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I have a cylinder full of water with diameter $D$ with a round opening on the bottom with diameter $d$. The water is friction-free and incompressible. Now I need a relationship for the efflux speed $v$ with which water exits the cylinder and I shouldn't use the approximation $d \ll D$, but formulate a general relationship.

Ok. So what I thought is to equate the Bernoulli law on the top of the cylinder with that on the bottom of the cylinder which gives me $v=\sqrt{2gh}$. Solved. How does the speed of efflux depend on the diameter of the efflux hole? I googled quite a lot and all I could find was the above relationship...

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While calculating velocity of efflux you use bernoulli's theorem as follows :

At the top of container :$ P + \frac{\rho {v_1}^2}{2} + \rho gh_1 = k$
At the efflux : $P_a + \frac{\rho {v_2}^2}{2} + \rho g h_2 = k$

$P - P_a + \rho g (h_1 - h_2) = \frac{\rho ({v_2}^2 -{v_1}^2)}{2} $

Now, in accordance with equation of continuity,
$ v_1 A_1 = v_2 A_2$
$ v_1 (\pi D^2) = v_2 (\pi d^2) $
$ v_1 = v_2 \frac{d^2}{D^2}$
If $ d<<D$ , then $v_1 = 0$.

This gives us :

$v_2 = \sqrt{2gh + \frac{2(P-P_a)}{\rho}}$

Also if top of container is open, or pressure inside container is $P = P_a$ , then $v_2 = \sqrt{2gh}$

If the condition $d<<D$ was not given, we would have to put value of $v_1$ along with $v_2$ in bernoulli's equation.

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If you are concerned about efflux speed you need not worry about the diameter of efflux hole. However If you are worried about Volume rate flow of fluid you need the diameter of the hole.

In that case ${dV/dt}$ =$ d * v $ ; where d is the diameter of the hole, and the v is the efflux speed.

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  • $\begingroup$ But why would they give me the diameters to work with if I don't need them? The problem says nothing about volume/mass rate - just speed. $\endgroup$ – user37600 Jan 18 '14 at 13:24
  • $\begingroup$ the volume rate flow coming down in the container must be equal to the volume rate flow exiting through the hole. Use this... for an incompressible liquid... $\endgroup$ – Sagnik Jan 18 '14 at 16:27

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