Given velocity vector the pathlines is a bunch of curves , parametrized by the stem points at zero-time , and solving the following equation:
\begin{cases} \displaystyle {\frac {d{\vec {r}}_s(t)}{dt}}={\vec {u}}_s({\vec {r}}_s,t) \\[1.2ex] {\vec {r}}_s(t_{0})={\vec {r}}_{s0} \end{cases} |