Question $1.[10$ Marks$]$ The ideal flow around any object can be considered as a superposition of a

uniform flow and a set of singularities such as vortex, doublet, source, and sink. From large distances,

all singularities appear to be located near the origin $(x,y)=(0,0)$

The superposition of the following elementary flows:

$(1)$ A uniform flow of a magnitude $U$ directed along the $x-$axis.

$(2)$ A line source $($sink$)$ with a volume flow rate per unit depth $($strength$)$ of $m.$

$(3)$ A line vortex of counterclockwise circulation $.$

$(4)$ A doublet $($a source$-$sink pair$)$ on the $x-$axis, of strength $.$

yields the velocity potential as

$(r,)=$Urcos$+\frac{m}{2}lnr+\frac{}{2}+\frac{}{r}cos$

$($a$)[3$ Marks$]$ Find the velocity components and velocity magnitude as $r.$ Are they consistent with

what you may expect to obtain?

b$)[5$ Marks$]$ Calculate the circulation, $-o{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$vec$(V)*dvec(s),$ around a circular path of radius $R,(>1)$ centered

at the origin.

c$)[2$ Marks$]$ As explained in class, the circulation can also be obtained using the velocity potential as

circulation $=-o{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$vec$(V)*dvec(s)=-o{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$vec$(grad)*dvec(s)=-o{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}d$

Evaluate the circulation using $(1\mathrm{.}2).$ Do the different approaches give an identical result?