A rectangular microcircuit chip floats on a thin layer of air of constant thickness $h,$ above a porous surface, maintained by air being blown through the porous surface at speed $q.$ The chip has width $b$ and is sufficiently long in the direction perpendicular to the figure that there is no flow in the $z$ direction. The flow is incompressible and frictional effects may be neglected.

$($Take $$ to be the density of the fluid, $p_a$ the atmospheric pressure and $g$ the gravitational acceleration.$)$

We begin by finding an expression for the speed $U(x)$ of the flow in the $x$ direction in the gap under the chip assuming the flow is uniform in the $y$ direction. Consider the conservation of volume in a thin vertical rectangular region of height $h$ lying between $x$ and $x+x$ represented by this sketch:

The mass flux into this fixed region can be written as:

$($Clear my choice$)$

$qx+U(x)h-U(x+x)h$

$qx-U(x)h+U(x+x)h$

$qx+U(x)h+U(x+x)h$

Now $U(x+x)$ can be expanded as

$U(x+x)=U(x)+x{U}^{\text{'}}(x)+$dots

and so the mass flux into the region can be rewritten as

$($Clear my choice$)$

$x(q+U^{\text{'}}(x)h)$

$x(q-U^{\text{'}}(x)h)$

$x(q+U(x))$

Now use incompressibility to conclude

$($Clear my choice$)$

$q=-U^{\text{'}}(x)h$

$q=U^{\text{'}}(x)h$

$q=-U(x)$

Hence, we obtain $U(x)$ by using symmetry to note that $U(0)=0$ :

$U(x)=$