Wednesday, 28 March 2018

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:


Cube equilibrium


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.


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