What is a venturi-meter ? Explain its construction and working.

The venturi meter is a device to measure the flow speed of incompressible fluid.

It consists of a tube with a broad diameter and a small contraction at the middle as shown in figure.A manometer in the form of a U-tube is also attached to it, with one arm at the broad neck point of the tube and the other at construction which is known as throat.

The manometer contains a liquid of density $\rho_{m}$.

At point $1$, cross section area is $\mathrm{A}$ and velocity of liquid is $v_{1}$. At point $2$, cross section area is $a$ and velocity of liquid is $v_{2}$. According to equation of continuity,

$\mathrm{A} v_{1}=a v_{2}$

$\therefore v_{2}=\frac{\mathrm{A} v_{1}}{a}$

Writing Bernoulli's equation at point $1$,$2$ .

$\mathrm{P}_{1}+\frac{1}{2} \rho v_{1}^{2}+\rho g h,=\mathrm{P}_{2}+\frac{1}{2} \rho v_{2}^{2}+\rho g h_{2}$

$\therefore \mathrm{P}_{1}-\mathrm{P}_{2}=\frac{1}{2} \rho v_{2}^{2}-\frac{1}{2} \rho v_{1}^{2}$

$\therefore \mathrm{P}_{1}-\mathrm{P}_{2}=\frac{1}{2} \rho \frac{\mathrm{A}^{2} v_{1}^{2}}{a^{2}}-\frac{1}{2} \rho v_{1}^{2}$