$|$Ideal Flow Around a Sphere

For the steady potential flow around a sphere with approach velocity $u_{}($see Fig. $2\mathrm{.}6-1),$

the stream function and velocity potential are

$=+\frac{v_{}{R}^3}{2r}si{n}^2-\frac{v_{}{r}^2}{2}si{n}^2$

$=-\frac{v_{}{R}^3}{2r^2}cos-v_{}rcos$

The stream function is related to the velocity components, as indicated in Table $4\mathrm{.}2-1,$ and

If the velocity potential is related to the velocity components by $v=-$grad$,$ or specifically in

spherical coordinates

$v_{}=-\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}$

$v_0=-\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}}$

a$.$ Show that. Eqs. $4.$H$-1,2$ give $v_z=v_{}$ far from the sphere. $($Fint: $v_z=v_{r}cos-$

$v_{}sin.)$

b$.$ Show that the velocity at any point on the surface of the sphere is $v_{}=-\frac{3}{2}{v}_{}sin.$

c$.$ Show that the pressure distribution on the surface of the sphere is $p-p_{}=$

T $(\frac{v_{}^2}{2})(1-\frac{0}{4}si{n}^2).$

Con you drow the atreomlimes?