The horizontal bottom of wide vessel with an ideal fluid has a round orifice of radius `R_1` over which a round closed cylinder is mounted, whose radius `R_2gtR_1` (figure). The clearance between the cylinder and the bottom of the vessel is very small, the fluid density is `rho`. Find the static pressure of the fluid in the clearance as a function of the distance r from the axis of the orifice (and the cylinder), if the height of the fluid is equal to h.

Explanation

Water flows through the small clearance into the orifice. Let d be the clearance. Then from the equation of continuity
`(2piR_1d)v_1=(2pird)v=(2piR_2d)v_2`
or `v_1R_1=vr=v_2R_2` (1)
where `v_1`, `v_2` and v are respectively the inward radial velocities of the fluid at 1, 2 and 3.
Now by Bernoulli's theorem just before 2 and just after it in the clearance
`p_0+hrhog=p_2+1/2rhov_2^2` (2)
Applying the same theorem at 3 and 1 we find that this also equals
`p+1/2rhov^2=p_0+1/2rhov_1^2` (3)
(since the pressure in the orifice is `p_0`)
From Eqs. (2) and (3) we also hence
`v_1=sqrt(2gh)` (4)
and `p=p_0+1/2rhov_1^2(1-(v/v_1)^2)`
`=p_0+hrhog(1-(R_1/r)^2)` [Using (1) and (4)]