First slide

Viscous flow/Laminar flow

Question

Two drops of the same radius are falling through air with a steady velocity of 5 cm per sec. If the two drops coalesce, the terminal velocity would be

Moderate

A

10 cm per sec
B

2.5 cm per sec
C

$5 \times {(4)}^{1 / 3} cm$ per sec
D

$5 \times \sqrt{2} cm$ per sec

Solution

If two drops of same radius r coalesce then radius of new drop is given by R
$\frac{4}{3} {πR}^{3} = \frac{4}{3} {πr}^{3} + \frac{4}{3} {πr}^{3} \Rightarrow R^{3} = 2 r^{3} \Rightarrow R = 2^{1 / 3} r$
If drop of radius r is falling in viscous medium then it acquire a critical velocity v and $v \propto r^{2}$
$\begin{array}{l} \frac{v_{2}}{v_{1}} = {(\frac{R}{r})}^{2} = {(\frac{2^{1 / 3} r}{r})}^{2} \\ \Rightarrow v_{2} = 2^{2 / 3} \times v_{1} = 2^{2 / 3} \times (5) = 5 \times (4)^{1 / 3} cm / s \end{array}$

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