= Streakline
{wiki}
A <streakline> at observation time $T$ is the set of present positions of all <fluid elements> that passed through one fixed release point $\mathbf a$ during a specified history of injection. For continuous release over $\tau_0\leq\tau\leq T$, solve a different <initial value problem> for each release time $\tau$:
$$
\frac{\partial\mathbf X}{\partial t}(t;\tau,\mathbf a)=\mathbf u(\mathbf X(t;\tau,\mathbf a),t),\qquad \mathbf X(\tau;\tau,\mathbf a)=\mathbf a,\qquad \mathcal S_T=\{\mathbf X(T;\tau,\mathbf a):\tau_0\leq\tau\leq T\}.
$$
Thus a <pathline> varies $t$ for one particle; a <streakline> varies $\tau$ while holding $T$ fixed. Interrupted release gives only the corresponding subsets of the <streakline>. A moving nozzle requires a time-dependent release position instead, so the standard fixed-point definition should not silently be applied to it. Continuous release makes <streaklines> easy to observe and useful for showing where material from one source has reached. https://web.mit.edu/13.021/demos/lectures/lecture2.pdf[MIT Marine Hydrodynamics, lecture 2, section 1.2]{ref}
In the analytical <velocity field> $(U,at)$ of <pathline>, a particle released at the origin at time $\tau$ follows
$$
x(t;\tau)=U(t-\tau),\qquad y(t;\tau)=\frac a2(t^2-\tau^2),\qquad t\geq\tau.
$$
Observe all particles together at time $T$, after continuous release from $\tau=0$:
$$
x=U(T-\tau),\qquad y=\frac a2(T^2-\tau^2),\qquad y=\frac{aT}{U}x-\frac{a}{2U^2}x^2,\qquad 0\leq x\leq UT.
$$
For $U=1\ \mathrm{m\,s^{-1}}$, $a=1\ \mathrm{m\,s^{-2}}$ and $T=2\ \mathrm s$, the <streakline> is the downward-curving <parabola> $y=2x-x^2/2$ in metre coordinates. Releases at $\tau=0,1,2\ \mathrm s$ occupy $(2,2)$, $(1,1.5)$ and $(0,0)\ \mathrm m$, respectively. At $x=1\ \mathrm m$, the <streakline> lies above the <pathline> $y=x^2/2$ and below the origin's <streamline> $y=2x$. All particles nevertheless have the same present <velocity> $(1,2)\ \mathrm{m\,s^{-1}}$. The <streakline> slope $dy/dx=a\tau/U$ generally differs from the present <velocity> slope $aT/U$; they agree at the nozzle. The oldest particle is the downstream endpoint, which explains why the <pathline> and <streakline> meet there without being the same curve.
\Image[/stream-path-streak-lines-release-family.png]
{title=Analytical release families observed at one and two seconds}
Each panel fixes one observation time. Coloured dots are the present positions of particles with the indicated release times; dashed <pathlines> show those particles' earlier histories. The black <streakline> connects present positions of the continuously released family, including release times between the sampled dots. After $T=1\ \mathrm s$, particles already present keep moving and new ones enter at the origin, so the later <streakline> is not a frozen record of the earlier one. Injection beginning only at $\tau=1\ \mathrm s$ would show just $0\leq x\leq1\ \mathrm m$ of the $T=2\ \mathrm s$ curve. Both panels are diffusion-free analytical models, not experimental photographs.
Kline's contracting-channel demonstration switches from marking an element to continuous bubble production at a marking wire: the resulting trace is a <streakline>. The apparatus therefore changes the curve being displayed by changing the marking history, even with the same channel. A wire produces a family of traces; conceptually each fixed emitting location supplies its own <streakline>. Individual bubbles are markers rather than perfect <fluid elements>: Jean Hertzberg's teaching notes describe <hydrogen> production by <electrolysis> at a thin <cathode> wire and warn that larger bubbles have more <buoyancy>. https://web.mit.edu/hml/ncfmf/05FV.pdf[S. J. Kline, Flow Visualization film notes, figure 8]{ref} https://www.flowvis.org/Flow%20Vis%20Guide/particles-3-in-water/[Jean Hertzberg and Flow Vis course students, Particles 3: In Water]{ref}
A dye version uses a small fixed injection port upstream in a transparent water channel, a dilute premix of dye with the channel water, illumination and a fixed <optical camera>. Start continuous injection at a recorded time $\tau_0$ and take a short exposure at $T$: the visible filament approximates a <streakline>, not the accumulated <pathline> of one particle. Repeat with the same port and imaging geometry after changing the flow with time; compare the filament against local directions estimated from short displacements of sparse markers at $T$. Agreement in a <steady flow> and possible disagreement when the flow changes are the predictions being tested; no measured curves are claimed here. The protocol follows Hertzberg's advice to use small ports, little injected volume and a dilute premix with matched <density> and <viscosity>. Check for nozzle disturbance by reducing injection rate, for <buoyancy> by watching the premix in still water, and for <diffusion> by comparing filament widths at different ages. Keep exposure short enough to suppress motion blur. Dye <concentration> approximates a <passive scalar> only when it has negligible feedback on the <velocity field>; finite port width, <diffusion> and out-of-plane motion limit how literally a visible filament can be read as a geometrical <streakline>. https://www.flowvis.org/Flow%20Vis%20Guide/do-not-disturb-dyes-in-liquids/[Jean Hertzberg and Flow Vis course students, Dye Techniques 1: Do Not Disturb]{ref}
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