Summary

DescriptionStreaklines and pathlines animation.gif	English: The plot shows fluid flow under the velocity field ${\displaystyle \mathbf {u} (\mathbf {x} ,t)=(1,xt)}$. The red dot represents a particle released from the origin at time ${\displaystyle t=0}$, tracing out its pathline in red. As it moves, it leaves behind a trail of blue ink (released from the origin), which starts on the pathline, but is carried up by the fluid flow. The velocity is represented twice, with flow vectors, and the grey dashed streamlines, which connect the flow vectors. Deutsch: Die Grafik zeigt Fluid-Strömung in einem Geschwindigkeitsfeld ${\displaystyle \mathbf {u} (\mathbf {x} ,t)=(1,xt)}$. Der rote Punkt wird zum Zeitpunkt ${\displaystyle t=0}$ im Ursprung ins Feld gesetzt und zeichnet eine rote Bahnlinie. In seiner Bewegung hinterlässt er eine blaue Spur, die Streichlinie, die mit der Bahnlinie im Ursprung beginnt, aber die Punkte der Bahnlinie zeigt, wie sie von der Strömung des Fluids mitgenommen werden. Die Geschwindigkeit wird zweifach dargestellt; mit Strömungs-Vektoren und mit grau gestrichelten Stromlinien, die die Strömungs-Vektoren verbinden.
Date	5 February 2009
Source	Own work   This diagram was created with Mathematica.
Author	Fi1Kaiv8
Other versions	Streaklines and pathlines animation (low).gif

Source code

p = Join[{
   If[x < t, 1/3 x^3, 100],
   If[x < t, 1/2 t x^2 - 1/6 x^3, 100]},
  Table[1/2 (x^2) t + y0, {y0, -10, 40, 0.9}]];
Export["plot.gif",
Table[Plot[p, {x, 0, 10.6},
  PlotRange -> {0, 10.6},
  AspectRatio -> 1,
  PlotRangeClipping -> True,
  PlotStyle ->
   PadRight[{Directive[Thickness[Medium], Red],
     Directive[Thickness[Medium], Green]}, 60,
    Directive[Gray, Dashed]],
  Exclusions -> {t},
  ExclusionsStyle -> None,
  ImageSize -> 500,
  AxesLabel -> {x, y},
  Epilog ->
   {Line[Flatten[
      With[{\[Epsilon] = 0.025},
       Table[{{x - \[Epsilon]/Sqrt[1 + x^2 t^2],
          y - x t \[Epsilon]/Sqrt[1 + x^2 t^2]}, {x + \[Epsilon]/
            Sqrt[1 + x^2 t^2],
          y + x t \[Epsilon]/Sqrt[1 + x^2 t^2]}}, {x, 0.1, 4,
         0.1}, {y, 0.1, 4, 0.1}]]
      , 1]],
    PointSize[0.015], Red, Point[{t, 1/3 t^3}]}
  ], {t, 0, 1.6, 0.025}],
AnimationRepetitions -> Infinity,
"DisplayDurations" -> PadLeft[{1.5}, 65, 0.1]
]

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