A solid sphere, of radius R acquires a terminal velocity v1 when falling (due to gravity) through a viscous fluid having a coefficient of viscosity η. The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, v2, when falling through the same fluid, the ratio (v1/v2) equals:

Engineering

Physics

Viscosity and Terminal Velocity New

Question

A solid sphere, of radius R acquires a terminal velocity v₁ when falling (due to gravity) through a viscous fluid having a coefficient of viscosity η. The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, v₂, when falling through the same fluid, the ratio (v₁/v₂) equals:

1/27

1/9

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$\frac{4}{3} {πR}^{3} = 27 \times \frac{4}{3} {πr}^{3}$

$\Rightarrow r^{3} = \frac{R^{3}}{3^{3}} \Rightarrow r = \frac{R}{3}$

$V_{1} = \frac{(ρ_{0} - ρ_{liq .}) \frac{4}{3} {πR}^{3} g}{6 πηR}$

$V_{2} = \frac{(ρ_{0} - ρ_{liq .}) \frac{4}{3} π {(\frac{R}{3})}^{3} g}{6 πη (\frac{R}{3})} = \frac{(ρ_{0} - ρ_{liq .}) \frac{4}{3} {πR}^{3} g \times \frac{1}{27}}{6 πηR \times \frac{1}{3}}$

$V_{2} = \frac{V_{1}}{9}$