Question

A bubble having surface tension T and radius R is formed on a ring of radius b (b << R). Air is blown inside the tube with velocity v as shown. The air molecules collide perpendicularly with the wall of the bubble and stops. Calculate the radius at which the bubble separates from the ring.

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is A

Force due to inflow,
$F_1=\frac{dp}{dt}$
=$v\frac{dm}{dt}$
= $v\times \rho Av$
=$\rho A{v}^2$
Force of adhesion = force due to surface tension

$F_{T}sin\theta =F_1$ (when the bubble breaks off)
$F_T=2T\times 2\pi b$
$\therefore 2T\times 2\pi bsin\theta =\rho A{v}^2$
$\sin \theta =\frac{b}{R^{\prime }}A=\pi b^2$
$\Rightarrow 2T\times 2\pi b\times \frac{b}{R}=\rho \pi b^2{v}^2$
$\Rightarrow \frac{4T{b}^2}{R}=\rho b^2{V}^2$
$\Rightarrow R=\frac{4T}{\rho v^2}$