Stream + Dipole: circles and spheres

2D:

$$\begin{align}\phi = & Ux + \frac{\mu x}{2\pi r^2}, \mbox{\ where \ } x = r\cos \... ...al r} = \cos \theta \left( {U-\frac{\mu }{2\pi r^2}} \right) \notag \end{align}$$

So $V_r = 0$ on $r = a = \left. {\sqrt {\frac{\mu }{2\pi U}} } \right\}$ (which is the K.B.C. for a stationary circle radius a) or choose $\mu = 2\pi Ua^{2}$.

Steady flow past a circle (U,a):

$$\begin{array}{l} \phi = U\cos \theta \left( {r + \frac{a^2}{... ...ial velocity } \\ \end{array}} \right. \\ \end{array} \notag$$

Illustration of the points where the flow reaches maximum speed around the circle.

3D:

$$\begin{array}{l} \phi = Ux + \frac{\mu }{4\pi }\frac{\cos \t... ...{\mu }{2\pi U}} \mbox{\ or \ } \mu = 2\pi Ua^3 \\ \end{array}$$

Steady flow past a sphere (U, a):

$$\begin{array}{l} \phi = \mbox{U}\cos \theta \left( {r + \fra... ...eta = \frac{\pi }{2} \\ \end{array}} \right. \\ \end{array}$$

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