In 1923 it was published Stability of a viscous liquid contained between two rotating cylindersa revolutionary text signed by the mathematician and physicist Geoffrey Ingram Taylor (1886, London – 1975, Cambridge). The research captured in it opened new ways to understand the patterns that appear in a flow, such as those observed in the movement of the ocean. A century later, the work continues to provide a solid foundation for a wide range of scientific studies and practical applications. For example, it is key to understanding how turbulence develops from a stable flow, that is, how it goes from an ordered flow to a chaotic and continuously changing flow.
The article represents the beginning of the so-called theory of hydrodynamic stability. This branch of physics and mathematics tries to understand instabilities in a fluid, small perturbations that cause the flow to deviate from its initial state and evolve towards a different configuration. They are formed, for example, by the friction of a moving fluid and a solid surface and can amplify over time and cause notable changes in the flow, such as the appearance of vortices or eddies.
For centuries, physicists and mathematicians tried to find a criterion to detect the moment at which instabilities appear, based on the properties of the fluid and the equations and parameters that describe its movement. The Lord Rayleigh theory was a first step to resolve this issue: it offered a theoretical model to predict the stability of a fluid without goo —that is, it does not present resistance to flow—. In particular, he showed that the flow is stable as long as a certain condition is satisfied—the square of the angular momentum per unit mass of the fluid increases outward. This means that if only the inner cylinder rotates, the flow is unstable, while if only the outer cylinder rotates, it is stable.
In 1890, the physicist Maurice Couette published his doctoral thesis about the friction between a moving fluid and a solid surface. In order to measure viscosity, he designed an experimental device made up of two concentric cylinders—one inner, fixed, and one outer, rotating—with liquid between. By rotating the outer cylinder, a flow was generated in the fluid and the friction produced, that is, its viscosity, could be quantified. His contribution was so important that, soon, Couette's name was associated with the flows he studied. Almost at the same time, and independently, Arnulph Mallockmaster in constructing experiments, described centrifugal instability which occurs when the inner cylinder rotates and the outer cylinder remains stationary.
The next step was to join these experimental observations with Rayleigh's mathematical formalization. This was precisely what Geoffrey Ingram Taylor did, who realized that, in the case of non-viscous fluids, experiments agreed with theory. Taylor, grandson of the famous mathematician George Boole, wrote: “It seems doubtful that we can fully understand the instability of fluid flow without obtaining a mathematical representation of the motion of a fluid in some particular case in which actual stability can be observed.” Therefore, to begin to understand – with the mathematical tools of Rayleigh – the instabilities of Couette flow, it was essential to find a good example to analyze.
Taylor considered that not just one, but two concentric cylinders rotate. He built the device and with it, and using mathematical tools, predicted the stability of the fluid. To do this, he linearized the Navier-Stokes equations—which describe the behavior of a fluid—. That is, he assumed that the perturbations were small enough to neglect part of the terms in the equation, and he found solutions to the equations that correspond to the instabilities observed in the experiments.
The equations also allow us to determine whether the instability increases – that is, if the unstable flow can give rise to more complex shapes – or decreases over time – that is, if the flow will remain stable, without changes. Thus, Taylor was able to theoretically describe the behavior of the fluid, based on its properties and the rotation speeds of the two cylinders.
In 1923 he published in Philosophical Transactions of the Royal Society TO his work. The close agreement between their theoretical and experimental results was unprecedented in the history of fluid mechanics, so the paper is described as the first convincing evidence of the applicability of mathematical approaches to predicting fluid stability.
His work connected mathematics, physics and engineering and, in the following years, was used in numerous studies related to stability, astrophysical and geophysical flows, nonlinear dynamics or the fundamental aspects of turbulence. Since then, this flow—called Taylor-Couette—has been widely studied and has provided a valuable platform to explore and understand the basic principles that govern the behavior of rotating fluids.
Proof of this is the biannual meeting which has been celebrated for more than 40 years, where a growing international community gathers interested in the diverse flow patterns observed in the Taylor-Couette apparatus. Last summer, the meeting took place in Barcelona, where the 100th anniversary of Taylor's publication was the excuse to continue talking about instabilities, vortexes and fascinating geometric patterns.
Jezabel Curbelo She is a Ramón y Cajal researcher at the Department of Mathematics of the Universitat Politècnica de Catalunya (UPC) and a member of the Institute of Mathematics of the UPC (IMTech) and the Center for Mathematical Research (CRM).
Agate Timón García—Longoria She is coordinator of the Mathematical Culture Unit of the Institute of Mathematical Sciences (ICMAT).
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
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