The development of a free laminar jet from a fully developed Poiseuille profile to a self-similar Gaussian flow is investigated theoretically. While the far-field self-similar state is well understood, the near-field transition region presents a significant analytical challenge due to the interplay of axial convection and radial viscous diffusion. This study introduces a first-principles analytical model based on a linearly decaying reference velocity. The model determines the jet's velocity decay rate, $k$, through a self-consistent framework where its evolution is required to match an optimal trajectory defined by a Galerkin-constrained guidance model. Built on a two-mode Laguerre-Gaussian expansion, the model reveals a universal linear scaling for the dimensionless centerline velocity decay and demonstrates that the transition length, $z_t$, scales linearly with the Reynolds number. Predictions for both centerline velocity and full velocity profiles show excellent agreement with numerical simulations over a diverse range of Reynolds numbers $(\mathrm{Re}=20–1500)$. This work establishes a robust and physically insightful framework for describing the evolution of laminar jets with finite Poiseuille inlets.

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• Convection
• Diffusion
• Laminar flows
• Wakes & jets
• Viscosity
• Navier-Stokes equation
• Self-similarity