Weld-Zone Dye Field: Advected FSW Material Flow Map

This is a 2D field-native counterpart to the 3D friction-stir-welding particle simulator. Rather than tracking individual spheres through a perspective camera, the same steady tool-frame velocity field — solid-body rotation inside the pin, a decaying vortex in the stirred zone, a small outward extrusion term and the background traverse flow — is used to advect a continuous dye-concentration field on a fixed 2D grid, using a semi-Lagrangian back-trace scheme. Striped "lanes" of dye stream in from upstream and get wound around the pin in real time, so the whole stir zone's onion-ring banding and the advancing/retreating asymmetry are visible at once as a continuous map, tracked live against the same dimensionless advance ratio λ = ωR/v that governs how thoroughly the weld is stirred.