First slide

Viscous flow/Laminar flow

Question

A spherical ball falls through viscous medium with terminal velocity v. If this ball is replaced by another ball of the same mass but half the radius, then the terminal velocity will be (neglect the effect of buoyancy.)

Easy

A

v
B

2v
C

4v
D

8v

Solution

$m = \frac{4}{3} π r^{3} ρ$
Keeping m constant, if r is halved, $ρ$ will increased by a factor of 8.
$\begin{array}{l} Now, v_{0} \propto r^{2} ρ \end{array}$
$N e w v e l o c i t y i s v_{0}' \propto \frac{r^{2}}{4} (8 ρ) or v_{0}' \propto 2 r^{2} ρ$
$Dividing, \frac{v_{0}'}{v_{0}} = 2 or v_{0}' = 2 v_{0}$
$or v_{0}' = 2 v$

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