Contd...

being a differential element in the liquid column and is . The external work done is due to the pressure difference a part of which is required to overcome viscous friction in the limbs of the manometer. Assuming a parabolic profile for velocity with a mean value , the expression for work done can be written as

The equation governing is then obtained after some algebraic manipulation as

where Hence the U-tube manometer is a second order system with a natural frequency

and a damping ratio

For a manometer response without oscillations we require . As an example consider a column of length m and water as the working fluid. Since = 80 10^-5 Pa. S and kg/m³ , the condition results in mm. Using tubes of larger radii will result in manometer oscillations.