Question
Microchips are supported on a thin air film on a smooth horizontal surface during one stage of the manufacturing process. The chips are $11.7 \mathrm{mm}$ long and $9.35 \mathrm{mm}$ wide and have a mass of 0.325 g. The air film is 0.125 mm thick. The initial speed of a chip is $V_{0}=1.75 \mathrm{mm} / \mathrm{s} ;$ the chip slows as the result of viscous shear in the air film. Analyze the chip motion during deceleration to develop a differential equation for chip speed $V$ versus time $t .$ Calculate the time required for a chip to lose 5 percent of its initial speed. Plot the variation of chip speed versus time during deceleration. Explain why it looks as you have plotted it.
Step 1
Step 1: First, we start with the shear stress equation, which is given by $\tau = \mu \frac{du}{dy}$, where $\mu$ is the dynamic viscosity, $du$ is the change in velocity, and $dy$ is the change in height. Show more…
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A rectangular computer chip floats on a thin layer of air, $h=0.5 \mathrm{mm}$ thick, above a porous surface. The chip width is $b=40 \mathrm{mm},$ as shown. Its length, $L,$ is very long in the direction perpendicular to the diagram. There is no flow in the $z$ direction. Assume flow in the $x$ direction in the gap under the chip is uniform. Flow is incompressible, and frictional effects may be neglected. Use a suitably chosen control volume to show that $U(x)=q x / h$ in the gap. Find a general expression for the $(2 \mathrm{D})$ acceleration of a fluid particle in the gap in terms of $q, h, x,$ and $y$ Obtain an expression for the pressure gradient $\partial p / \partial x$. Assuming atmospheric pressure on the chip upper surface, find an expression for the net pressure force on the chip; is it directed upward or downward? Explain. Find the required flow rate $q$ $\left(\mathrm{m}^{3} / \mathrm{s} / \mathrm{m}^{2}\right)$ and the maximum velocity, if the mass per unit length of the chip is $0.005 \mathrm{kg} / \mathrm{m} .$ Plot the pressure distribution as part of your explanation of the direction of the net force.
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