"The hydrogen atom of fluid dynamics"--introduction to the Taylor-Couette flow for soft matter scientists.

The flow between concentric cylinders is routinely used in soft matter studies. In many cases, the purpose of the setup is rheometric: the idea is to relate macroscopic changes in material properties to microscopic changes in the structure of the material. The correspondence between the modifications of the microscopic structure and the macroscopic flow often relies on viscometric assumptions, which require the flow to be at least laminar. Flow instabilities are usually neglected because the viscosities of the materials are high and the geometries are small, such that the creeping flow approximation can be used. Nonetheless, the phenomenology of viscoelastic instabilities that emerged in the last twenty years warns us that flows can become turbulent without inertia, in particular flows between concentric cylinders. Given the strong similarities between inertial hydrodynamic instabilities and viscoelastic instabilities, a general knowledge of the former is advised for any researcher working on complex fluids. In this tutorial review, we focus on the inertial instability of isothermal and incompressible Newtonian fluids flowing between concentric cylinders. We highlight important aspects that can guide the study and control of instabilities in complex fluids in general.