Air $(1\mathrm{.}23\frac{kg}{m^3})$ flows past an object in a $2-$m$-$diameter pipe and exits as a free jet as shown in the figure below. The velocity and pressure upstream are uniform at $V=18\frac{m}{s^{\text{'}}}$ and $p=53\frac{N}{m^2},$ respectively. At the pipe exit the velocity is non$-$uniform as indicated. The shear stress along the pipe wall is negligible. $($a$)$ Determine the head loss associated with a fluid streamline as it flows over a solid body from the upstream region of uniform velocity to the wake region at the exit plane of the pipe. $($b$)$ Determine the force that the air puts on the object.

Assume $V_1=22\mathrm{.}7\frac{m}{s},V_2=4\frac{m}{s}.$

Head loss is associated with changes in static pressure head, dynamic pressure head, elevation and mechanical head in a flow.

$($a$)$ Does the total head increase or decrease as it flows over the body?

$($b$)$ What is the decrease in pressure head from a location upstream of the object to a location at the exit plane in the wake of the object? $\frac{?}{{?}_{\frac{?}{?}}P}=$

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$($c$)$ What is the dynamic pressure head upstream of the object? $P_d=$

$($c$)$ What is the dynamic pressure head upstream of the object? $P_d=$ i

$($d$)$ What is the dynamic pressure head of the fluid streamline in the wake region downstream of the object? $P_w=$

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$($e$)$ What is the decrease in dynamic pressure head from a location upstream of the object to a location at the exit plane in the wake of the object? $P_{dyn}=$

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$($f$)$ What is the change in elevation from a location upstream of the object to a location at the exit plane in the wake of the object? $z=$

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$($g$)$ What is the change in mechanical head, due to drag from a location upstream of the object to a location at the exit plane in the wake of the object? $h_m=$

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$($h$)$ What is the magnitude of the head loss associated with the object? $h_L=$