Details

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).

Reference

[1] B. R. Munson, T. H. Okiishi, and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed., Hoboken, NJ: John Wiley & Sons, 2010.