Slip length estimation and flow reconstruction from micro-particle image velocimetry via functional uniqueness guarantees

Slip length, a fundamental measure of solid–fluid interaction, plays a crucial role in the inversion and reconstruction of flow-field parameters. However, near-wall errors in micro-particle image velocimetry make gradient-based estimation brittle and hinder robust data assimilation under unknown boundary conditions. To address these challenges, we establish—via functional–analytic arguments grounded in Sobolev embeddings and the Lax–Milgram theorem—a monotone, invertible mapping between the channel's maximum velocity and the slip length for a given geometry and Reynolds-number range. The mapping is numerically calibrated and turns the ill-posed near-wall-gradient inversion into a well-posed inference that relies on core-flow measurements and physical laws. We then reconstruct complete two-/three-dimensional (2D/3D) velocity fields using a physics-informed neural network (PINN) that does not enforce explicit boundary conditions. Experiments in rectangular microchannels show that the slip lengths inferred from the mapping agree closely with independent pressure-drop measurements. In 2D reconstructions, the PINN achieves a relative L2 velocity error of 3.56% and yields slip length estimates more accurate than common regression baselines (random forest, support vector regression, gradient boosting trees). In 3D, the error is 8.95%, surpassing the recent artificial intelligence velocimetry method. This study introduces a new pathway for efficient slip-length quantification in engineering settings and advances flow-field assimilation of realistic microscale flows under slip boundary conditions.