Question

Which one of the following options is TRUE?

• A. Pathlines and streaklines are the same in an unsteady flow, and streamlines are tangential to the local fluid velocity at a point.
• B. Streamlines are perpendicular to the local fluid velocity at a point, and streamlines and streaklines are the same in a steady flow.
• C. Pathlines and streaklines are the same in an unsteady flow, and streamlines and streaklines are the same in a steady flow.
• D. Streamlines are tangential to the local fluid velocity at a point, and streamlines and streaklines are the same in a steady flow.

Hint:

In steady flow, streamlines and streaklines are the same, and both are tangential to the local velocity at every point.

Answer 1

The Correct Option is D

In fluid mechanics, the different types of lines (pathlines, streaklines, and streamlines) represent different ways of analyzing fluid flow: Streamlines are lines that are tangential to the velocity of the fluid at every point in the flow field. In a steady flow, the streamlines do not change over time, and they are indeed tangential to the local velocity at every point. This makes statement (D) true. Pathlines represent the actual path traveled by individual fluid particles over time. In steady flow, pathlines and streamlines are the same, but in unsteady flow, they can differ. Streaklines represent the loci of particles that have passed through a given point in the flow, and they can also differ from streamlines in unsteady flow.

Step 1: Analysis of the options.
- (A) is incorrect because pathlines and streaklines are not necessarily the same in an unsteady flow. - (B) is incorrect because streamlines are not perpendicular to the local fluid velocity; they are tangent to it. - (C) is incorrect because while streamlines and streaklines can be the same in steady flow, they are not the same in unsteady flow. - (D) is correct because streamlines are tangential to the velocity field, and in steady flow, streaklines and streamlines are indeed the same.

Final Answer: \text{(D) Streamlines are tangential to the local fluid velocity at a point, and streamlines and streaklines are the same in a steady flow.}