P$3\mathrm{.}128$ A venturi meter, shown in Fig. P$3\mathrm{.}128,$ is a carefully designed constriction whose pressure difference is a measure of the flow rate in a pipe. Using Bernoulli's equation for steady incompressible flow with no losses, show that the flow rate $Q$ is related to the manometer reading $h$ by

$Q=\frac{A_2}{\sqrt[\phantom{2}]{1-(\frac{D_2}{D_1}{)}^4}}\sqrt[\phantom{2}]{\frac{2gh({}_M-)}{}}$

where ${}_M$ is the density of the manometer fluid.

P$3\mathrm{.}128$