PX285 - H6a - stream line

a stream line is a curve, generally in 3D, which at a fixed time at each location is tangential to the fluid velocity, $\vec{u}$
let $d \vec{r}$ be parallel to a stream line, ie: $d \vec{r} ∥ \vec{u}$ , or, $d \vec{r} = d λ \vec{u}$ , where $d λ$ is a small scalar
$d x = d λ u_{x}$
$d y = d λ u_{y}$
$d z = d λ u_{z}$

d λ = \frac{d x}{u_{x}} = \frac{d y}{u_{y}} = \frac{d z}{u_{z}}

\frac{d x}{u_{x}} = \frac{d x}{- 2 y} = \frac{d y}{x} = \frac{d y}{d u_{y}}

\int x d x = \int - 2 y d y ⟹ \frac{x^{2}}{2} = - y^{2} + c

the solution belongs to a family of ellipses, with different ellipses for different $c$ values
therefore, it is a deformed vortex