Streamlines are not the paths particles take
Draw a curve through a moving fluid and it can mean three quite different things. Which one it means depends on how it was made, and the three coincide only when the flow is not changing in time.
Almost every flow photograph ever published is one of the other two, and captioned as the first.
The three curves
A streamline is a curve that is everywhere tangent to the velocity field, at one instant. It is a snapshot of direction: if the whole flow were frozen right now, this is where a particle at each point would be heading. Nothing travels along a streamline; it is a description of an instant.
A pathline is the track a single fluid particle actually follows over time. It is a long exposure of one speck of dust. It has a beginning and an end and it records history.
A streakline is the line joining all the particles that have passed through one fixed point. It is what a smoke wire or a dye injector produces, and it is what nearly every visualisation in a textbook actually shows.
Why they coincide in steady flow
If the velocity field does not change with time, the three are the same curve. That is not obvious and the reason is worth having.
A particle at a point moves in the direction of the velocity there. If it moves a short distance, it arrives somewhere else — and if the flow is steady, the velocity there is the same as it was a moment ago. So the particle continues to follow the direction field, which is precisely the streamline through its starting point.
Every particle passing through a given point therefore follows the same subsequent path, which makes the streakline identical too.
The moment the field changes with time, all three arguments fail together. The particle arrives somewhere whose velocity has since changed; it no longer follows the frozen direction field; and particles released at different times take different routes.
A worked contrast
Picture a flag on a pole in a gusting wind, and a smoke source just behind the pole.
The streamline pattern at any instant is whatever the air is doing right then: a set of curves sweeping past the pole in the current direction of the gust. Change the instant and the whole pattern changes with it.
A pathline is what one smoke particle does over its whole life. It leaves the source, gets carried downstream, and is deflected up or down as gusts arrive — so it wanders, and its shape records the sequence of gusts it happened to meet.
The streakline is the visible smoke plume. It is made of particles released at many different times, each of which met a different history of gusts, so the plume waves about in a pattern that resembles neither of the other two.
The plume is the thing anybody actually sees, and it is the least direct of the three. Its shape at one instant encodes the history of the flow over the preceding several seconds, which is why a waving smoke plume looks like a wave travelling — and why nothing in the air is actually travelling that way.
The material derivative
The formal expression of the difference is worth meeting, because it is where the distinction stops being a matter of drawing and becomes part of the equations.
Something measured at a fixed point changes for two reasons: because the whole field is changing, and because the fluid arriving at that point has come from somewhere with a different value.
The first term is what a fixed probe sees. The second is what a particle experiences by moving. Their sum is what happens to a particle, and it is called the material derivative.
That second term is why a flow can be steady and still accelerate fluid. Water in a steady narrowing pipe speeds up as it goes, but at any fixed point the velocity never changes at all. Nothing depends on time, and every particle is accelerating.
Which is precisely the distinction this essay is about, written as an operator: a streamline picture knows about being zero, and a pathline knows about not being.
Why it matters
This is not pedantry, and there are three practical consequences.
Photographs lie about instants. A smoke visualisation of a wing is a streakline pattern accumulated over the exposure. In steady flow that is fine. In separated or shedding flow it is a record of the past, and reading it as an instantaneous velocity field is simply wrong.
Streamlines cannot cross and pathlines can. Two streamlines crossing would mean two velocities at one point, which is impossible. A single particle can perfectly well cross its own earlier track, and in an unsteady flow it often does.
A closed streamline is not a closed orbit. Recirculation in a picture does not mean fluid is going round and round; in an unsteady flow it may mean nothing of the kind.
The last one matters directly for separated flow, where the recirculating region drawn in a steady figure is a genuinely different object from the fluid motion in an unsteady wake.
What the solver computed
The figure uses a deliberately simple unsteady field — a uniform stream with a transverse component that swings sinusoidally in space and time. It is not a solution of anything physical, and the site says so on the figure.
That is a deliberate choice. The point being made is kinematic rather than dynamic: it is about the definitions of the three curves, and a field simple enough to reason about makes the distinction visible where a real unsteady flow would bury it in detail.
The streamlines are integrated through the field frozen at . The pathlines are integrated in time, with the field updated at every step. Both use the same integrator; only the treatment of time differs, and that difference alone produces the two families of curve.
Elsewhere on this site every field is a solution, and the note under each figure says which model produced it. This one is the exception, and it is labelled.
Steady is a choice of frame
A subtlety that catches people, and it is worth knowing because it turns unsteady problems into steady ones.
Whether a flow is steady depends on who is watching. The flow round an aircraft is wildly unsteady to somebody standing on the ground — a wing arrives, the air is thrown about, the wing leaves. To an observer riding on the wing, the same flow is perfectly steady.
So a great many problems are made tractable by changing frame: work in the aircraft’s frame, get a steady flow, and use every result that requires steadiness — including Bernoulli’s equation, which needs it.
The catch is that the three curves are frame-dependent too. Streamlines in the wing’s frame and streamlines in the ground’s frame are entirely different pictures of the same physics, and a figure that does not say which frame it is drawn in has left out something essential.
What a visualisation is really showing
Given all this, it is worth being explicit about the common laboratory techniques and which curve each produces.
Smoke or dye from a fixed probe gives streaklines. This is the most common technique and the most commonly mislabelled.
Particles with a short exposure give short segments of pathlines, which approximate streamlines if the exposure is short compared with the flow’s timescale.
Particles with a long exposure give pathlines outright.
Particle image velocimetry measures the velocity field directly at an instant, from which true streamlines can be computed. It is the only one of the four that gives streamlines without an assumption.
So the honest caption on most tunnel photographs is “streaklines”, and the honest reading is that in steady flow they may be treated as streamlines and otherwise may not.
The streamtube, and why it is useful
One construction depends entirely on getting this right, and it is the workhorse of elementary fluid mechanics.
Take a closed curve in the flow and follow the streamlines through every point on it. The surface they sweep out is a streamtube, and it has one defining property: no fluid crosses it. Velocity is tangent to a streamline by definition, so it is tangent to the tube’s wall, so nothing passes through.
That makes a streamtube a pipe with no pipe — a control volume whose sides need no accounting, so mass in equals mass out and the tube’s cross-section alone sets the speed.
The catch is that this holds for streamtubes, in the instantaneous streamline sense. In unsteady flow the tube itself changes shape from one instant to the next, so it is not a fixed pipe and fluid that was inside it a moment ago may be outside it now. The convenient “no flow through the walls” picture is an instantaneous statement.
For steady flow — which is most of the elementary treatment — the tube is fixed and the picture is exactly right. It is one more thing that quietly requires steadiness.
Why the confusion is worth fixing
A closing argument for caring, since in steady flow the distinction makes no difference and most figures are steady.
It matters because the interesting cases are exactly the unsteady ones. A wing beginning to lift, a wake beginning to shed, a gust arriving, a stall developing — every one of these is a transient, and every one is where the three curves separate. The distinction is dispensable precisely where nothing much is happening, and indispensable where the physics is.
It also matters because it is a special case of a habit this whole site runs on: knowing what a picture is a picture of. A streakline photograph and a streamline plot look alike and are different objects, in the same way that a beautiful ideal-flow solution and a real flow look alike and are different objects.
Getting into the habit of asking costs nothing and prevents a category of error that no amount of care with the arithmetic will catch.
Where the model stops
The figure’s field is not a solution. It is chosen to make a kinematic point clearly, and it is labelled as schematic.
Only two of the three curves are drawn. Streaklines need particles released continuously from a fixed point over a period, which is a third integration and would crowd the figure past legibility.
Two-dimensional. In three dimensions streamlines can be knotted and the distinctions get harder rather than easier.
Nothing here is about turbulence, where all three curves become erratic and the useful description is statistical rather than geometric.
Reading a figure on this site
Every flow picture here is steady, and that is a deliberate restriction rather than an oversight.
Steady flow means the three curves coincide, so a streamline figure can be read as a particle path without qualification. It also means Bernoulli applies, that a streamtube is a genuine pipe, and that the picture describes a situation rather than an instant in a sequence.
The cost is that everything genuinely unsteady is out of reach. A shedding wake, a starting vortex, a gust encounter, a stalling wing in the act of stalling — none of them can be drawn honestly as a steady figure, and this site does not try.
That restriction is stated on every figure in the model note, which is why those notes say “steady” so often. It is not filler; it is the hypothesis that licenses reading the picture the obvious way.
The one that has no name in common use
For completeness there is a fourth curve, and it is the one a reader is most likely to have made without knowing it.
A timeline is the curve formed by a set of particles that were in a straight line at one instant and have since been carried by the flow. Release a row of bubbles from a wire all at once and watch what happens to the row.
It is the technique behind hydrogen-bubble visualisation, and it shows something the other three do not: the shear. Where the flow is faster the line runs ahead, where it is slower it lags, and the distortion of the line is a direct picture of the velocity gradient — which is exactly the quantity the boundary layer is made of.
A timeline in a uniform flow stays straight. A timeline crossing a boundary layer bends over sharply near the wall, and the bend is the profile.
Who separated them, and when
The distinction is old and the vocabulary is nineteenth-century, settling into roughly its modern form with the growth of experimental hydrodynamics. Osborne Reynolds’s dye-filament experiments of 1883, which established the number that carries his name, are streakline visualisations, and he was careful about what they showed.
The confusion is not really a historical error; it is a persistent practical one. Experimentalists have always known the difference and captions have always blurred it, because “streamlines” reads better than “streaklines” and in the steady case it is harmless.
The habit worth keeping
One question, asked of every flow picture, catches nearly everything this essay is about: is this flow steady, and in whose frame?
If the answer is yes, the three curves coincide, the picture can be read as particle paths, a streamtube is a pipe, and Bernoulli is available. Almost everything convenient about elementary fluid mechanics follows from that one property.
If the answer is no, the picture is a record of something and the reader has to find out what. A smoke photograph of a shedding wake is beautiful and is not a velocity field. A long-exposure particle image is not an instant. And a caption saying “streamlines” is, more often than not, saying something the technique could not have produced.
The question is worth asking of the figures here too, which is why every one of them carries a note saying which model produced it and under what assumptions. A picture without that note is asking to be trusted on its appearance, and appearance is exactly what cannot be trusted in this subject.
The ladder from here
Nearby: streaklines drawn properly, with continuous release; the material derivative, which is the formal expression of the difference between watching a point and following a particle; and flow visualisation techniques compared.
Then across to mass conservation, which is what makes a streamtube a useful object in the first place, and to the separated flows where getting this distinction wrong does the most damage.