Question

Derive equation of continuity.

Answer 1

Consider a non-viscous liquid in streamlined flow through a tube AB at the varying cross-section.
Let $a_1$,$a_2$ be the area of cross-section of the tube at A and B.
$v_1,v_2$ = velocity of flow of liquid at A and B.
Volume of the liquid entering per second at $A=a_1{v}_1$
Mass of liquid entering per second at $A=a_1{v}_1{\rho }_1$
Similarly mass of liquid leaving per second at $B=a_2{v}_2{\rho }_2$
Assuming, no loss of liquid in the tube and
steady flow, then the mass of liquid entering at A/sec = mass of liquid leaving at B/sec.
$\Rightarrow a_1{v}_1{\rho }_1=a_2{v}_2{\rho }_2$
Assuming liquid is incompressible,
${\rho }_1={\rho }_2$
$\Rightarrow a_1{v}_1=a-2v_2$
$av=constant$
This is the equation of continuity.