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I don't think this is as large of a problem as it seems. Consider the steady state assumption in more detail. $v_2$ is the final velocity going through the hole after sufficient time has passed for the acceleration to occur. Right after you open the hole everything is stationary. There is an acceleration phase which is normally assumed to take a short ...

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Bernoulli's principle is a simplification of the Navier–Stokes equations, namely it assumes constant density and steady state. In the situation where $A_2$ has almost the same surface area as $A_1$ it will be hard to satisfy the assumption that the system is steady state. In real life there will also be some frictional losses, but if you neglect those then ...

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Yes, the flow rate is related to the pump pressure by the Darcy-Weisbach equation: $$\Delta P = f_d \frac{L}{D} \frac{\rho v^2}{2}$$ where $L$ is the length of the pipe, $D$ is the diameter, $\rho$ is the water density, $v$ is the flow velocity and $f_D$ is a fudge factor called the Darcy friction factor. $f_D$ varies with the pipe diameter, density and ...

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