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?

Answer

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.