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Earlier, I asked this question:

Bernoulli's equation + Torricelli's law: does the speed of the fluid change if we change the area of the hole but not the height?

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I now understand: in summary, the velocity at the exit hole v2 stays the same according to Torricelli's Law, the cross sectional area of the top of the container A1 stays the same, so an increase in the area A2 of the exit hole would lead to an increase in the velocity v1 at the top of the water. Because the exit hole area increases but exit velocity stays the same, the volume flow rate increases.

Here is the continuity equation:

$$A_1v_1 = A_2v_2\iff v_1 = \frac{A_2}{A_1}\cdot v_2$$

What I don't understand is this: I get why the volume flow rate increases, but is it allowed to? Don't the continuity equation and conservation of mass in fluids state that the volume flow rate of a fluid through the same container must stay constant? Or, to be concise, $Av = \textrm{constant}$?

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The volume flow A·v=constant holds along the path of the flow for an unchanged flow geometry. The constant is not constant in time and it is not the same constant when you change your flow geometry by increasing the area A2 of your hole. Then you have different a different volume flow A·v=constant2.

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To apply the continuity equation to this container you have to consider the flow rate through the container, compared to the combined flow rate of the spigots. Increasing the diameter of one of the spigots will increase the flow rate out of the spigot. It will also increase the rate at which you lose level in the container, or the flow rate through the container.

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