Effect of diffusion on chaotic advection in Stokes flow

Abstract

The effect of molecular diffusion has been studied both numerically and experimentally for the case of Stokes flow between two eccentric cylinders. The parameters (eccentricity, time periods, and directions of rotation of the cylinders) are so chosen that for the nondiffusive case, we obtain a mixed phase space of regular and chaotic regions. This enables us to study the relative effects of molecular diffusion in the two kinds of regions of the same flow. The Lagrangian description of a diffusing particle has been modeled by the use of Langevin equation (as done by Aref and Jones in Ref. 1) in which the instantaneous velocity of a particle has a deterministic component (as a result of the flow solution) and a stochastic component (arising out of the Brownian motion of the molecules). The stochastic component is assumed to be a Gaussian process with zero mean and variance proportional to the diffusivity. The numerical investigation is carried out by locating circular blobs of particles in various regions of the given phase space. Each blob is represented by about 1000 evenly distributed points inside a small circle. With a given value of diffusivity, D, our two-cylinder system is stirred by completing three cycles of motion followed by three reverse cycles. For a deterministic case, the particles are expected to return to their initial locations after flow reversal, as has been observed for the case with D=0. For D>0, the mean-square separation of the particles from their initial locations has been evaluated with the idea of finding an effective diffusivity for each blob location. The mean-square separations for the stirred cases are then compared with that of the unstirred case for the same value of D. It has been found that the mean-square separation in a regular region is slightly greater than that of the unstiffed case, whereas in the chaotic region it is greater than that of the corresponding unstirred case by an order of magnitude. The reason for this enhancement of diffusion by stirring, as explained by Aref and Jones,1 is that stirring causes an increase of intermaterial area for diffusion to act over a longer interface. Since stretching is identified to be the cause of enhancement of diffusion, Lyapunov exponents have been evaluated for various regions of the flow. In the regular regions, the Lyapunov exponents have been found to be nearly zero, and it is found that the greater the separation, the greater the exponent in that region. A correlation between the two quantities has been established. However, the Lyapunov exponents in the chaotic regions have been found to be highly positive, indicating a high rate of stretching. It is also found that there is a variation of separation within the chaotic region, indicating a difference in the degree of chaos between various regions within the chaotic sea. Corresponding experimental observations, made by using the same apparatus as in Chaiken et al.,2 show good agreement with the numerical results.