First slide

Viscous flow/Laminar flow

Question

A marble of mass x and diameter 2r is gently released in a tall cylinder containing honey. If the marble displaces mass y (< x) of the liquid, then the terminal velocity is
proportional to

Moderate

A

$x + y$
B

$x - y$
C

$\frac{x + y}{r}$
D

$\frac{x - y}{r}$

Solution

$\begin{array}{l} v_{0} = \frac{2}{9} \frac{r^{2} (ρ - ρ') g}{η} \\ Now, x = \frac{4}{3} π r^{3} ρ or ρ \propto \frac{x}{r^{3}} \\ Similarly, ρ' \propto \frac{y}{r^{3}} \\ ∴ v_{0} \propto \frac{x - y}{r} \end{array}$
$I I m e t h o d x g - y g = 6 {πηrv}_{T} \Rightarrow v_{T} = \frac{(x - y) g}{6 πηr}$

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