A thin square plate of side $2\ m$ is moving at the interface of two very viscous liquids of viscosities ${\eta _1} = 1$ poise and ${\eta _2} = 4$ poise respectively as shown in the figure. Assume a linear velocity distribution in each fluid. The liquids are contained between two fixed plates. $h_1 + h_2 = 3\ m$ . A force $F$ is required to move the square plate with uniform velocity $10\ m/s$ horizontally then the value of minimum applied force will be ........ $N$
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$\mathrm{f}=0.1 \times 4 \times \frac{10}{\mathrm{h}_{1}}+0.4 \times 4 \times \frac{10}{\mathrm{h}_{2}}$
$=\frac{0.1 \times 4 \times 10}{3-h_{2}}+\frac{0.4 \times 4 \times 10}{h_{2}}$
for $f_{\min } \frac{d f}{d h_{2}}=0$
$\mathrm{h}_{2}=2 \mathrm{m}$
$\mathrm{F}_{\min }=12 \mathrm{N}$
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