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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.

Step by step solution

01

Derive Velocity

Velocity \(U(x)\) can be derived from mass flow rate, which is constant and does not depend on \(x\). Thus, \[U(x)=\frac{q}{h}\] where \(q\) is the flow rate.

02

Derive acceleration

Acceleration of a fluid particle in the gap, which is two dimensional, can be given by: \[a = \frac{\partial U}{\partial t} + U \frac{\partial U}{\partial x}\] Since the flow is steady, \(\frac{\partial U}{\partial t} = 0\). Hence, acceleration becomes \[a = U \frac{\partial U}{\partial x} = \frac{q}{h} \frac{\partial qx/h}{\partial x} = \frac{q^2}{h^2}\]

03

Find pressure gradient

The equation of motion in the x direction is: pressure gradient = fluid density * acceleration, hence \[\frac{\partial p}{\partial x} = \rho * a = \rho * \frac{q^2}{h^2}\]

04

Find net pressure force

The upper surface undergoes atmospheric pressure while the lower surface undergoes a pressure distribution determined by the position \(x\). To find the net force, integrate the pressure distribution over the lower surface. The sign will determine the direction of the net force.

05

Find the required flow rate and maximum velocity

The force balance for the chip can be represented as the sum of forces (pressure and gravitational forces) equal to zero. Substituting the values and rearranging the equation would yield a value for \(q\). The maximum velocity can be determined by substituting \(x=0\) in \(U(x) = qx/h \) because at \(x=0\), \(U(x)\) is maximum.

06

Plot the pressure distribution

The pressure at the lower surface of the chip at a distance \(x\) from the left edge can be represented by \(p(x) = p0 - \rho * q^2/h^2 * x\). Using this equation, plot the pressure distribution.

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