Bernoulli's Equation — Question 4

Question 4

A finite reservoir has a moving free surface

Use a quasi-steady, lossless model for water draining from a vertical tank of constant area $A$ through an outlet of area $a$ at its bottom elevation. The free surface and jet are both at atmospheric pressure. Treat their velocities as uniform and let $h$ be their height difference. Define $r=a/A<1$.

1. Using continuity and Bernoulli at an instant, derive the outlet speed including free-surface motion.

2. For $r=0.20$ and $g=10\,\mathrm{m/s^2}$, estimate the time to drain from $h=1.0$ to $0.25\,\mathrm{m}$. Explain why the limit $r\to1$ signals failure of the approximation.

Question 4 - Solution

A finite reservoir has a moving free surface

(a) Derivation and calculation

With $r=a/A$, continuity gives $Av_s=av_o$, so $v_s=rv_o$. Instantaneous quasi-steady Bernoulli gives$$gh+\tfrac12v_s^2=\tfrac12v_o^2\Rightarrow gh=\tfrac12(1-r^2)v_o^2.$$$$\boxed{v_o=\sqrt{\frac{2gh}{1-r^2}}},\qquad\frac{dh}{dt}=-r\sqrt{\frac{2gh}{1-r^2}}.$$

(b) Result and interpretation

$$\Delta t=\frac{\sqrt{1-r^2}}r\sqrt{\frac2g}(\sqrt{h_i}-\sqrt{h_f}).$$$$\boxed{\Delta t=\frac{\sqrt{.96}}{.20}\sqrt{\frac2{10}}(1-.50)=1.09545\,\mathrm s}.$$$$r\to1^-:\quad v_o\to\infty,\quad\Delta t\to0.$$These singular predictions flag failure of the quasi-steady approximation; unsteady dynamics must then be retained.

Physical check

For $r\ll1$, the outlet-speed expression approaches $\sqrt{2gh}$ for a large reservoir.