# Equation of Continuity

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.

## More Questions on Equation of Continuity

## Related Topics

## References

- IIT JEE Physics by Jitender Singh and Shraddhesh Chaturvedi