Lattice–Boltzmann study of the transition from quasi-two-dimensional to three-dimensional one particle hydrodynamics

Abstract

We report the results of a study designed to categorize the hydrodynamics of a quasi-two-dimensional system as either a 2D or 3D fluid. The characterization is based on the asymptotic decay of the velocity autocorrelation functions for different modes of motion and different boundary conditions at the enclosing walls. Our results show that for the case of no-slip boundary conditions the long time decay corresponds to neither 2D or 3D behaviour, nor anything in-between. The no-slip walls cause the long time tail of the velocity autocorrelation functions to have an exponential decay, more in agreement with Langevin model predictions. The free-slip boundary conditions create no-friction conditions for the tangential flow, and no-slip conditions for the perpendicular flow. The tangential component of the flow behaves like flow in 2D, but the perpendicular flow is hindered. As a result, the effect of the free-slip walls on the particle motion depends on the particular mode of motion. For perpendicular rotation where there is no flow perpendicular to the walls and for parallel translation where the perpendicular flow component is weak, we retrieve a 2D like decay for the autocorrelation functions. When the greatest part of the flow is in the perpendicular direction, the distinction between the no-slip and free-slip boundary is small, as in perpendicular translation. For the case of parallel rotation, the situation is more complicated as the perpendicular and tangential flows are more or less balanced, thus the no-slip boundary conditions slow down and speed up the autocorrelation decay. In the net result, the slowing down shows up in the increased diffusion coefficient, but the autocorrelation function shows a faster long time decay. As the plate separation decreases, the balance between the two contributions shifts and the diffusion coefficient begins to fall with decreasing separation.