Physics - Fluid Question with Solution | TestHub

Consider a body of density ${\rho }_b$ kept in density ${\rho }_{\mathcal{l}}$ whose viscosity is $\eta$ and terminal velocity V. then



${\overrightarrow{F}}_{viscous\mathrm{ }}+{\overrightarrow{F}}_{mg}+{\overrightarrow{F}}_{Buoyancy}=0$
${\overrightarrow{F}}_{viscous}+{\rho }_b\frac{4}{3}\pi R^3\left(-\hat{j}\right)+{\rho }_l\frac{4}{3}\pi R^3\left(\hat{j}\right)=0$
$\therefore \mathrm{ }{\overrightarrow{F}}_{viscous}=\left({\rho }_b-{\rho }_{\mathcal{l}}\right)\frac{4}{3}\pi R^3\left(\hat{j}\right)\Rightarrow 6\pi \eta RV=\left({\rho }_b-{\rho }_{\mathcal{l}}\right)\frac{4}{3}\pi R^3\mathrm{ }$
$\therefore$ if ${\rho }_b>{\mathrm{\rho }}_1$ then ${\overrightarrow{F}}_{viscous}\uparrow V\propto \frac{1}{\eta }$ & if ${\rho }_b<{\rho }_{\mathcal{l}}$
${\overrightarrow{F}}_{viscous}\mathrm{ }\downarrow$
As per given diagram we can say
${\sigma }_2>{\sigma }_1;{\rho }_1<{\sigma }_1\mathrm{ }\&\mathrm{ }{\sigma }_2>{\sigma }_2$
$\Rightarrow {\rho }_2>{\sigma }_2>{\sigma }_1>{\rho }_1$
$\therefore$ if we put P in $L_2$ where $\left|{\overrightarrow{V}}_P\right|\propto \frac{1}{{\eta }_2}$ when ${\rho }_1<{\sigma }_1$
$\therefore {\overrightarrow{F}}_{viscous}\mathrm{ }\downarrow$
$\therefore {\overrightarrow{V}}_P\mathrm{ }\uparrow$
$\therefore$ if we put Q in $L_1$ where $\left|{\overrightarrow{V}}_Q\right|\propto \frac{1}{{\eta }_1}$ when ${\rho }_2<{\sigma }_1$
$\therefore {\overrightarrow{F}}_{viscous}\mathrm{ }\uparrow$
$\therefore {\overrightarrow{V}}_P\mathrm{ }\downarrow$
$\Rightarrow \frac{\left|{\overrightarrow{V}}_P\right|}{\left|{\overrightarrow{V}}_Q\right|}=\frac{{\eta }_1}{{\eta }_2}$ & ${\overrightarrow{V}}_P.{\overrightarrow{V}}_Q<0$