= Solution
Let $\zeta=\zeta(z,t)$ be the inverse <conformal map>. With $Q$ real and nonzero, the <function> $Q\log|\zeta|$ is <harmonic> away from the source, vanishes on the moving <boundary>, and has the required logarithmic <singularity>. Indeed $F(\zeta,t)=a_1(t)\zeta+O(\zeta^2)$ with $a_1\ne0$, so $\log|\zeta|=\log|z|-\log|a_1|+o(1)$ near the origin. Subtracting this candidate from the physical <velocity potential> leaves a bounded <harmonic function> near the source. Its <singularity> is removable, and the difference has zero <boundary values>. The <maximum principle for harmonic functions> makes it identically zero. Therefore
$$
\boxed{\phi(z,t)=Q\log|\zeta(z,t)|=\operatorname{Re}\bigl(Q\operatorname{Log}\zeta(z,t)\bigr).}
$$
Only the <real part> is needed, so the multivalued <imaginary part> of the logarithm causes no ambiguity in $\phi$.
Differentiate $F(\zeta(z,t),t)=z$ while holding the physical point $z$ fixed:
$$
\zeta_t=-\frac{F_t}{F_\zeta},\qquad
\phi_t=-Q\operatorname{Re}\frac{F_t}{\zeta F_\zeta}.
$$
Also the local analytic <complex potential of a two-dimensional potential flow> has <derivative>
$$
w_z=\frac{Q}{\zeta F_\zeta},\qquad |\nabla\phi|^2=|w_z|^2=\frac{Q^2}{|\zeta F_\zeta|^2}.
$$
The distinction between differentiating at fixed $z$ and fixed $\zeta$ is essential because the <boundary> map depends on time. Substituting into the <boundary> evolution law and multiplying by $|\zeta F_\zeta|^2/Q$ gives
$$
\operatorname{Re}\bigl(F_t\overline{\zeta F_\zeta}\bigr)=Q.
$$
Taking the conjugate inside a <real part> leaves it unchanged, so the desired <conformal map equation for planar Darcy growth> is
$$
\boxed{\operatorname{Re}\left(\zeta F_\zeta\overline{F_t}\right)=Q,\qquad |\zeta|=1.}
$$
As a normalization check, parameterize the <boundary> by $\zeta=e^{i\theta}$. Its outward <normal vector> times arclength is $\zeta F_\zeta\,d\theta$, so its <area> growth rate is $\int_0^{2\pi}\operatorname{Re}(F_t\overline{\zeta F_\zeta})d\theta=2\pi Q$, exactly the specified source <flux>. For an expanding circular map $F=R(t)\zeta$, the equation reduces to $R\dot R=Q$.
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