PROBLEM $8\mathrm{.}2$

There are numerous occasions in which a fairly uniform free$-$stream flow encounters a long circular cylinder

aligned normal to the flow. Examples include air flowing around a car antenna, the wind blowing against a flag

pole or telephone pole, wind hitting electrical wires, and ocean currents impinging on the submerged round beams

that support oil platforms. In all these cases, the flow at the rear of the cylinder is separated, unsteady, and usually

turbulent. However, the flow in the front half of the cylinder is much more steady and predictable. In fact, except

for a very thin boundary layer near the cylinder surface, the flow field may be approximated by the following steady,

two$-$dimensional velocity components in the $x-y$ or $r-$ plane:

$u_r=$Vcos$(1-\frac{a^2}{r^2}),u_{}=-$Vsin$(1+\frac{a^2}{r^2})$

Determine the stagnation point if it exists.

Determine if the flow is compressible or incompressible. $($must use volumetric rate change$)$

Determine if the flow is rotational or irrotational. If it is rotational, is it convective with the fluid?

Show that the acceleration is $($you are allowed to use a symbolic software to simply, show proof that you used

it$)$:

vec$(a)=2\frac{a^2}{r^3}{V}^2(1-\frac{a^2}{r^2}-2si{n}^2)hat(e{)}_r+2V^2\frac{a^2}{r^3}sin2hat(e{)}_{}$

Determine if the fluid is accelerating on the surface of the cylinder. If so$,$ in what direction? Consider the

following angles

$=0,\frac{}{2},,3\frac{}{2}$

Note that on the surface $r=a.$

Determine if the fluid is accelerating on the surface of the cylinder at $r=2a,=45.$ If so$,$ in what direction?

Work in cylindrical coordinates.