The Leaking Hourglass · AI Generator #GEN-4D60D826

Physics · fluid-mechanics

Problem

An hourglass contains fine sand that flows through a circular opening of diameter

d = 2.0\;\text{mm}

at the bottom of the upper chamber. The upper chamber has a uniform circular cross-section with diameter

D = 4.0\;\text{cm}

. Initially, the sand forms a column of height

h_0 = 12\;\text{cm}

above the opening. As the sand flows out, it behaves like a viscous fluid with density

\rho = 1500\;\text{kg/m}^3

and dynamic viscosity

\mu = 0.50\;\text{Pa}\cdot\text{s}

. The flow through the opening follows Poiseuille's equation for flow through a short circular orifice, where the volume flow rate is given by:

Q = \frac{\pi d^4 \rho g h}{128 \mu L}

where

h

is the current height of sand above the opening and

L = 1.5\;\text{mm}

is the effective length of the opening. Find the time required for the sand level to drop from the initial height

h_0

to half that height,

h_0/2