= Streamline
{wiki}
A <streamline> at a specified instant $T$ is a curve whose <tangent line> follows the <velocity field> at that same frozen instant everywhere along the curve. One convenient <integral curve> parameterization is
$$
\frac{d\boldsymbol\gamma}{ds}=\mathbf u(\boldsymbol\gamma(s),T),\qquad T\text{ fixed}.
$$
Here $s$ is an auxiliary curve parameter, with units of seconds in this convention; advancing $s$ does not advance physical time. Any regular reparameterization preserves the tangent geometry. Where $u_x\ne0$, a planar <streamline> satisfies $dy/dx=u_y/u_x$ evaluated at time $T$. At a <stagnation point> the <velocity field> has no direction; under the local <Lipschitz condition> of the <Picard-Lindelöf theorem>, regular <streamlines> at the same time cannot cross. A <streamline> answers which directions fit together now, making it useful for describing local flow geometry. https://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec02/tutorial/[MIT fluid mechanics modules, tutorial on streamlines, streaklines and pathlines]{ref}
In the analytical <velocity field> $(U,at)$ introduced under <pathline>, the <streamline> through the origin at time $T$ is
$$
y=\frac{aT}{U}x.
$$
At $T=2\ \mathrm s$ this is $y=2x$ when both coordinates are expressed in metres. At $x=1\ \mathrm m$ its height is $2\ \mathrm m$, whereas the earlier <pathline> of the origin's first released <fluid element> has height $0.5\ \mathrm m$. Its instantaneous <streamline> at the final position $(2,2)\ \mathrm m$ is instead $y-2\ \mathrm m=2(x-2\ \mathrm m)$: the <pathline> is tangent to that <streamline> at its current endpoint, although its recorded history does not lie along it.
\Image[/stream-path-streak-lines-three-curves.png]
{title=Analytical comparison of a frozen-time streamline, one pathline and a streakline}
The green <streamline> uses the <velocity field> at $T=2\ \mathrm s$ throughout; the orange <pathline> uses times $0\leq t\leq2\ \mathrm s$; the purple <streakline> uses different release times at the single observation time $T=2\ \mathrm s$. The grey arrows all show the instantaneous <velocity> direction at $T$. These are exact analytical curves with chosen units, not measured dye shapes.
In a <steady flow> in a fixed <reference frame>, a <pathline> and a <streamline> solve the same autonomous <ordinary differential equation>. Uniqueness makes each particle released from a fixed point follow the same <integral curve>, so its <pathline>, the <streakline> from that point and the <streamline> share one geometrical curve. Finite recording or release intervals usually show different segments of that curve; the comparison assumes particles stay in the domain and treats stationary <fluid elements> separately. Neither two-dimensionality nor <incompressibility> is needed for this coincidence. Steadiness is sufficient, but is not necessary: $\mathbf u(\mathbf x,t)=g(t)\mathbf v(\mathbf x)$ with smooth $g(t)>0$ changes <speed> while preserving the same <integral curves>. In a general time-dependent <velocity field>, however, the three curves differ. Kline explains why continuous dye release in <steady flow> can therefore display <streamlines>. https://web.mit.edu/hml/ncfmf/05FV.pdf[S. J. Kline, Flow Visualization film notes, section II]{ref}
For two-dimensional <incompressible flow>, the convention $\mathbf u=(\partial_y\psi,-\partial_x\psi)$ gives $\mathbf u\cdot\nabla\psi=0$, so regular <streamlines> are <level sets> of the <stream function> at each frozen time. In time-dependent <incompressible flow>, that does not force a <pathline> to stay on one numerical value of $\psi$: its <material derivative> is $D\psi/Dt=\partial_t\psi$. The distinction is the difference between a geometric snapshot and a parcel's history.
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