Q$3$: A potential flow against a flat wall $($Fig$.2$a$)$ can be described by the stream function $=$Axy, where $A$ is a constant for the flow strength. This flow is called the stagnation flow. By adding a source of strength $m$ at Point $O,$ the stagnation flow against the wall with a "bump" can be generated as illustrated in Fig. $2$b$.$

rig. $2$

$($a$)$ Find an expression for the velocity potential $$ of the stagnation flow in Fig. $2$a$.(3$ marks$)$

$($b$)$ Find the velocity $u_x$ and $u_y$ along the $y-$axis for the flow in Fig. $2$b$.(3$ marks$)$

$($c$)$ Find the resulted bump height $h$ as a function of A and $m$ in Fig. $2$b$.(3$ marks$)$

$($d$)$ If the fluid density is $,$ find the pressure increase from point