Question bank — Streamlines, pathlines, streaklines

False — at a crossing the fluid would need two different velocity directions at once, but $\vec{v}$ has exactly one value per point. Crossings are only allowed where $\vec{v}=0$ (stagnation points), where "direction" is undefined.

True — if the field never changes in time, the direction a particle "sees" at each point is always the same one the streamline drew, so it can never peel off. This is the boxed golden rule on the parent note: steady flow ⇒ streamlines = pathlines = streaklines.

Not necessarily — they are tangent at that instant (both follow the local $\vec{v}$), but they generally diverge afterward. They can even coincide over a stretch by coincidence; unsteadiness only permits difference, it doesn't force it everywhere.

True — the particle whose release time equals the current instant ($\tau =t$) is still sitting at the source $(x_p,y_p)$, so the source point is one end of the streakline at every moment.

False — by construction the velocity is tangent to the streamline, so its component across the line is zero. This is exactly why streamlines act like impermeable "walls" in flow reasoning.

True — in unsteady flow the whole field redraws itself each instant, so the snapshot changes. In steady flow the pattern is frozen forever.

True — different particles can pass through the same point at different times. Pathlines are time-parametrised trails, so a spatial crossing just means "same place, different clock reading," which is fine.

False — the general rule is that at a regular point ($\vec{v}\ne 0$) each streamline must follow the single well-defined direction of $\vec{v}$ there, so two curves sharing that point would share that same tangent and the same continuation, forcing them to be the identical streamline. Tangency of distinct streamlines is therefore impossible except at a stagnation point ($\vec{v}=0$), where the direction is undefined.

Error — the streamline is defined at one instant, so the geometry it captures is "which way do the arrows point at this frozen moment." Mathematically that means the field values $u(x,y,t)$ and $v(x,y,t)$ must be evaluated at a single fixed $t$: the equation is an ODE in the space variables $x,y$ only, with $t$ entering as a constant parameter. If you let $t$ advance while integrating, you're no longer relating $dx$ to $dy$ at one snapshot — you've smuggled in the field's time-changes and produced neither a streamline nor a pathline.

Error — a plume is a streakline: it strings together many different parcels released at many different times $\tau$, all viewed now. One parcel's actual route is its pathline, generally a different curve in unsteady wind.

Error — that's the streamline ODE (a spatial relation). A pathline needs the time ODEs $\frac{dx}{dt}=u,\frac{dy}{dt}=v$ integrated forward from $(x_0,y_0)$ at $t_0$. Time is the driving variable, not a frozen label.

Error — they coincide only in steady flow. Assuming it in general silently drops the "history" that makes unsteady dye patterns differ from single-particle trails. Check $\partial \vec{v}/\partial t=0$ first.

Error — it shows the instantaneous direction everywhere. In unsteady flow, by the time the particle reaches the next point the field there has changed, so the particle's real route (its pathline) departs from the streamline it started tangent to.

Half-error — circular streamlines mean the snapshot is circular; whether particles trace full circles depends on the flow being steady. For $\vec{v}=(-y,x)$ the flow is steady, so here it's fine — but the reasoning "streamline shape = particle path" is only valid because of steadiness, not automatically.

Because you inject marker continuously at one spot and photograph it — that visible ribbon is by definition all particles that passed the source, i.e. a streakline. See Flow visualization techniques (dye, smoke, PIV). Getting streamlines instead requires measuring the whole velocity field at one instant.

A pathline follows a particle through time, so time is the thing advancing; a streamline is a single-instant snapshot, so time is held fixed and only space ($x,y$) varies. The differing role of $t$ is the deepest structural difference between the two.

With $\partial \vec{v}/\partial t=0$ the field has no memory of when — every particle arriving at a point follows the identical direction the field always held, so the snapshot, the single trail, and the many-particle streak all trace the same locus. This is the golden rule.

Because a streamline is defined at one instant over all space, while a particle's route is defined over time. In unsteady flow the field changes underneath the moving particle, so its trail (pathline) generally leaves the streamline it was tangent to. The two answer different questions.

Because "all particles that passed the source" means one particle per release time $\tau$; each obeys its own pathline, and we photograph them together at the current $t$. Freezing $t$ and sweeping $\tau$ threads them into one visible curve.

A flow tube (streamtube) is a bundle of streamlines forming a closed surface — imagine a hose made of neighbouring streamlines, sketched as the shaded band in the figure below. Because no fluid crosses a streamline, no fluid can leak out through the tube's walls, so every bit of mass that enters one end must exit the other — that sealed-wall bookkeeping is the continuity equation applied to the tube.

It becomes undefined — there's no direction to be tangent to, so multiple streamlines may meet there. Stagnation points are the only places streamlines legitimately intersect.

A single point — with zero velocity it never moves, so its "trail" is just the stagnation point itself. Its streakline (if that point is a source) is also just that point.

Yes — $\vec{v}$ has no explicit $t$, so $\partial \vec{v}/\partial t=0$, the flow is steady, and the golden rule makes pathlines = streamlines = circles. Particles genuinely orbit the origin. See Steady vs unsteady flow.

Just the source point — no particle has had time to travel away yet, so the streak has zero length and grows outward as time passes.

All three are identical horizontal straight lines — steadiness plus a constant field means snapshot, single trail, and dye streak are the same set of parallel lines. A clean sanity check of the golden rule.

No — at any frozen instant every point shares one direction, so streamlines are parallel straight lines. But pathlines can curve, because a particle experiences the direction changing over time as it moves. This isolates unsteadiness as the sole cause of the difference.