$(35$ minutes$)$ The velocity components for a two$-$dimensional incompressible jet flow impinging on a wall $($see the sketch on the left of the figure below$)$ are given as follows:

$\backslash [$

u$=$k x ; $\backslash$quad v$=-$k y

$\backslash ]$

where $\backslash ($ k $\backslash )$ is a positive constant.

a$.$ Does this flow have a velocity potential? Justify your answer and find an expression for the velocity potential if it exists.

b$.$ Determine the stream function.

c$.$ Plot the streamlines for $\backslash ($ k$=1\backslash ).$

d$.$ Show that the pressure gradient on the surface of the wall is in the form of

$\backslash [$

$\backslash$frac$\{\backslash$partial p$\}\{\backslash$partial x$\}=$c x

$\backslash ]$

where $\backslash ($ c $\backslash )$ is a constant.

When a source with a strength of $\backslash (\backslash$Lambda $\backslash )$ is added at the stagnation point of the flow field described above, the resultant flow field corresponds to a jet flow around a bump with a height of $\backslash ($ h $\backslash )($see the sketch on the right of the figure below$).$ By using a polar coordinate system $\backslash (($r$,\backslash$theta$)\backslash )$ :

e$.$ Find the velocity components of the resultant vector field.

f$.$ Derive an expression for the bump height in terms of the constant $\backslash ($ k $\backslash )$ and the strength of the source $(\backslash (\backslash$Lambda $\backslash )).$