Unsteady three-dimensional numerical simulations to investigate mixing performance enhancement by chaotic advection driven by MHD in a disk-ring microfluidic device are conducted. The externally applied potential of 1V and the magnetic field intensity of 0.36T are smaller compared to those in other works on MHD-driven microfluidics, and the maximum velocity is also correspondingly smaller (~0.5mm/s vs. 10 mm/s).  We demonstrate that it is still possible for the described MHD device to achieve good mixing by means of chaotic advection judiciously created by the choice of the disk electrode placement and potential switching schemes. By using an on/off switching scheme at two symmetrically placed disk electrodes on a diameter of the circular microfluidic cavity floor, flow patterns similar to that of the “blinking vortex” model proposed by previous investigators have been created. For smaller switching periods, the flow patterns are regular and the mixing performance is poor. By increasing the switching period, more chaotic motion is generated, thus improving the stretching and deformation of material lines and enhancing mixing performance. By having two disk electrode pairs, and each pair switched according to a prescribed scheme, more complex chaotic advection has been generated that led to greater mixing. Analyses of the simulation results have been aided by a powerful post-processing tool that enabled graphical representation of the results such as Poincaré maps and animation of the flow. Post-processing provided Poincaré maps, concentration of numerical particles, and stretching and deformation of material lines. These different methods of post-processing helped confirm the accuracy of the simulations and post-processing techniques.  Below, we describe one of the post-processing techniques known as vector stretching.

In the early stages the fluid filaments must stretch and fold efficiently in order to decrease their width.  One characteristic of chaotic motion is the exponential growth of initial conditions, or an asymptotic divergence of initial conditions. Therefore, it is reasonable to expect that the vectors initially seeded into the flow may be stretched to a length several orders of magnitude greater than their initial length if the flow is chaotic. And, it would be of interest to know where the most stretching occurs since these would also be the locations of efficient mixing. The stretching of an infinitesimal arbitrary vector on that fluid element then can be computed by knowing the deformation tensor. The stretching of the fluid element at time t can be defined as the ratio of the length magnitude |dx| at time t to the length magnitude |dx₀|, at time t = 0.

l = | dx | / |dx₀ |                                                                   (1)

The method follows the Lagrangian view point, since the infinitesimal fluid element moves to a new position after time interval dt, and the time dependent deformation tensor F(t) is computed upon the covariant derivative of velocity vector locally. Once we obtain the deformation tensor at each time step, the new deformed vector dx can be calculated. The stretching ratio λ can be then obtained from Eq. (1). First we discretize the flow domain into 200×200 cells each with sides 30µm long, then homogeneously seed into the flow domain a large number of passive particles each tagged with a random initial vector, and then compute the stretching of each vector. We can count the number of particles in each cell and compute the mean stretching of the vectors to approximate the stretching value for that cell. 70,000 passive particles are initially seeded on a plane parallel to the cell floor. We can plot how much the vector attached to a particle initially located in a cell will stretch. We plot the stretching value only after several periods. The temporal evolution of the root-mean-square value λ_rms on a semi-log plot for a 1s-10s range of the period T for the same 10 number of periods showed that λ_rms increases exponentially with the switching period.

     Though excellent works on MHD-induced mixing by chaotic advection have been reported previously by other researchers, our present work brings additional insights facilitated by the power of CFD simulations, high resolution graphics and powerful post-processing of the simulation results. The present results are based on numerical simulation of the unsteady, three-dimensional Navier-Stokes equations without relying on simplifying assumptions. The differences between the present results and the simulations of other investigators may be due to our choice of the full Navier-Stokes equations for our simulations.  The present work lays out a promising approach to gain insight into the complex optimal problem of designing Lab-on-a-Chip (LOAC) devices to optimize mixing and flow control.