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I was reading about free vortexes and came across this equation that describes the profile of a free vortex $z_{r}= z_{\infty} + \dfrac{k^2}{2gr^2}$

how do you get this relationship? I can't get the expression.

I would be very grateful if you recommend bibliography for this relationship and the free vortices.

Source: equation

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You haven't written the equation from the source you give correctly, the $z_\infty$ and $z_r$ should swap.

Consider a free vortex flow, where fluid is draining through a hole in the base of an infinite reservoir. We will follow a streamline from a point far away from the hole, to a closer point.

By a continuity argument, the velocity at a distance $r$ from the hole is

$$ v=\frac{k}{r}, $$

where $k$ is a constant. This means that very far away from the hole we can say that the velocity is effectively equal to 0. Using Bernoulli along our streamline from a point at infinity to a closer point, we have

$$ \rho g z_\infty=\rho g z_r + \frac{\rho v^2}{2}, $$

where $z_\infty$ is the height of the free surface very far from the hole and $z_r$ is the height at a closer point. Substituting for $v$ using our first equation and rearranging, we find that

$$ z_\infty = z_r + \frac{k^2}{2gr^2}. $$

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  • $\begingroup$ Thank you very much! $\endgroup$ Mar 17, 2021 at 21:29

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