A fluid in a tank that rotates at a constant angular velocity about an axis (in this case about the
axis), rotates with the tank as a rigid body. The pressure gradient is written in cylindrical coordinates:
,
,
,
.
For this type of rigid body rotation, pressure is a function of
and
:
,
.
Integrating the last equation yields the pressure distribution:
,
.
Along an isobaric surface (surface of constant pressure)
, so:
,
,
integrating both sides yields the equation for surfaces at constant pressure:
,
since
, the equation is simplified to:
,
where
is the pressure in the fluid (Pa),
is the radial distance from the center (m),
is distance from the tank bottom (m),
is fluid density (
),
is angular velocity (rad/s),
is specific weight (
),
is acceleration due to gravity,
is a constant estimated as
(
), where
is fluid height (m).
[1] B. R. Munson, T. H. Okiishi, and W. W. Huebsch,
Fundamentals of Fluid Mechanics, 6th ed., Hoboken, NJ: John Wiley & Sons, 2010.
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