Geometrical Distinction between Streamline, Pathline, and Streakline

Introduction:
In fluid mechanics, streamline, pathline, and streakline are three important concepts used to describe the motion of fluid particles. While they have different definitions and characteristics, there is no geometrical distinction between them in certain flow conditions. Let's explore these flow conditions and understand why there is no geometrical distinction between the streamline, pathline, and streakline.

a) Steady Flow:
In steady flow, the velocity of the fluid particles at any point remains constant over time. This means that the fluid particles follow the same path at all times. In this case, the streamline, pathline, and streakline coincide with each other.

- Streamline: A streamline is an imaginary line that is tangent to the velocity vector of the fluid at every point. It represents the instantaneous direction of the flow at each point.
- Pathline: Pathline is the actual path followed by a specific fluid particle as it moves through the flow field. It is obtained by tracking the motion of a specific fluid particle over time.
- Streakline: Streakline is a continuous line formed by fluid particles that have passed through a specific point in the flow field at different times. It represents the history of fluid particles that have flowed through a particular location.

In steady flow, since the velocity at each point remains constant, the streamline, pathline, and streakline are identical and can be represented by the same line.

b) Uniform Flow:
Uniform flow is a type of steady flow where the velocity of the fluid particles remains constant not only at each point but also in magnitude and direction throughout the flow field. Similar to steady flow, there is no geometrical distinction between the streamline, pathline, and streakline in uniform flow.

c) Laminar Flow:
Laminar flow is characterized by smooth and orderly fluid motion with well-defined layers of fluid sliding over each other. In this flow condition, the streamline, pathline, and streakline may not necessarily coincide due to the presence of velocity gradients within the flow field. Therefore, there is a geometrical distinction between these concepts in laminar flow.

d) Irrotational Flow:
Irrotational flow is a flow condition where the fluid particles do not rotate as they move. In this case, the streamline, pathline, and streakline are again indistinguishable from each other due to the absence of fluid rotation.

Conclusion:
In summary, there is no geometrical distinction between the streamline, pathline, and streakline in steady flow and uniform flow conditions. This is because in steady flow, the velocity remains constant at each point, and in uniform flow, the velocity remains constant in magnitude and direction throughout the flow field. However, in laminar flow where velocity gradients exist, and in irrotational flow where there is no fluid rotation, there can be a geometrical distinction between these concepts.