Streamlines, streaklines, and pathlines

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:

By definition, different streamlines at the same instant in a flow do not intersect, because a fluid particle cannot have two different velocities at the same point. Similarly, streaklines cannot intersect themselves or other streaklines, because two particles cannot be present at the same location at the same instant of time; unless the origin point of one of the streaklines also belongs to the streakline of the other origin point. However, pathlines are allowed to intersect themselves or other pathlines (except the starting and end points of the different pathlines, which need to be distinct).

Streamlines and timelines provide a snapshot of some flowfield characteristics, whereas streaklines and pathlines depend on the full time-history of the flow. However, often sequences of timelines (and streaklines) at different instants—being presented either in a single image or with a video stream—may be used to provide insight in the flow and its history.

If a line, curve or closed curve is used as start point for a continuous set of streamlines, the result is a stream surface. In the case of a closed curve in a steady flow, fluid that is inside a stream surface must remain forever within that same stream surface, because the streamlines are tangent to the flow velocity. A scalar function whose contour lines define the streamlines is known as the stream function.

Dye line may refer either to a streakline: dye released gradually from a fixed location during time; or it may refer to a timeline: a line of dye applied instantaneously at a certain moment in time, and observed at a later instant.

Streamlines are defined by

where "${\displaystyle \times }$" denotes the vector cross product and ${\displaystyle {\vec {x}}_{S}(s)}$ is the parametric representation of just one streamline at one moment in time.

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