= Solution
Neglect the <inertia> of the <sphere> and balance the applied <force> against <Stokes drag>. The <sphere> translates with $\mathbf U=\mathbf F_1/\zeta$, where $\zeta=6\pi\mu a$. The applied <force> is also the net <force> transmitted to the fluid; the fluid's drag on the <sphere> has the opposite sign.
Translate the origin to $\mathbf r_1$. In the solution of part <2/a>, retain the terms of order $a/r$ and discard those of order $a^3/r^3$. With $\mathbf n=(\mathbf x-\mathbf r_1)/r$,
$$
\begin{aligned}
\mathbf u(\mathbf x)
&\sim\frac{3a}{4r}(I+\mathbf n\mathbf n^T)\mathbf U\\
&=\boxed{\frac1{8\pi\mu r}(I+\mathbf n\mathbf n^T)\mathbf F_1}.
\end{aligned}
$$
This is the <Stokeslet>, or equivalently the <Oseen tensor> $\mathsf G(\mathbf r)=(I+\mathbf n\mathbf n^T)/(8\pi\mu r)$ acting on $\mathbf F_1$. The omitted finite-size <velocity> is of order $a^2|\mathbf F_1|/(\mu r^3)$, so the displayed term is the leading far field for $r\gg a$. The singular <Stokeslet> must not be evaluated at its own centre to obtain a particle's self mobility.
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