The mixing of passive scalars in three-dimensional fluid flows is a poorly understood problem. Due to it's wide scale applicability to variety of real flows, it's starting to attract alot of attention. The traditional approach to these problems came from turbulent flows where both effective stirring and diffusion took place. However, work on the "ABC" flow (Arnold,Beltrami,Childress) and the Blinking Vortex flow (Aref) showed that mixing can occur even in laminar flows (low Reynolds #) via chaotic advection. The central idea of chaotic advection is to design flows with nonintegrable trajectories in order to efficiently stir the fluid. This spawned a variety of two dimensional work however, flows in three dimensions are relatively uncovered.

Here, we design several models of three-dimensional flows that are physically motivated yet mathematically simple. The models consist of alternatively active convection rolls in different directions. The models we design are idealized, however there is experimental precedent for the existence of these flows. In this talk, I discuss the case of two orthogonal blinking rolls which are seen in rayleigh-benard convection experiments in a binary fluid. A perturbation on that system is also applied and we measure the resulting mixing via Lyapunov exponents and diffusion coefficients. Then a blinking roll model based on the Kuppers-Lortz instability is introduced and we study the resulting mixing. This model consists of alternatively active rolls at 120 degrees to one another. Initial results are shown for the mixing in this case. Plans for future research will also be discussed.

Committee members: Keith Julien, James Meiss, James Curry, Meredith Betterton, Jeff Weiss (PAOS)

One can construct maps of systems that display roll switching phenomena in rayleigh benard convection experiments. The figure below shows an example of rolls whose axes intersect at angle pi/2-epsilon. This figure shows some of the extraordinary dynamics for T_1=7, T_3=5, and epsilon=.08 for randomly chosen initial conditions in the fundamental rhombic cell. A normal form expansion about the elliptic fixed point at the intersection of roll axes, can be used to show that the dynamics in that region are generally simple. As we move away from those regions, the motion becomes increasingly chaotic in a three-dimensional sense. This is demonstrated by the blue, light blue, and red orbits.

Mixing Movie (13 MB, .avi)

These pictures a sequence as J decreases. Here you see a period 3 bifurcation. The blue marker is a fixed point. The green markers are stable period 3 points, and the purple points are unstable period 3 points. As J decreases, there comes a J value such that the blue, green, and purple points coincide. This occurs between pictures (b) and (c). The immediate areas surrounding the blue point are regular regions with predictable behavior. The areas outside of this are chaotic zones.