. The velocity is given by the gradient of the potential: \[ \mathbf{v} = \nabla \phi \] For \( \phi = Ax^3 \), we find \( v_x = \frac{\partial \phi}{\partial x} = 3Ax^2 \). The flow is not incompressible since the velocity depends on \( x \), and there is no constant velocity field. Hence, statement (A) is false.
Step 2: Check if the flow is irrotational.
A flow is irrotational if the curl of the velocity is zero. The curl of a 1D velocity field (in the x-direction) is zero, indicating that the flow is irrotational. Therefore, statement (B) is true.
Step 3: Check for local acceleration.
Local acceleration refers to the rate of change of velocity with respect to time. Since the velocity depends on \( x \), the velocity field can change with time if the position changes. This indicates local acceleration. Thus, statement (C) is true.
Step 4: Check for convective acceleration.
Convective acceleration is the acceleration due to the change in velocity as a fluid particle moves through space. Since the velocity field is a function of \( x \), there is convective acceleration present. Thus, statement (D) is true.
Step 5: Conclusion.
Thus, the correct answers are (B) and (D).