Problem 28: Radial Spreading of Fluid
1. Conservation of Mass:
The rate at which mass enters the system is $\mu$ (kg/s). The fluid is incompressible with density $\rho$. The volume flow rate is $Q = \frac{\mu}{\rho}$.
The rate at which mass enters the system is $\mu$ (kg/s). The fluid is incompressible with density $\rho$. The volume flow rate is $Q = \frac{\mu}{\rho}$.
2. Volume of Spreading Fluid:
The fluid completely fills the tube and then spreads radially in the gap of height $h$. Let $r$ be the radius of the fluid spread at time $t$. The volume of fluid in the gap is a cylinder of radius $r$ and height $h$. $$ V(t) = \pi r^2 h $$
The fluid completely fills the tube and then spreads radially in the gap of height $h$. Let $r$ be the radius of the fluid spread at time $t$. The volume of fluid in the gap is a cylinder of radius $r$ and height $h$. $$ V(t) = \pi r^2 h $$
3. Relate Volume to Time:
Since flow rate is constant: $$ V(t) = Q \times t $$ $$ \pi r^2 h = \frac{\mu}{\rho} t $$
Since flow rate is constant: $$ V(t) = Q \times t $$ $$ \pi r^2 h = \frac{\mu}{\rho} t $$
4. Solve for Radius $r$:
$$ r^2 = \frac{\mu t}{\pi \rho h} $$ $$ r = \sqrt{\frac{\mu t}{\pi \rho h}} $$
$$ r^2 = \frac{\mu t}{\pi \rho h} $$ $$ r = \sqrt{\frac{\mu t}{\pi \rho h}} $$
Answer:
$$ r = \sqrt{\frac{\mu t}{\pi \rho h}} $$