Particle Paths

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

$$x(t), y(t), z(t),$$

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

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

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

$${dx\over ds} = {v_{x}\over v}, {dy\over ds} = {v_{y}\over v}, {dz\over ds} = {v_{z}\over v},$$