What is equation of continuity ? Derive it.

Explanation

It states that, as the area of cross section of the tube becomes larger, then speed of liquid becomes smaller and vice versa.
i.e: av = constant
where a = area of cross section
v= speed of liquid.
Proof: Consider a liquid flowing through a tube of varying cross-sectional areas.
Let `a_(1)`= Cross-sectional area at end A.
`a_(2)` = Cross-sectional area at end B
`v_(1)`= Velocity of the liquid at end A
`v_(2)`= Velocity of the liquid at end B
`rho_(1)`= Density of the liquid at end A
`rho_(2)`= Density of the liquid at end B.
`:.` Volume of liquid entering at A per second,
`V_(1)=a_(1)v)(1)" "......(i)`
Volume of liquid leaving at B per second,
`V_(2)=a_(2)v_(2)" "........(ii)`
Mass of liquid entering A per second,
`m_(1)=` volume per second `xx` density
`m_(1)=a_(1)v_(1)rho_(1)" "......(iii)`
Mass of liquid leaving B.per second,
`m_(2)`= volume per second `xx` density
`m_(2)=a_(2)v_(2)rho_(2)" ".......(iv)`
Since there is no sink or source between end A and B. Therefore, Mass entering at A = Mass leaving at B
Using (iii) and (iv), we get
`a_(1)v_(1)rho_(1)=a_(2)v_(2)rho_(2)`
Since, liquid is incompressible, therefore
`:.rho_(1)=rho_(2)=rho`
`:.a_(1)v_(1)rho=a_(2)v_(2)rhoimpliesa_(1)v_(1)=a_(2)v_(2)`
or av=constant. This is equation of continuity.