= Solution
The proposed <mixing length> can be motivated as a local phenomenological estimate: on either half of a self-similar jet, the centreline supplies a distinguished reference plane, and an eddy at distance $|y|$ might retain momentum over a fixed fraction $C_1|y|$ of that transverse distance. This gives
$$
\boxed{l=C_1|y|.}
$$
It is compatible with the dilation $x\mapsto ax$, $y\mapsto ay$ and introduces no independent outer length. In the <turbulent plane jet similarity> regime its typical value at $|y|\sim\delta$ is $C_1\delta$, so the scaling from part (b) has $\delta/L\sim C_1^2$.
This argument is a modeling assumption, not a physical necessity: a jet centreline is not a wall that prevents turbulent eddies from crossing it. Thus a <mixing length> vanishing there cannot automatically be justified throughout the core. Parts (e) and (f) exhibit the precise difficulty: the resulting finite-velocity similarity profile has singular centreline shear and zero shear at a finite free edge, contradicting the printed edge definition. \b[The local dimensional motivation does not make all the stipulated global jet properties consistent.]
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