![CouetteTaylorSystem](https://upload.wikimedia.org/wikipedia/commons/3/3d/CouetteTaylorSystem.svg)
In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
, the Taylor–Couette flow consists of a viscous fluid confined in the gap between two rotating cylinders. For low angular velocities, measured by the
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
''Re'', the flow is steady and purely
azimuthal
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematically, ...
. This
basic state is known as circular
Couette flow
In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes a shear stress on the fluid and induces flow. ...
, after
Maurice Marie Alfred Couette
Maurice Marie Alfred Couette (9 January 1858, Tours – 18 August 1943, Angers) was a French physicist known for his studies of fluidity.
Couette is best known for his contributions to rheology and the theory of fluid flow. He designed a concent ...
, who used this experimental device as a means to measure
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
. Sir
Geoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as " ...
investigated the stability of Couette flow in a ground-breaking paper. Taylor's paper became a cornerstone in the development of
hydrodynamic stability
In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these insta ...
theory and demonstrated that the
no-slip condition
In fluid dynamics, the no-slip condition for viscous fluids assumes that at a solid boundary, the fluid will have zero velocity relative to the boundary.
The fluid velocity at all fluid–solid boundaries is equal to that of the solid boundary. C ...
, which was in dispute by the scientific community at the time, was the correct boundary condition for viscous flows at a solid boundary.
Taylor showed that when the angular velocity of the inner cylinder is increased above a certain threshold, Couette flow becomes unstable and a secondary steady state characterized by axisymmetric toroidal vortices, known as Taylor vortex flow, emerges. Subsequently, upon increasing the angular speed of the cylinder the system undergoes a progression of instabilities which lead to states with greater spatio-temporal complexity, with the next state being called wavy vortex flow. If the two cylinders rotate in opposite sense then spiral vortex flow arises. Beyond a certain Reynolds number there is the onset of
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
.
Circular Couette flow has wide applications ranging from desalination to
magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...
and also in viscosimetric analysis. Different flow regimes have been categorized over the years including twisted Taylor vortices and wavy outflow boundaries. It has been a well researched and documented flow in fluid dynamics.
[
]
Flow description
A simple Taylor–Couette flow is a steady flow created between two rotating infinitely long coaxial cylinders. Since the cylinder lengths are infinitely long, the flow is essentially unidirectional in steady state. If the inner cylinder with radius
is rotating at constant angular velocity
and the outer cylinder with radius
is rotating at constant angular velocity
as shown in figure, then the azimuthal velocity component is given by
:
where
:
.
Rayleigh's criterion
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...
studied the stability of the problem with inviscid assumption i.e., perturbing
Euler equations
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
. The criterion states that ''in the absence of viscosity the necessary and sufficient condition for distribution of azimuthal velocity
to be stable is''
:
''everywhere in the interval; and, further, that the distribution is unstable if
should decrease anywhere in the interval.'' Since
represents angular momentum per unit mass, of a fluid element about the axis of rotation, an alternative way of stating the criterion is: ''a stratification of angular momentum about an axis is stable if and if only it increases monotonically outward.''
Taylor vortex
![Taylor-Couette Streamlines Re=950](https://upload.wikimedia.org/wikipedia/commons/f/f2/Taylor-Couette_Streamlines_Re%3D950.png)
Taylor vortices (also named after Sir
Geoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as " ...
) are
vortices
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
formed in rotating Taylor–Couette flow when the
Taylor number
In fluid dynamics, the Taylor number (Ta) is a dimensionless quantity that characterizes the importance of centrifugal "forces" or so-called inertial forces due to rotation of a fluid about an axis, relative to viscous forces.
In 1923 Geoffre ...
(
) of the flow exceeds a critical value
.
For flow in which
:
instabilities in the flow are not present, i.e. perturbations to the flow are damped out by viscous forces, and the flow is steady. But, as the
exceeds
, axisymmetric instabilities appear. The nature of these instabilities is that of an exchange of stabilities (rather than an overstability), and the result is not turbulence but rather a stable secondary flow pattern that emerges in which large toroidal vortices form in flow, stacked one on top of the other. These are the Taylor vortices. While the
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
of the original flow are unsteady when
, the new flow, called ''Taylor–Couette flow'', with the Taylor vortices present, is actually steady until the flow reaches a large
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
, at which point the flow transitions to unsteady "wavy vortex" flow, presumably indicating the presence of non-axisymmetric instabilities.
The idealized mathematical problem is posed by choosing a particular value of
,
, and
. As
and
from below, the critical Taylor number is
Gollub–Swinney circular Couette experiment
In 1975, J. P. Gollub and
H. L. Swinney published a paper on the onset of turbulence in rotating fluid. In a Taylor–Couette flow system, they observed that, as the rotation rate increases, the fluid stratifies into a pile of "fluid donuts". With further increases in the rotation rate, the donuts oscillate and twist and finally become turbulent. Their study helped establish the
Ruelle–Takens scenario in turbulence,
which is an important contribution by
Floris Takens
Floris Takens (12 November 1940 – 20 June 2010) was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems.
Together with David Ruelle, he predicted that fluid turbulence could develop through a strange attra ...
and
David Ruelle
David Pierre Ruelle (; born 20 August 1935) is a Belgian mathematical physicist, naturalized French. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term '' strange attractor'', and developed a ...
towards understanding how hydrodynamic systems transition from stable flow patterns into turbulent. While the principal, governing factor for this transition is the
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
, there are other important influencing factors: if the flow is open (meaning there is a lateral up- and downstream) or closed (flow is laterally bound; e.g. rotating), and bounded (influenced by wall effects) or unbounded (not influenced by wall effects). According to this classification the Taylor–Couette flow is an example of a flow pattern forming in a closed, bounded flow system.
References
Further reading
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{{DEFAULTSORT:Taylor-Couette flow
Fluid dynamics
Fluid dynamic instabilities