Railway line

This article explains the technical term pathway in fluid mechanics ; for the same term from the transport sector, see line (traffic) or railway line .

Fluid flow in a velocity field.

Spark trajectory lines , recording of flying sparks with long exposure times

A railway line ( engl. Pathline ) draws in the fluid mechanics the path (the trajectory ) of a single particle in a flow field by. A railway line is clearly created when the trajectory of a spark is recorded over an open fire with a long exposure time.

Railway lines, along with streamlines , stroking lines and timelines, are part of the “Characteristic Lines” visualization concept .

The graphic shows fluid flow in a velocity field . The red point is placed in the field at the point in time at the origin and draws a red path line. In its movement it leaves a blue trail, the streak line that begins with the trajectory at the origin, but shows the points of the trajectory line as they are carried along by the flow of the fluid. The speed is shown twice; with flow vectors and with gray dashed streamlines that connect the flow vectors. ${\displaystyle \mathbf {u} (\mathbf {x} ,t)=(1,xt)}$ ${\displaystyle t=0}$

calculation

Railway lines are defined by

${\displaystyle {\begin{cases}\displaystyle {\frac {\mathrm {d} {\vec {x}}}{\mathrm {d} t}}={\vec {u}}({\vec {x}},t)\\[1.2ex]{\vec {x}}(t_{0})={\vec {x}}_{0}\end{cases}}}$

where we follow the motion of a particle in the flow.

Note that at the point the curve is parallel to the velocity vector of the flow, the velocity vector being evaluated at the position of the particle at the time . ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {u}}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle t}$