= Solution
Orient the body contour counterclockwise. The outward normal from the body satisfies $(n_x,n_y)ds=(dy,-dx)$, so pressure produces complex force $F_x-iF_y=-i\oint p\,d\bar z$. <Bernoulli's theorem> gives $p=p_\infty-\rho|w'|^2/2$; the constant term integrates to zero. On an impermeable boundary the velocity is tangent, so $w' dz$ is real, equivalently $w'dz=\overline{w'}d\bar z$. Consequently $|w'|^2d\bar z=(w')^2dz$ and
$$
\boxed{F_x-iF_y=\frac{i\rho}2\oint_{\partial B}(w'(z))^2dz.}
$$
Choose the square-root branch analytic off the slit $[-a,a]$ with $\sqrt{z^2-a^2}\sim z$ at infinity. Continuation from $x>a$ through the upper half-plane gives the upper-face value $+i\sqrt{a^2-x^2}$; continuation through the lower half-plane gives the lower-face value $-i\sqrt{a^2-x^2}$. Both have the required square and the sign is fixed by the branch normalization.
Differentiate the potential:
$$
w'(z)=\cos\alpha-i\sin\alpha\,\frac z{\sqrt{z^2-a^2}}.
$$
At infinity, $z/\sqrt{z^2-a^2}=1+a^2/(2z^2)+O(z^{-4})$, so $w'(z)^2=e^{-2i\alpha}+O(z^{-2})$ has no $z^{-1}$ term. Deforming a contour around the plate to a large circle in its analytic exterior therefore gives zero integral. The force is $\boxed{F_x=F_y=0}$. The contour formulation includes the endpoint behavior and avoids assigning separate divergent endpoint pressure contributions.
For $0<\alpha<\pi/2$, use the two face values of the square root. The upper tangential velocity is $\cos\alpha-x\sin\alpha/\sqrt{a^2-x^2}$, vanishing at $\boxed{x=a\cos\alpha}$ on the upper face. The lower value is $\cos\alpha+x\sin\alpha/\sqrt{a^2-x^2}$, vanishing at $\boxed{x=-a\cos\alpha}$ on the lower face. Squaring the equation $w'=0$ gives only these candidates, and the branch signs locate the correct faces. At $\alpha=\pi/2$ both face <stagnation points> have coordinate zero. At $\alpha=0$ the exterior velocity is identically one, so there is no finite <stagnation point>; the endpoints are merely limits of the two locations as the angle tends to zero.
\Image[/past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2009-as2-flat-plate-flow.png]
{title=Potential-flow streamlines around the plate at angle pi over four}
The plotted streamlines are contours of the imaginary part of the potential, with the two boundary <stagnation points> marked. The far-<field> velocity is $(\cos\alpha,\sin\alpha)$, giving a flow from the lower left toward the upper right at the plotted angle.
Back to article page
