The Draining Hourglass · AI Generator #GEN-1BDF51A8

Physics · fluid-mechanics

Problem

A cylindrical hourglass contains a viscous honey-like fluid with viscosity

\eta = 2.4\;\text{Pa}\cdot\text{s}

and density

\rho = 1400\;\text{kg/m}^3

. The upper chamber has radius

R = 6.0\;\text{cm}

and the connecting tube has radius

r = 1.5\;\text{mm}

and length

L = 8.0\;\text{mm}

. Initially, the fluid level in the upper chamber is

h_0 = 12.0\;\text{cm}

above the top of the connecting tube. As the fluid drains, the height

h(t)

of fluid above the connecting tube decreases. The flow through the narrow tube is laminar and follows Poiseuille's law. Surface tension effects can be neglected. (a) Find the initial flow rate

Q_0

(in

\text{m}^3/\text{s}

) when

h = h_0

. (b) Derive a differential equation for

h(t)

. (c) Find the time

t_{1/2}

(in seconds) for the fluid height to drop to half its initial value.

Related theory

fluid-mechanics

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