First slide

Viscous flow/Laminar flow

Question

A thin plate AB of large area A is placed symmetrically in a small gap of height 'h' filled with water of viscosity $η_{0}$ and the plate has a constant velocity v by applying a force F, as shown in the figure. If the gap is filled with some other liquid of viscosity 0.75 $η_{0}$ where should the plate be placed in the gap, so that the plate can again be pulled at the same constant velocity V, by applying the same force F. (Take h = 20 cm)

Moderate

Solution

$\begin{array}{l} Case I: y_{1} = y_{2} = h / 2 \\ \frac{\partial u}{\partial y} = \frac{v}{h / 2} = \frac{2 v}{h} \\ Viscous force = \frac{η_{0} 2 V}{h} A \end{array}$
The plate has got two sides.
$\begin{array}{l} \begin{aligned} Total force F = \frac{4 η_{c} v a}{h} \\ Case II: F = \frac{η v A}{y} + \frac{η v A}{y - h} = \frac{η v A h}{y (h - y)} \\ \frac{η v A h}{y (h - y)} = \frac{4 n^{2} A v}{h}; \frac{3}{16} h^{2} = y (h - y) \end{aligned} \\ Solving we get y = \frac{h}{4}, \frac{3 h}{4} \end{array}$
Hence, for minimum distance from top y=20/4= 5 cm

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