Streamlines vs Streaklines vs Pathlines: Differences

Three ways to draw the same flow — identical only when the flow is steady.

Fluid KinematicsMechanical EngineeringFree preview
⏱️ About 16 min

Drop a stream of dye into a moving river and it traces a curve. Sketch the river's velocity arrows end-to-end and you get another curve. Follow a single floating leaf and you get a third. To the eye these curves often look the same — and in calm, steady water they actually are the same. But the moment the flow becomes unsteady, those three curves pull apart and tell three different stories. They have three different names — <em>streamline</em>, <em>streakline</em>, and <em>pathline</em> — and knowing which one your dye, your sketch, or your leaf actually represents is the difference between understanding a flow and misreading it completely.

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The big idea: A <strong>streamline</strong> is a curve everywhere tangent to the instantaneous velocity field — a snapshot of where the flow is headed right now. A <strong>pathline</strong> is the actual trajectory of a single identified particle over time. A <strong>streakline</strong> is the locus of every particle that passed through a particular injection point (the shape a streak of dye takes). A fourth, the <strong>timeline</strong>, joins particles released at the same instant. The crucial fact: in <em>steady</em> flow all of these coincide and there is nothing to keep straight; in <em>unsteady</em> flow they generally differ, and you must know which one your observation corresponds to. A streamline through a point is found by integrating <em>dx/u = dy/v</em>.
🎯 By the end, you'll be able to
  • Define streamline, pathline, streakline, and timeline
  • State precisely when the three flow lines coincide and when they differ
  • Derive and integrate the streamline equation dy/dx = v/u for a 2D field
  • Identify which flow line a given experimental observation (dye streak, particle track, snapshot) represents
📎 Helpful to know first

Four flow lines, defined

Because a velocity field is a tangle of arrows, engineers summarize it with curves. There are four standard ones, each defined by a different rule.

A streamline is a curve that is everywhere tangent to the velocity field at a single instant in time. Picture freezing the flow and walking along a path that always points the way the arrows point — that path is a streamline. A streamline is a property of the field at one moment, so a snapshot of streamlines is the standard way to draw a flow.

A pathline is the actual trajectory traced by one identified fluid particle over some interval of time — the path a single floating leaf follows. This is a Lagrangian object (Lesson 1): it follows one particle.

A streakline is the locus of all particles that have passed through a particular fixed point — the shape a continuously injected streak of dye or smoke takes at this instant. Each dot of dye is at a different age, but together they trace the streakline.

A timeline is the curve formed by a row of particles all released at the same instant; it shows how an initially straight material line deforms. Timelines are used less often but appear in the next lesson's picture of a deforming fluid element.

In unsteady flow the streamline, pathline, and streakline through the same injection point P are three different curves. The streamline is tangent to the current velocity field; the pathline is one particle's actual history; the streakline is the locus of all particles that passed through P. velocity at t₁,t₂,t₃ streamline (tangent now) pathline (one particle) streakline (all particles from P) P injection Unsteady flow: the three curves through the same point P are genuinely different. In steady flow they would all coincide.
Unsteady flow is where the distinction bites. Through the same injection point P, the streamline (tangent to the field now), the pathline (one particle's history), and the streakline (all particles from P) are three genuinely different curves. In steady flow they would all lie on top of one another.
⚠️ The misconception this lesson exists to kill

Many students absorb 'streamline', 'pathline', and 'streakline' as three synonyms for 'a line showing the flow'. They are not. They coincide only in steady flow — when the velocity field at every fixed point never changes. The instant the flow becomes unsteady, the three pull apart, and a photograph of dye (a streakline) is no longer the same curve as the field's streamlines. Wind-tunnel smoke and river dye are streaklines; a CFD contour plot is streamlines. In unsteady flow you cannot read one off the other.

The streamline equation

Because a streamline is everywhere tangent to the velocity, its slope must equal the ratio of the velocity components. In two dimensions, with velocity (u, v), a small step (dx, dy) along the streamline must be parallel to (u, v), which forces dy/dx = v/u. This is the streamline equation, and finding a streamline is the business of separating variables and integrating it. At a stagnation point (where u = v = 0) the slope is undefined and streamlines can meet — the only place they are allowed to cross.

\[ \frac{dx}{u}=\frac{dy}{v}\quad\Longleftrightarrow\quad \frac{dy}{dx}=\frac{v}{u} \]
The 2D streamline equation. A step (dx, dy) along a streamline must be parallel to the local velocity (u, v), so the slopes match. Integrating dy/dx = v/u (after separating variables) gives the family of streamlines for the field. Where u = v = 0 (a stagnation point) the slope is indeterminate and streamlines may meet.
🎮 Interactive: a live velocity field and its streamlines LIVE
Predict first: In the source-sink preset the streamlines run smoothly from source to sink — and because that flow is steady, a dye streak (a streakline) would trace exactly the same curves. Switch presets and watch the streamlines re-form.

An interactive 2D velocity-field visualizer showing live integrated streamlines and animated tracer particles for a source-sink pair, flow past a cylinder, and a vortex, with a mouse-position speed readout.

The flow_field simulator integrates short streamline segments from a grid of seed points and floats tracer particles along them. Three presets: a source-sink pair, potential flow past a cylinder, and a vortex. In the 'show vorticity' mode (used in Lesson 4) regions of rotation light up.
📝 Worked example: A steady 2D velocity field is V = (2x) i − (2y) j m/s. Find the equation of the streamline passing through the point (x₀, y₀) = (2, 1).
  1. Apply the streamline equation: dy/dx = v/u = (−2y)/(2x) = −y/x.
  2. Separate variables: dy/y = −dx/x.
  3. Integrate: ln y = −ln x + C, so ln(xy) = C, hence xy = constant.
  4. Evaluate the constant at (2, 1): xy = 2 × 1 = 2.
  5. So the streamline through (2, 1) is the hyperbola xy = 2.
  6. Sanity check: this field has a stagnation point at the origin, and the streamlines are rectangular hyperbolas asymptotic to the axes — the classic stagnation-point (impinging-jet) flow pattern.
✓ The streamline is xy = 2 (a rectangular hyperbola).
✏️ Practice: For the steady field V = (2x) i + (2y) j m/s, the streamlines are straight rays y = (y₀/x₀) x. The streamline through (1, 2) is y = 2x. Find the y-coordinate at which this streamline crosses x = 3.
Solution
  1. Streamline through (1, 2): y = 2x.
  2. At x = 3: y = 2(3) = 6.
✏️ Practice: A steady field is V = (3) i − (2y) j m/s (a uniform horizontal wind of 3 m/s with a vertical shear). Find the streamline slope dy/dx at the point where y = 4.
Solution
  1. Streamline equation: dy/dx = v/u = (−2y)/3.
  2. At y = 4: dy/dx = −(2 × 4)/3 = −8/3 = −2.67.

Check your understanding

1. In STEADY flow, the streamline, pathline, and streakline through a point are:
When the field at every fixed point is time-independent, all three constructions collapse onto one curve. Only in unsteady flow do they separate — which is exactly why the distinction matters mainly for unsteady flows.
2. A streamline is, by definition:
A streamline is a snapshot concept: at one instant, it is the curve whose tangent everywhere matches the local velocity. The pathline follows one particle; the streakline collects all particles from one injection point.
3. The 2D streamline equation is dy/dx = v/u. Streamlines are allowed to cross ONLY at:
Two distinct streamlines cannot cross, because the velocity has a single direction at each point. The sole exception is a stagnation point, where the velocity is zero and the streamline slope dy/dx = v/u is indeterminate — streamlines can meet (and often do) there.
🔎 Related lessons
Describing Fluid Motion: Lagrangian vs Eulerian Velocity Field & the Material Derivative · soon Vorticity & Circulation · soon Euler's Equation Along a Streamline
✅ Key takeaways
  • A streamline is tangent to the instantaneous velocity field; a pathline is one particle's trajectory; a streakline is the locus of all particles from one injection point.
  • Streamline, pathline, and streakline coincide in steady flow and generally differ in unsteady flow — the central distinction of flow visualization.
  • The 2D streamline equation is dy/dx = v/u; integrating it (after separating variables) gives the streamline family.
  • Streamlines cannot cross except at a stagnation point (u = v = 0), where the slope is indeterminate.
➡️ You can now both read and draw a velocity field. The remaining kinematic question is what the field does to a particle caught in it — how fast that particle accelerates, even when the field itself looks perfectly steady. Answering that needs one new operator: the material derivative.
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