# Is equation of continuity valid even against gravity?

Consider a tube having variable cross section and is placed vertically. The upper part it has a cross section of $4\ \mathrm{mm^2}$ and the lower part has a cross section of $2\ \mathrm{mm^2}$.

If water enters per second through upper part (say at a rate of $V\ \mathrm{cm^3}$) will equation of continuity be valid to calculate velocity at the lower part. (I don't think so because gravity will accelerate the fluid).

And if it is valid then does it means that no matter in what orientation the tube is placed ( whether horizontal or vertical) velocity at smaller cross section will be the same?

Here with equation of continuity I specifically mean this equation $A_1v_1=A_2v_2$. Will this be valid in both the cases.

Consider this figure for reference • Loop Back, the continuity equation that you used applies only to incompressible flow (e.g., liquids). Having said that, the continuity equation describes conservation of mass, which is independent of gravitational fields. Regardless of elevation changes, the continuity equation still applies. – David White Aug 26 '18 at 18:20

## 2 Answers

Yes, the equation of continuity is always valid, because otherwise it is implied that mass is being created or destroyed. Even in the case where you have a phase change (e.g. vaporization) or a chemical reaction, continuity would still be valid (as long as you include the relevant terms for the additional phase or chemical species).

In your example, the effect of the orientation of the tube is that when it is vertical, you will have to include a gravitational term in the momentum equation when calculating the velocity. Consider Bernoulli's Equation (for incompressible water, along a streamline):

$$\frac{v^2}{2}+gz+\frac p\rho =constant$$

When the tube is vertical, the $gz$ term will come into play ($z$ is the height). So, if you have the same pressure drop as the horizontal case, the gravity term will act to increase $v$. Conversely, to achieve the same $v$, you wouldn't have to have as much of a pressure drop. Gravity acts in a similar way to pressure on momentum.

• Comments are not for extended discussion; this conversation has been moved to chat. – David Z Aug 26 '18 at 21:41

Pascals law does not apply here. It is only true for static situations.

• Ok so how do I correct bernoulli theorum then. And what about equation of continuity – user203191 Aug 25 '18 at 17:48
• If equation of continuity is valid then it won't make any difference between a horizontal tube and vertical tube. But don't you think in vertical tube gravity will change fluid's velocity – user203191 Aug 25 '18 at 17:49
• @Loop Back Not if you specify the entering velocity, and if $P$ remains greater than the fluid's vapour pressure (if it gets slower than that, cavitation will occur). – mike stone Aug 25 '18 at 18:54