Question

The horizontal bottom of a wide vessel with an ideal fluid has a round orifice of radius R₁, over which a round closed cylinder is mounted, whose radius R₂ > R₁. The clearance between the cylinder and the bottom of the vessel is very small, the fluid density is ρ. 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.

Answer 1

P_atm + \rho g h

Answer 2

The static pressure in a fluid at rest depends only on the depth from the free surface. Since the fluid is ideal and static, and the clearance has a very small vertical gap, the pressure within the clearance is the same as the pressure at the bottom of the vessel. The pressure at the bottom is given by $P = P_{atm} + \rho g h$, where $P_{atm}$ is the atmospheric pressure, $\rho$ is the fluid density, $g$ is the acceleration due to gravity, and $h$ is the height of the fluid. Therefore, the static pressure in the clearance is $P_{atm} + \rho g h$. If gauge pressure is considered, it is $\rho g h$. The pressure does not depend on the radial distance $r$ or the radii $R_1$ and $R_2$.