## Quantification of chaotic mixing in microfluidic systems

##### Abstract

Periodic and chaotic dynamical systems follow deterministic equations such as Newton's laws of motion. To distinguish the difference between two systems, the initial conditions have an important role. Chaotic behaviors or dynamics are characterized by sensitivity to initial conditions. Mathematically, a chaotic system is defined as a system very sensitive to initial conditions. A small difference in initial conditions causes unpredictability in the final outcome. If error is measured from the initial state, the relative error grows exponentially. Prediction becomes impossible and finally, chaotic systems can come to become stochastic system.
To make chaotic motion, the number of variables in the system should be above three and there should be non-linear terms coupling several of the variables in the equation of motion. Phase space is defined as the space spanned by the coordinate and velocity vectors. In our case, mixing zone is phase space. With the above characteristics
- the initial condition sensitivity of a chaotic system, our plan is to find most efficient chaotic stirrer. In this thesis, we present four methods to measure mixing state based on the chaotic dynamics theory.
The Lyapunov exponent is a measure of the sensitivity to initial conditions and can be used to calculate chaotic strength. We can decide the chaotic state with one real number and measure efficiency of the chaotic mixer and find the optimum frequency.
The Poincare section method provides a means for viewing the phase space diagram so that the motion is observed periodically. To do this, the trajectory is sectioned at regular intervals. With the Poincare section method, we can find 'islands' considered as bad mixed zones so that the mixing state can be measured qualitatively.
With the chaotic dynamics theory, the initial length of the interface can grow exponentially in a chaotic system. We will show the above characteristics of the chaotic system to prove as fact that our model is an efficient chaotic mixer.
The final goal for making chaotic stirrer is how to implement efficient dispersed particles. The box counting method is focused on measurement of the particles dispersing state. We use snap shots of the mixing process and with these snap shots, we devise a plan to measure particles' dispersing rate using the box-counting method.