Chaotic Advection

Over the years the importance of mixing in industrial applications has driven researchers to study and implement several new methods and enhancements to conventional mixing methods. It has been seen that mixing is quantitatively better in the turbulent flow regime. But the control prolems assosiated with the turbulent flow has rendered it improbable for practical applications. As the microfluid analysis has gained momentum, the new challenge was to improve the mixing in laminar regime. The idea of lagrangian chaos and chaotic advection has made this a reality.Intense research on this matter was carried out by Ottinow and Wiggins. Despite the very low Reynold's number and the laminar regime, the fluid particle ends up being chaotic. Following the premise set we can now dwell into the details of the Chaotic Advection/Mixing and then see how an engineer can exploit this to the benefit of the society.

The first and foremost step for understanding this phenomenon is tho know what chaos is ? So answering that that question as an engineer, it is nothing but the sensitivity to initial conditions. The best example in real life is the day today weather. Say a meteorologist is following the temperature of a particular place. He can follow the temperature for 10 years but still cannot come up with a prediction for the next day temperature. Reason the system we are considering is totally chaotic. There are several books available on this that you can go through. Coming to a specific example, The Gas Chamber.

Construct a simple system: take a box, a simple solid rectangular solid. Within this box, place a gaseous substance. Heat the box, sit back, and watch. What happens to the gas? Everyone knows that warm gases rise while cooler gases sink; and initially, the portions of the gas closest to the walls of the box will become heated and rise. At certain temperatures, the gas will begin to form cylindrical rolls spaced like jelly rolls lying lengthwise in the box. On one side of the box, the gas rises, and on the other, it sinks; the rising gases move to one side and carry warmer gases up with them; as the gas cools, it falls on the other side of the box. With a regularly applied temperature, a smooth box interior, and a system completely closed with regards to the gas itself, it might be expected that the circular motion of the moving gas should be regular and predictable. Nature, however, is neither regular nor predictable. It turns out that the motion of the gas is chaotic. The rolls do not simply roll around and around in one direction like a steamroller; they roll for a while

in one direction, and then stop and reverse directions. Then, seemingly at random, the gas reverses direction again; these changes continue at unpredictable times, at unpredictable speeds.

How can Fluid Mechanics enthusiast exploit this? We must be familiar with the term Stream function. Its a general thumb of law that the stream function is time dependant, then system invariably turns out to be chaotic. I will explain this with a specific example.

We all know about the RayleighBenard Convection. There are 2 parallel plates with the hot one at the bottom and the cold one at the top. The convection currents formed in between the plates is known as the Rayleigh Bernard convection. To see how this is realised, we can See the contour plots of the 2 stream functions with and without chaos. Illustration 1 give the contour of a stream function that is time independent convective currents given by,
$ \phi (x,y) = K \sin x \sin y $

To differentiate it from a system with inherent chaos let us see the contours formed by a time

dependent model of the stream function for RB convection current.
\[ \phi (x,y,t) = K_{1} \s in \left (x + K_{2} \cos t \right ) \sin y \]

have a look at illustrations 2 and 3.

Illustration 1: Stream function of system without chaos

Analyse the contour in illustration 2 and 3. The seperatrix (neutral point) oscillates about a position x=3.14 in the system analysed. The position of the seperatrix is dependent on time. So what is the use of this system being chaotic?

Its the ease of mixing. The basic reason why we turn to chaotic advection is that it brings about good mixing in laminar. Microfluidics is a fast growing research field for applied sciences. With nanotechnology getting a huge boost for R&D purpose, such efficient mixing at very small channels. (as diameter reduces drastically, so does the reynolds number). What I have given here is just a introduction to a vast topic of chaotic advection. One might wonder how having a time dependent stream function will affect the mixing this much? Its for you to explore. People who are interested to do further research in this Field can contact me for papers. One of the pioneers in this field is one Dr. John Otttino. He is from the Northwestern university. More info here .

Illustration 2: Contours of a time dependent stream function at t=0 .








Illustration 3: At a later time t=40