Effects of diffusion and inertia on chaotic mixing: A computational and experimental study

The role of molecular diffusion and transient velocities in the overall mixing of particles by chaotic advection produced in a periodic journal-bearing flow is studied numerically and experimentally. For the numerical study, the Lagrangian description of diffusing particles is modelled by the use of Langevin equation, in which the effective velocity of a diffusing particle is expressed as the sum of the deterministic velocity (as a result of the flow solution) and a stochastic component arising out of the Brownian motion of the molecules.; The stochastic component is modelled as a Gaussian process with zero mean and variance proportional to the diffusivity, D. Two separate cases are considered in obtaining the deterministic velocity. In the first case, the flow solution is obtained analytically by making Stokes approximation. Numerical experiments are performed by stirring a blob of particles for a few cycles and then reversing the flow. It is observed that the mean square separation (after flow reversal) of the diffusing particles initialized in chaotic regions are a few times greater than those initialized in regular regions. The main causes of enhancement of separation are identified to be stretching and folding of material lines. Corresponding experimental observations are made by taking photographs of blobs of dye before and after stirring, and by analyzing the pictures by digital image processing. For a few cycles, there is a good agreement between the numerical results and the experimental observations. But, for longer stirring, the final distribution of particles after flow reversal predicted numerically deviates from the corresponding experimental results. The cause of this deviation is believed to be the omission of the transient terms from the governing momentum equation. Hence, in the second case, the effect of diffusion is studied in the presence of transient velocities. The flow solution is obtained for the transient Navier-Stokes equation in the eccentric annulus geometry by a finite difference method using the streamfunction-vorticity formulation and ADI method in a curvilinear coordinate system. On including the effects of transient velocities in the numerical model, there is a striking agreement with experimental observations even for higher numbers of cycles of stirring. The transient effects, however, are not significant if the advection is not chaotic.