Derive the equation of continuity of steady flow of incompressible fluid.

Explanation

If in steady flow , we draw streamline of every particle ,then creation form like tube.This tube is known as tube of slow .No any particle goes outside from this tube and no anyy other particle enter in this tube.

In above figure tube of flow is shown and shows points , P,R and Q on the tube.
Suppose `v_(p),v_(Q)andv_(R)` is the velocity of fluid particle at point P,Q and R respectively and cross sectional areas are `A_(P),A_(Q)andA_(R)`.
Since flow is steady so number of particles passing point P,Q and R are same. ,
Mass of fluid at cross section area `A_(P)` with velocity `v_(P)` in `Deltat` time.
`m_(P)=A_(P)v_(P)Deltat_(rho)`, where `rho=` density of fluid
The mass of fluid passing through Q and R are respectively ,
`m_(Q)=A_(Q)v_(Q)Deltatrhoandm_(R)=A_(R)v_(R)Deltatrho`
now in `Deltat` time mass entering is same as leaving from flow of tube.
`thereforem_(p)=m_(R)=m_(Q)`
`thereforeA_(P)v_(P)Deltat_(rho)=A_(R)v_(R)Deltat_(rho)=A_(Q)v_(Q)Deltat_(rho)`
`thereforeA_(P)v_(P)=A_(R)v_(R)=A_(Q)v_(Q)`
is the equation of continuity for steady flow. It is a statement for conservation of mass in incompressible liquid.
In general Av = constant `vprop(1)/(A)`
Av gives the volume flux or flow rate and remains constant throughout the pipe of flow .
At narrower portions where the streamlines are closely spaced , velocity increases and its vice versa . Widely spaced streamlines indicate regions of low speed and closely spaced streamlines indicates regions of high speed.