Problem $73.$ A flat plate of length $L$ and width $W$ is mounted in a wind tunnel with the

plate lying in the horizontal $xy-$plane and the $z-$axis pointing upward $($see figure below$).$

The flow is in the $x-$direction and the width of the plate runs transverse to the flow in the

$y-$direction. Take $x=0$ to be the leading edge of the plate. Let $P_{}$ denote the upstream

wind$-$tunnel pressure. There is no flow over the bottom and sides of the plate.

The walls of the wind tunnel are designed to give an approximate pressure profile along

the surface of the plate $(z=0)$ that falls off linearly with downstream distance and is

independent of the transverse $y-$direction. The form of the pressure distribution along

the plate is given by:

$P(x)=P_{}(1-a\frac{x}{L}),$ where $a>0is$ a dimensionless parameter.

Since the flow field is one$-$dimensional in the $x-$direction, the shear stress distribution

is also independent of $y$ over the plate $($i$.$e$.,$ there is no cross flow$).$ The shear stress

distribution is known to be of the form: ${}_w(x)=P_0(1+\frac{x}{L}{)}^{-\frac{1}{2}},$ where $P_0$ has units of

pressure.

For the given pressure distribution $P(x)$ and shear stress distribution ${}_w(x),$ calculate

the magnitude and direction of the net aerodynamic force on the surface.

Hint: The net force due to pressure and shear stress is found by integrating the pressure

and shear stress over the upper surface of the plate. In both cases, the differential area

is given by $dA=Wdx.$

Figure $4$: Flow over a flat plate.