2D corner flow

Potential function:

$$\phi = r^\alpha \cos (\alpha \theta), \notag$$

Stream function:

$$\psi = r^\alpha \sin (\alpha\theta) \notag$$

Velocity vector components (polar coordinates):

$$\begin{array}{l} u_r = \frac{\partial \phi }{\partial r} = \a... ... n\pi , n = 0,\pm 1,\pm 2,\ldots \hspace{1.5in}\\ \end{array}$$

1.

interior corner flow - stagnation point origin: $\alpha >$ 1.

e.g. $\alpha$ = 1, $\theta _0 = 0,\pi ,2\pi,\ \ \ u = 1,\ v = 0$

2.

Exterior corner flow, $\vert v \vert \to \infty$ at origin: $\alpha < 1 (\raise.5ex\hbox{$\scriptstyle 1$ }\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$ } \le \alpha < 1$). $\theta _0 = 0,\frac{\pi }{\alpha }$ only. Since we need $\theta _0 \le 2\pi$, we therefore require $\frac{\pi }{\alpha } \le 2\pi ,$ i.e. $\alpha \ge 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2$ only.
e.g. $\alpha = 1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2, \theta _0 = 0, 2\pi$ ( $\raise.5ex\hbox{$\scriptstyle 1$ }\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$ }$ infinite plate, flow around a tip)

$\alpha = 2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3, \theta _0 = 0,\frac{3\pi }{2}$ ($90^{o}$ exterior corner)

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