  For integers $a\ge 2b>0$, a \emph{circular $a/b$-flow} is a flow that takes
values from $\{\pm b, \pm(b+1), \dots, \pm(a-b)\}$. The Planar Circular Flow
Conjecture states that every $2k$-edge-connected planar graph admits a circular
$(2+\frac{2}{k})$-flow. The cases $k=1$ and $k=2$ are equivalent to the Four
Color Theorem and Gr\"otzsch's 3-Color Theorem. For $k\ge 3$, the conjecture
remains open. Here we make progress when $k=4$ and $k=6$. We prove that (i)
{\em every 10-edge-connected planar graph admits a circular 5/2-flow} and (ii)
{\em every 16-edge-connected planar graph admits a circular 7/3-flow.} The dual
version of statement (i) on circular coloring was previously proved by
Dvo\v{r}\'ak and Postle (Combinatorica 2017), but our proof has the advantages
of being much shorter and avoiding the use of computers for case-checking.
Further, it has new implications for antisymmetric flows. Statement (ii) is
especially interesting because the counterexamples to Jaeger's original
Circular Flow Conjecture are 12-edge-connected nonplanar graphs that admit no
circular 7/3-flow. Thus, the planarity hypothesis of (ii) is essential.
