# Taylor Couette Flow

Taylor-Couette flow is the flow of a viscous fluid sheared in the gap between two rotating coaxial cylinders. Sometimes called the “hydrogen atom of hydrodynamics,” Taylor-Couette flow has a rich phenomenology and has long been a testbed for hydrodynamic stability theory and an important tool in the study of nonlinear dynamics and pattern formation theory. Because of its simplicity, the Taylor-Couette system is an ideal platform for teaching students about these important physics topics, but also about the fabrication of scientific apparatus.

Figure 1. Basic Taylor-Couette cell.

The basic apparatus consists of two coaxial cylinders. The gap between them is filled with fluid and the inner cylinder is made to rotate by a stepper motor (see Fig. 1). For low rotation rates, viscosity causes the fluid to flow as might be predicted from symmetry. Fluid elements follow circular paths and the flow field has no axial or azimuthal dependence. However, as the rotation rate of the inner cylinder is increased the flow suddenly undergoes an instability and a cellular pattern of toroidal vortices called Taylor rolls emerges. As the rotation is increased further, the Taylor rolls themselves become unstable giving way to a progressively more complicated flow states as shown in Fig. 2, eventually leading to turbulence.

Figure 2. As the rotation rate of the inner cylinder is increased (left to right), the flow undergoes bifurcations from Taylor rolls to progressively more complex and eventually turbulent flows.

During the first day of the Immersion participants will learn how to build a small Taylor-Couette apparatus and how the design and construction of the apparatus can be implemented as a series of instructional modules on digital electronics, computer-aided design (CAD), conventional and modern fabrication methods (soldering, machining, laser cutting, 3D printing, etc. as appropriate to participant’s home institutions).

During the second part of the Immersion (2 days), participants will learn how the Taylor-Couette system can be used to demonstrate a host of important topics in fluid mechanics and nonlinear dynamics as stand-alone classroom demonstrations, laboratory exercises, or extended student projects. Some of the topics that will be explored include flow visualization, Reynolds similarity, flow instabilities, the time reversibility of low Reynolds number flows, the coexistence of attractors in nonlinear systems, the Ruelle-Takens route to chaos, and attractor reconstruction from time series. For the 2019 ALPhA/PICUP Immersion at the University of Wisconsin-River Falls, special attention will be paid to computational analysis of digital images, including spectral and time series analysis. We will also cover numerical modeling of the advection of passive tracers in circular Couette flow using GlowScript and numerical modeling of the primary instability using Daedalus, a Python-based package for computational fluid dynamics.

Participants should bring their lab notebook and a flash drive.

There are no special safety considerations besides the usual hazards involved in working with hand and machine tools.
The apparatus can be constructed as student projects for ~\$200 per unit. Most of the parts can be re-used, so the design and construction of the apparatus can be implemented as a project-based activity in subsequent semesters at significantly lower cost. While the Taylor-Couette system can be used to perform most of the experiments as a standalone unit, additional equipment ranging in cost from \$200 to \$2000 is necessary to conduct some of the experiments.