$2\mathrm{.}2.$ A simple $2-$D converging nozzle can be represented by the following schematic. It starts with a

center line velocity $)=(0,y$ flowing toward the converging portion of the channel. Depending

on the channel area $A(x),$ one can generate different velocity variation along the streamwise direction if

the area variation is gradual, $A(x)=A_0-x.$ In this case, a simple linear profile is generated:

$u(x,y)=U_0+x,$ where $$ is a constant. Note: higher order terms, such as ${}^2{x}^2,$ etc.., have been

neglected.

$($a$)$ Determine the vertical velocity component, $v(x,y),$ if the flow

is steady and incompressible.

$($b$)$ Determine the change in length of a thin fluid element $($with an

original length of $L$ and a height of $2h)$ along the centerline of the

channel. The element initially $(t=0)$ located at $x=0$ while moving along

the $x-$axis for a distance of $5L.$ Let's determine the extended length of

the element after the travel. Hint: find the position of two points, $A$&$B,$

along the element where $A$ is the leftmost point and $B$ is the rightmost

point of the element. Find their difference, $l,$ after $5L$ travel. Use the

dilatation along $x-$axis definition $\frac{l}{lt}$ to verify the relation ${}_{}=\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}.$

$($c$)$ Repeat part $($b$)$ but focusing on the $y-$axis dilatation by assuming the element has an initial

height of $2h.$ Note: due to symmetry, only need to deal with the top half of the element.

Determine height change after $5$ L travel then verify the relation: ${}_{yy}=\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{y}}.$