# Pressure in a fluid

If a fluid is flowing along a vertical line with a constant velocity, will the pressure at every point be the same and irrespective of height?

• What are your thoughts? – Chet Miller Mar 17 '16 at 17:57
• I suspect that it will be the same. Consider three infinitesimal layers of fluid touching each other. If the hydrostatic pressure(as i suspect that you are referring to it) from the middle layer of fluid that acts on the lower layer is greater than the pressure acted from the upper to the middle layer, then the lower layer would get pushed more than the middle, so(if you think about it in non-fluid Newtonian mechanics terms) the middle and lower layers will tend to detach. So, the pressure must be the same. But,having little experience with fluids, i am not sure if my reasoning is correct. – TheQuantumMan Mar 17 '16 at 18:15
• Is pressure at every point in a static column constant? Is $v=0$ a valid constant velocity? – BowlOfRed Mar 17 '16 at 19:27

The pressure must be different because the fluid is in equilibrium (moving at a constant velocity), but the force of gravity is acting on it downwards. This can only be balanced by a pressure difference:

Resolving forces vertically for a cube of fluid with cross-sectional area $A$ and height $\Delta z$: $$(p + \Delta P)A = pA + \rho g A \, \Delta z$$

The areas cancel, and so do the main pressure terms, leaving:

$$\Delta p = \rho g \Delta z$$

which is pascal's law.

In order for it to flow with constant velocity (and assuming incompressible), it must have constant area, as in a vertical pipe. So @alexdavey is right, for non-viscous fluid or slow velocity.

If the fluid is viscous, there will be wall drag. If the velocity is downward, the drag will counteract gravity, so there will be a terminal velocity. In that case, the pressure will not depend on $z$.