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Particle Paths

For a given flow velocity ${\bf v}(x,y,z,t)$, the equations of the paths of the fluid elements,

\begin{displaymath}
x(t), y(t), z(t),
\end{displaymath}

the `particle paths', are given by the solutions of the coupled differential equations

\begin{displaymath}
{dx\over dt} = v_{x}(x,y,z,t), \hbox{ }{dy\over dt} =
v_{y}(x,y,z,t), \hbox{ }{dz\over dt} =
v_{z}(x,y,z,t).
\end{displaymath}

By comparison, the equation of the streamlines (1.3) are

\begin{displaymath}
{dx\over ds} = {v_{x}\over v}, {dy\over ds} = {v_{y}\over v}, {dz\over
ds} = {v_{z}\over v},
\end{displaymath}



Andrew Wright
2002-09-16