= Solution
{c}
Differentiating the axisymmetric <stream function> gives
$$
v_r=\frac{2C}{r}\cos\theta,\qquad
v_\theta=-\frac C r\sin\theta.
$$
The normal <kinematic boundary condition> fixes $C=Ua/2$, so
$$
\boxed{\Psi=\frac{Ua}{2}r\sin^2\theta,\qquad
v_r=\frac{Ua}{r}\cos\theta,\qquad
v_\theta=-\frac{Ua}{2r}\sin\theta.}
$$
Both <velocity> components vanish at infinity, and the axisymmetric <stream function> is regular at the polar axes. The tangential component of the <strain-rate tensor> vanishes everywhere:
$$
e_{r\theta}=\frac12\left(\frac{2C}{r^2}\sin\theta
-\frac{2C}{r^2}\sin\theta\right)=0.
$$
Thus the <stress-free boundary condition> is satisfied. The <vorticity> is $\omega_\phi=2C\sin\theta/r^2=Ua\sin\theta/r^2$, and the supplied dipolar <pressure> is consistent with the <Stokes equation>.
Use the <Newtonian fluid stress tensor> $\boldsymbol\sigma=-p\mathbf I+2\eta\mathbf e$. At the bubble surface,
$$
e_{rr}=-\frac{U}{a}\cos\theta,\qquad
\sigma_{rr}=-p_0-\frac{3\eta U}{a}\cos\theta,\qquad
\sigma_{r\theta}=0.
$$
The vertical force exerted by the liquid is the integral of the vertical <traction>:
$$
F_z=\int_{r=a}(\sigma_{rr}\cos\theta-\sigma_{r\theta}\sin\theta)\,dS
=-\frac{3\eta U}{a}\,2\pi a^2\int_0^\pi\cos^2\theta\sin\theta\,d\theta.
$$
Since the integral is $2/3$, the <drag> is downward and has magnitude
$$
\boxed{F=4\pi\eta aU.}
$$
The constant <pressure> integrates to zero force. The <pressure> perturbation contributes $4\pi\eta aU/3$ and the normal viscous <stress> contributes $8\pi\eta aU/3$; there is no tangential <traction> contribution.
For the free-interface qualification in part (b), the calculated normal <traction> at $\theta=0$ and $\theta=\pi$ differs by $6\eta U/a$. It cannot equal a single uniform gas <pressure> if every restoring effect is discarded. Restoring the external-minus-internal variation of <hydrostatic pressure> supplies a term $(\rho_w-\rho_a)ga\cos\theta$. It cancels the dynamic normal <traction> at the <terminal velocity> $U=(\rho_w-\rho_a)ga^2/(3\eta)$, consistent with a <spherical clean bubble without surface tension>. This distinguishes the drag calculation from the extra conditions needed for a freely rising bubble.
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