5.3 Streamlines, Pathlines, and Streaklines

Fluid Particle Trajectories

Fluid particle trajectories are the primary tools for understanding how fluid actually moves through space. Streamlines, pathlines, and streaklines each capture a different aspect of fluid motion, and knowing when to use each one is essential for flow visualization and analysis.

A critical fact to internalize: in steady flow, all three trajectories are identical. They only diverge from each other under unsteady conditions. This distinction shows up repeatedly in problems and exams.

Streamlines, Pathlines, and Streaklines

These three trajectory types answer different questions about the same flow.

Streamlines are a family of curves that are instantaneously tangent to the velocity vector at every point in the flow field. They answer: "If I froze the flow right now, which direction would fluid be heading at each location?"

• Streamlines are a snapshot of the entire velocity field at a single instant in time.
• A fluid element at any point will travel in the direction indicated by the streamline passing through that point, at that instant.
• Streamlines never cross each other in a well-defined velocity field (if they did, a fluid particle at the intersection would have two simultaneous velocity directions, which is physically impossible).

Pathlines trace the actual trajectory of a single fluid particle over a period of time. They answer: "Where has this specific particle been?"

• Think of attaching a tiny GPS tracker to one fluid particle and recording its position over time.
• A dye particle released into a river traces out a pathline as it moves downstream.

Streaklines connect the current positions of all fluid particles that have passed through a specific fixed point in the past. They answer: "Where are all the particles now that once passed through this point?"

• Continuously injecting dye or smoke at a fixed location produces a streakline. The visible trail you see at any instant is the streakline.
• Smoke rising from a chimney is a classic streakline: every smoke particle passed through the chimney opening, and the visible plume shows where all those particles are right now.

Visualization of Fluid Flow

Steady flow

In steady flow, the velocity field does not change with time. Because of this, all three trajectory types collapse into the same curves. A particle released from a point follows the same path as every particle before it, and the instantaneous velocity directions don't shift. Laminar flow in a straight pipe is a good example: streamlines, pathlines, and streaklines are all parallel lines along the pipe axis.

Unsteady flow

When the velocity field changes over time, the three trajectories diverge from one another:

• Streamlines shift from instant to instant because the velocity vectors themselves are changing. A streamline drawn at $t = 0$ may look completely different from one drawn at $t = 1$.
• Pathlines reflect the cumulative history of a particle that experienced different velocity fields at different times, so they won't match any single instantaneous streamline.
• Streaklines connect particles that were released at different times (and therefore experienced different velocity fields), so they form yet another distinct curve.

Flow around a bluff body with vortex shedding is a standard example where all three differ.

How to sketch each one

• Streamlines: At a given instant, draw curves that are tangent to the velocity vector at every point. No two streamlines cross.
• Pathlines: Pick a single particle and track its position through time, plotting each successive location.
• Streaklines: Mark a fixed release point, then plot the current positions of all particles that have passed through that point over time. Connect those positions.

Streamlines and Velocity Fields

The defining mathematical property of a streamline is that the velocity vector $\vec{v}$ is everywhere tangent to it. For a two-dimensional flow with velocity components $u$ and $v$, the streamline equation is:

$\frac{dx}{u} = \frac{dy}{v}$

This comes directly from the requirement that the streamline direction matches the velocity direction at every point.

Streamline patterns reveal key flow features:

• Sources and sinks: Streamlines diverge radially outward from a source or converge inward toward a sink. Flow draining into a sink (like water into a drain) shows converging streamlines.
• Vortices: Streamlines form closed circular or spiral patterns. A tornado is a dramatic real-world example.
• Stagnation points: Streamlines appear to meet or split at points where the local velocity is zero.

Closely spaced streamlines indicate higher velocity (the fluid is being squeezed through a smaller effective area), while widely spaced streamlines indicate lower velocity. This is directly visible in a converging nozzle, where streamlines bunch together as the cross-section narrows.

Fluid Particle Behavior Analysis

Pathline computation

To compute a pathline, you integrate the velocity field over time. Starting from position $\vec{x}_0$ at time $t_0$:

$\vec{x}(t) = \vec{x}_0 + \int_{t_0}^{t} \vec{v}(\vec{x}(t'), t') \, dt'$

This is an initial value problem: you need both the velocity field and the particle's starting position. In unsteady flow, pathlines can cross streamlines because the streamlines themselves are shifting over time. A particle in a von Kármán vortex street, for instance, may weave back and forth across the instantaneous streamline pattern.

Streakline formation

Streaklines are built by tracking the current positions of all particles continuously released from a fixed point. Each particle in the streak was released at a different time, so in unsteady flow, each one has experienced a different velocity history. This is why streaklines reflect the time-varying nature of the flow and generally differ from both streamlines and pathlines under unsteady conditions.

In steady flow, every particle released from the same point follows the same path, so the streakline, pathline, and streamline through that point all coincide.

Practical applications

Analyzing pathlines and streaklines provides insight into several engineering problems:

1. Mixing and dispersion: Pathlines reveal how fluid particles spread through a domain. Pollutant dispersion in the atmosphere, for example, is studied by tracking particle trajectories through a time-varying wind field.
2. Residence time distribution: In chemical reactor design, pathlines determine how long fluid elements spend inside the reactor, which directly affects reaction yield.
3. Recirculation and stagnation: Streaklines and pathlines help identify recirculation zones (where fluid gets trapped in a loop) and stagnation points (where velocity drops to zero), such as the separated flow region behind an obstacle.