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:

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. Pathlines are allowed to intersect themselves or other pathlines (except the starting and end points of the different pathlines, which need to be distinct). Streaklines can also intersect themselves and other streaklines.

Streamlines provide a snapshot of some flowfield characteristics, whereas streaklines and pathlines depend on the full time-history of the flow. Often, sequences of streamlines or streaklines at different instants, presented either in a single image or with a videostream, may provide insight to 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.

Streamlines are defined by $${\displaystyle {d{\vec {x}}_{S} \over ds}\times {\vec {u}}({\vec {x}}_{S})={\vec {0}},}$$ 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.

If the components of the velocity are written ${\displaystyle {\vec {u}}=(u,v,w),}$ and those of the streamline as ${\displaystyle {\vec {x}}_{S}=(x_{S},y_{S},z_{S}),}$ then $${\displaystyle {dx_{S} \over u}={dy_{S} \over v}={dz_{S} \over w},}$$ which shows that the curves are parallel to the velocity vector. Here ${\displaystyle s}$ is a variable which parametrizes the curve ${\displaystyle s\mapsto {\vec {x}}_{S}(s).}$ 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: $${\displaystyle {\frac {\partial c}{\partial t}}+c{\frac {\partial c}{\partial s}}=\nu {\frac {\partial ^{2}c}{\partial r^{2}}}-{\frac {1}{\rho }}{\frac {\partial p}{\partial s}}-g{\frac {\partial z}{\partial s}}}$$