Equation of Continuity
By Consider flow through a pipe of varying cross-sectional area. The fluid velocity is related to the cross-section area by \begin{align} A_1 v_1 =A_2 v_2 \end{align}
This is called equation of continuity. It is valid for incompressible fluids. The equation of continuity is the same as the conservation of mass.
Problems from IIT JEE
Problem from IIT JEE 2008
Statement 1: The stream of water flowing at high speed from a garden hose pipe tends to spread like a fountain when held vertically up, but tends to narrow down when held vertically down.
Statement 2: In any steady flow of an incompressible fluid, the volume flow rate of the fluid remains constant.
- Statement 1 is true, statement 2 is true; statement 2 is a correct ex- planation for statement 1.
- Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
- Statement 1 is true, statement 2 is false.
- Statement 1 is false, statement 2 is true.
Solution: The statement 2 is same as the continuity equation, $Av=\text{constant}$, where $A$ is the cross-section area and $v$ is the fluid velocity. When pipe is held vertically up, the energy conservation makes $v$ to decrease as water goes up and hence $A$ increases by continuity equation. It is the other way around when the pipe is held vertically down.
