Navier–Stokes Equations in Cylindrical Coordinates: A First-Principles Derivation and Non-Dimensional Analysis

This work presents a complete first-principles derivation of the incompressible Navier–Stokes equations in cylindrical coordinates. The formulation begins from kinematic relations between Cartesian and cylindrical coordinate systems, proceeds through differential mass and momentum conservation for a cylindrical control volume, incorporates curvature effects explicitly, and derives the Newtonian constitutive relations adapted to cylindrical geometry. The document consolidates the full derivation chain into a single, self-contained pedagogical reference with explicit intermediate steps. While no new physical laws are introduced, the contribution lies in presenting a detailed and structured derivation that is typically dispersed across multiple classical fluid mechanics texts. This manuscript is intended as an educational technical note for students and researchers studying incompressible flow, pipe flow, and curvilinear coordinate formulations of the Navier–Stokes equations.