= Solution
Use the <spherical symmetry> of the expansion, with fluid at rest at infinity and no superposed imposed flow. The radial <Laplace equation> gives
$$
\partial_r(r^2\phi_r)=0,
\qquad \phi_r=\frac{A(t)}{r^2},
\qquad \phi=G(t)-\frac{A(t)}r.
$$
The moving sphere is a material boundary: its <kinematic boundary condition> is $u_r(R,t)=\dot R$. Consequently $A(t)=R^2\dot R$, and the <spherically symmetric incompressible radial flow> has
$$
\boxed{\phi(r,t)=G(t)-\frac{R^2\dot R}{r},\qquad
\mathbf u=\frac{R^2\dot R}{r^2}\,\widehat{\mathbf r}}.
$$
Equivalently, its <volumetric flow rate> through any concentric sphere is $4\pi r^2u_r=4\pi R^2\dot R$, the rate of change of the enclosed <volume>.
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