A cylindrical vessel with water is rotated about its vertical axis with a constant angular velocity co. Find:

(a) the shape of the free surface of the water;

(b) the water pressure distribution over the bottom of the vessel

along its radius provided the pressure at the central point is equal to Po.

Explanation:

The question is one that examine the physical fundamental of mechanics of a cylindrical vessel .

We would use the Euler' equation and some coriolis and centripetal force formula.

The fig below explains it.

(a) The shape of the free surface of the water is a parabola of revolution as follows;

(b) The water pressure distribution over the bottom of the vessel is

   where r is the distance from the axis.

Explanation:

To solve the question, we solve the Euler's equation of the form

ω) = -ρu×ω = (P + ρ ║u║²) + ρg

When in uniform rotation, we have

u = , ω = where = rΩ and   = 2Ω

Therefore, u × ω = 2·r·Ω²·

From which the radial component of the vector equation is given as

-2·p·r·Ω² =

Therefore,

=

Integrating gives

P(r, z) =

By substituting the above into the z component of the equation of motion, we obtain;

Therefore

P(r, z) =

From the boundary conditions r = R and z = , we find C₃ as follows

P(r = R, z = )  =

Therefore   =  

From which we have

P(r, z) -   =   -  

P(r, z) -    =

We note that at the surface, the interface between the air and the liquid

P = , the shape of the of the free surface of the water is therefore;

,

Given that at r = 0 we have the height = h₀

Therefore,

The shape of the of the free surface of the water is a parabola of revolution.

(b) The water pressure distribution over the bottom of the vessel is given by

ρ × g × z  

= =