= Solution
{c}
The coefficient of the growing term is fixed by the far-field <complex potential of a two-dimensional potential flow>[complex potential]: $A=\beta e^{i\alpha}$. On $z=ae^{i\theta}$, the <no-penetration boundary condition> makes the cylinder a <streamline>, so the imaginary part of $Aa^2e^{2i\theta}+Ba^{-2}e^{-2i\theta}$ must be constant. Choosing the two terms to be complex conjugates makes that constant zero, giving
$$
\boxed{A=\beta e^{i\alpha},\qquad B=a^4\overline A=\beta a^4e^{-i\alpha},\qquad
w=\beta\left(e^{i\alpha}z^2+e^{-i\alpha}\frac{a^4}{z^2}\right).}
$$
This is the zero-<circulation> solution, also obtained by the <Milne-Thomson circle theorem>. Impermeability and the leading far field alone permit an additional circulation term $-i\Gamma\log z/(2\pi)$; the requested algebraic form implicitly selects $\Gamma=0$. No <no-slip boundary condition> is imposed on an <inviscid flow>.
The <streamfunction in polar coordinates> is
$$
\psi=\beta\left(r^2-\frac{a^4}{r^2}\right)\sin(2\theta+\alpha).
$$
It gives $v_r(a,\theta)=0$ and
$$
v_\theta(a,\theta)=-\left.\partial_r\psi\right|_{a}
=-4a\beta\sin(2\theta+\alpha),\qquad
\boxed{|v_\theta|=4a\beta|\sin(2\theta+\alpha)|.}
$$
The liquid still exerts normal <pressure> on each surface element even though tangential viscous <traction> is absent. For this <steady flow> with zero <vorticity>, the <Bernoulli equation> has one constant $C$ throughout the connected fluid region:
$$
p(a,\theta)=C-\frac12\rho|\mathbf v|^2
=C-8\rho a^2\beta^2\sin^2(2\theta+\alpha).
$$
There is no finite uniform far-field <pressure> to substitute for $C$, because this straining <velocity field> grows with radius. Taking $\mathbf n=(\cos\theta,\sin\theta)$ outward from the cylinder, the fluid <force> per unit axial width is
$$
\mathbf F=-a\int_0^{2\pi}p(a,\theta)\mathbf n\,d\theta=\boxed{\mathbf0}.
$$
Indeed $p(\theta+\pi)=p(\theta)$ while $\mathbf n(\theta+\pi)=-\mathbf n(\theta)$: diametrically opposite elements cancel exactly. Zero resultant <force> does not mean zero local <pressure> or zero deformation tendency.
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