A thin plate separates two liquids of coefficients of viscosity $\eta$ and $4\ \eta$ kept between two fixed plates as shown. If plate has to be pulled by applying minimum force then $\frac{d_2}{d_1}$ is
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$\mathrm{F}=\eta \mathrm{A} \frac{\mathrm{V}-0}{\mathrm{y}}+4 \eta \mathrm{A} \frac{\mathrm{V}-0}{\mathrm{d}-\mathrm{y}}$
$\frac{\mathrm{dF}}{\mathrm{dy}}=0$
$-\frac{\eta}{y^{2}}+\frac{4 \eta}{(d-y)^{2}}=0$
$\frac{\mathrm{d}-\mathrm{y}}{\mathrm{y}}=2$
$y=\frac{d}{3}, \Rightarrow d_{1}=\frac{d}{3}, d_{2}=\frac{2 d}{3}$
$\frac{\mathrm{d}_{2}}{\mathrm{d}_{1}}=2$
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