Statistical Reconstruction of Velocity Profiles for Nanoparticle Image Velocimetry

Velocities and Brownian effects at nanoscales near channel walls can be measured experimentally in an image plane parallel to the wall by evanescent wave illumination techniques [R. Sadr, M. Yoda, Z. Zheng, and A. T. Conlisk, J. Fluid Mech., 506 (2004), pp. 357–367], but the depth of field in this technique is difficult to modify. Assuming mobility of spherical particles dominated by hydrodynamic interaction between particle and wall, the out-of-plane dependence of the mobility and in-plane velocity are clearly coupled. We investigate such systems computationally, using a Milstein algorithm that is both weakand strong-order 1. In particle image velocimetry (PIV), image pairs are cross-correlated to approximate the mean displacement of n matched particles between two windows. For comparison, we demonstrate that a maximum likelihood algorithm can reconstruct the out-of-plane velocity profile, as specified velocities at multiple points, given known mobility dependence and perfect mean measurements. We then test this reconstruction for noisy measurements as might be encountered in experimental data. Physical parameters are chosen to be as close as possible to the experimental parameters while we consider three types of velocity profiles (linear, parabolic, and exponentially decaying).