Streamlines, streaklines, and pathlines

• Streamlines are a family of curves whose tangent vectors constitute the velocity vector field 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 (as in dye tracing) 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.

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.

Mathematical description

Streamlines

Streamlines are defined by $${\displaystyle \frac{d{\overrightarrow{x}}_S}{ds}\times \overrightarrow{u}({\overrightarrow{x}}_S)=\overrightarrow{0},}$$ where "${\displaystyle \times }$" denotes the vector cross product and ${\displaystyle {\overrightarrow{x}}_S(s)}$ is the parametric representation of just one streamline at one moment in time.

If the components of the velocity are written ${\displaystyle \overrightarrow{u}=(u,v,w),}$ and those of the streamline as ${\displaystyle {\overrightarrow{x}}_S=(x_S,y_S,z_S),}$ then $${\displaystyle \frac{d{x}_S}{u}=\frac{d{y}_S}{v}=\frac{d{z}_S}{w},}$$ which shows that the curves are parallel to the velocity vector. Here ${\displaystyle s}$ is a variable which parametrizes the curve ${\displaystyle s\mapsto {\overrightarrow{x}}_S(s).}$ 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: $${\displaystyle \frac{\mathrm{\partial }c}{\mathrm{\partial }t}+c\frac{\mathrm{\partial }c}{\mathrm{\partial }s}=\nu \frac{{\mathrm{\partial }}^{2}c}{\mathrm{\partial }{r}^2}-\frac{1}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }s}-g\frac{\mathrm{\partial }z}{\mathrm{\partial }s}}$$

The flow velocity in the direction ${\displaystyle s}$ of the streamline is denoted by ${\displaystyle c}$. ${\displaystyle r}$ is the radius of curvature of the streamline. The density of the fluid is denoted by ${\displaystyle \rho }$ and the kinematic viscosity by ${\displaystyle \nu }$. ${\displaystyle \frac{\mathrm{\partial }p}{\mathrm{\partial }s}}$ is the pressure gradient and ${\displaystyle \frac{\mathrm{\partial }c}{\mathrm{\partial }s}}$ the velocity gradient along the streamline. For a steady flow, the time derivative of the velocity is zero: ${\displaystyle \frac{\mathrm{\partial }c}{\mathrm{\partial }t}=0}$. ${\displaystyle g}$ denotes the gravitational acceleration.

Pathlines

Pathlines are defined by $${\displaystyle \{\begin{array}{l}{\displaystyle \frac{d{\overrightarrow{x}}_P}{dt}}(t)={\overrightarrow{u}}_P({\overrightarrow{x}}_P(t),t)\\ {\overrightarrow{x}}_P(t_0)={\overrightarrow{x}}_{P0}\end{array}}$$

The subscript ${\displaystyle P}$ indicates a following of the motion of a fluid particle. Note that at point ${\displaystyle {\overrightarrow{x}}_P}$ the curve is parallel to the flow velocity vector ${\displaystyle \overrightarrow{u}}$, where the velocity vector is evaluated at the position of the particle ${\displaystyle {\overrightarrow{x}}_P}$ at that time ${\displaystyle t}$.

Streaklines

Streaklines can be expressed as, $${\displaystyle \{\begin{array}{l}{\displaystyle \frac{d{\overrightarrow{x}}_{str}}{dt}={\overrightarrow{u}}_P({\overrightarrow{x}}_{str},t)}\\ {\overrightarrow{x}}_{str}(t={\tau }_P)={\overrightarrow{x}}_{P0}\end{array}}$$ where, ${\displaystyle {\overrightarrow{u}}_P(\overrightarrow{x},t)}$ is the velocity of a particle ${\displaystyle P}$ at location ${\displaystyle \overrightarrow{x}}$ and time ${\displaystyle t}$. The parameter ${\displaystyle {\tau }_P}$, parametrizes the streakline ${\displaystyle {\overrightarrow{x}}_{str}(t,{\tau }_P)}$ and ${\displaystyle t_0\le {\tau }_P\le t}$, where ${\displaystyle t}$ is a time of interest.