With a classic Schwarz-Christoffel mapping, we can map a square area in the complex plane to the unit disk (image below, left). I am wondering if there exists a mapping in the unit disk which can effecticely collapse the 3 o'clock and 9 o'clock quadrants of the unit circle (orange dashed line, left) to a single point to yield a field similar to what I have approximate sketched on the right.
I believe some complications could arise from the fact that the mapping is no longer 1-to-1 on the unit circle (as we collapse some stretches of the boundary to a point). If this is a precluding issue, I would alternatively be interested in a solution which reduces these edgepoints of the orange quadrants to a very small distance. Furthermore, if there were a method to obtain the right side field directly, this would also be sufficient.
Does such a mapping or field exist? If so, how would I go about constructing it?
I assume that by a "mapping" you mean "a nonconstant holomorphic mapping." Then such a mapping cannot exist: If $f$ is a holomorphic function on the open disk $D$ such that $f$ has constant radial limit on a nontrivial arc on $\partial D$ then $f$ is constant. This is a special case of Riesz uniqueness theorem. More generally (but this requires more work), if $D$ is any domain in ${\mathbb C}$ bounded by a Jordal curve, $A\subset \partial D$ is a nontrivial arc and $f: D\to {\mathbb C}$ is holomorphic with constant limit on $A$, then $f$ is constant.
Of course, if by a mapping you mean merely a smooth (even injective) mapping of an open round disk, then collapsing a boundary arc of a round disk is clearly possible.