page_title: Streamlines, streaklines, and pathlines
section_title: Streamlines
hierarchical_section_title: Streamlines, streaklines, and pathlines / Mathematical description / Streamlines
caption_attribution_description: English: The magnetic field of a bar magnet revealed by iron filings on paper. A sheet of paper is laid on top of a bar magnet and iron filings are sprinkled on it. The needle shaped filings align with their long axis parallel to the magnetic field. They clump together in long strings, showing the direction of the magnetic field lines at each point.Magyar: Rúdmágnes mágneses terének megjelenítése papírra szórt vasreszelékkel. A rúdmágnes tetejére egy papírlapot helyeztek, amelyre vasreszeléket szórtak. A tűalakú reszelékdarabkák tengelye a helyi mágneses térerősségvektorral párhuzamosan helyezkedik el. Hosszú fonalakká kapcsolódnak össze, így mutatva a mágneses erővonalak irányát minden egyes pontban. Español: El campo magnético de un imán se manifiesta por la distribución de las limaduras de hierro alrededor de dicho imán. Una hoja de papel reposa sobre un imán y las limaduras de hierro son esparcidas sobre el papel. Las limaduras de hierro se alinean con el campo magnético, de forma que muestran la dirección de las lineas de campo magnético en cada punto.
context_page_description: Streamlines, streaklines and pathlines are field lines in a fluid flow.
  They differ only when the flow changes with time, that is, when the flow is not steady.
  Considering a velocity vector field in three-dimensional space in the framework of continuum mechanics, we have that:
  Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction in which a massless fluid element will travel at any point in time.
  Streaklines are the loci of points of all the fluid particles that have passed continuously through a particular spatial point in the past. Dye steadily injected into the fluid at a fixed point extends along a streakline.
  Pathlines are the trajectories that individual fluid particles follow. These can be thought of as "recording" the path of a fluid element in the flow over a certain period. The direction the path takes will be determined by the streamlines of the fluid at each moment in time.
  Timelines are the lines formed by a set of fluid particles that were marked at a previous instant in time, creating a line or a curve that is displaced in time as the particles move.
context_section_description: Streamlines are defined by
  where "" denotes the vector cross product and  is the parametric representation of just one streamline at one moment in time.
  If the components of the velocity are written  and those of the streamline as  we deduce
  which shows that the curves are parallel to the velocity vector. Here  is a variable which parametrizes the curve   Streamlines are calculated instantaneously, meaning that at one instance of time they are calculated throughout the fluid from the instantaneous flow velocity field.
  A streamtube consists of a bundle of streamlines, much like communication cable.
  The equation of motion of a fluid on a streamline for a flow in a vertical plane is:
  The flow velocity in the direction  of the streamline is denoted by .  is the radius of curvature of the streamline. The density of the fluid is denoted by  and the kinematic viscosity by .  is the pressure gradient and  the velocity gradient along the streamline. For a steady flow, the time derivative of the velocity is zero: .  denotes the gravitational acceleration.
caption: The direction of magnetic field lines are streamlines represented by the alignment of iron filings sprinkled on paper placed above a bar magnet
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