Couette Current - Taylor

	Couette Current - Taylor
Named after	and

Couette-Taylor flow - in hydrodynamics, a viscous fluid flow arising under the action of viscous friction forces between two coaxial cylinders rotating at different speeds.

Changing the flow pattern with increasing cylinder rotation speed is one of the most common scenarios of transition to chaos ( turbulence ) in simple flows of liquid and gas.

Content

1 Flow pattern
2 History
3 notes
4 Literature

Flow pattern

At small angular velocities of rotation of the cylinders (with the corresponding Reynolds numbers), the flow is stable and purely circular with a speed

v=C_{1}r+C_{2}/r

{\ displaystyle v = C_ {1} r + C_ {2} / r}

( $r$ ${\ displaystyle r}$ - cylindrical coordinate, constant $C_{1},C_{2}$ ${\ displaystyle C_ {1}, C_ {2}}$ determined from the boundary conditions). If the cylinders rotate in one direction, then such a flow is stable under the condition (obtained by Taylor in 1923)

\omega _{1}b^{2}<\omega _{2}a^{2},

{\ displaystyle \ omega _ {1} b ^ {2} <\ omega _ {2} a ^ {2},}

Where $a$ ${\ displaystyle a}$ Is the radius of the outer cylinder. Sense received (1938) this stability criterion without restrictions on the relative sizes of cylinders used by Taylor.

The loss of stability by the flow (if the stability criterion is not fulfilled) is manifested in the fact that so-called Taylor vortices form during the flow. They fill the entire space between the cylinders, their rotation directions alternate. If the cylinders rotate in different directions, then two rows of vortices are formed, a row near the surface of the inner cylinder has a higher intensity.

A further increase in the rotation speed leads to the emergence of a very complex flow pattern — turbulent flow.

The various modes of the Couette – Taylor flow have their own names: Taylor’s rotating vortices, wave boundary flows, etc. ^[1]

The fluid flow in the annular space by two rotating cylinders under the action of a pressure gradient is called the Taylor - Dean flow.

History

The flow got its name (Couette circular flow) after Maurice Couette used a device of his kind designed for experimental measurement of fluid viscosity.

Sir Jeffrey Ingram Taylor investigated the stability of the Couette flow in 1923, this work became one of the most significant in the development of the theory of hydrodynamic stability. ^[2] Taylor showed that with an increase in the angular velocity of rotation of the inner cylinder above a certain threshold, a purely circular flow becomes unstable and a new stable state arises with axisymmetric toroidal vortices, known as Taylor vortices. With a further increase in the angular velocity of rotation of the cylinder, the flow passes into states of greater spatio-temporal complexity (perturbed vortex flow). If two cylinders rotate in opposite directions, then a vortex flow spirals.

Notes

↑ Andereck, CD; Liu, SS; Swinney, HL Flow regimes in a circular Couette system with independently rotating cylinders // Journal of Fluid Mechanics : journal. - 1986. - Vol. 164 . - P. 155-183 . - DOI : 10.1017 / S0022112086002513 . - .
↑ Taylor, GI Stability of a Viscous Liquid contained between Two Rotating Cylinders // Phil. Trans. Royal Society : journal. - 1923. - Vol. A223 , no. 605-615 . - P. 289—343 . - DOI : 10.1098 / rsta.1923.0008 . - .

Literature

Slezkin N.A. Dynamics of a viscous incompressible fluid. - M .: GITTL. - 1955. - 519 p.

[andereck-1] Andereck, CD; Liu, SS; Swinney, HL Flow regimes in a circular Couette system with independently rotating cylinders // Journal of Fluid Mechanics : journal. - 1986. - Vol. 164 . - P. 155-183 . - DOI : 10.1017 / S0022112086002513 . - .

[2] Taylor, GI Stability of a Viscous Liquid contained between Two Rotating Cylinders // Phil. Trans. Royal Society : journal. - 1923. - Vol. A223 , no. 605-615 . - P. 289—343 . - DOI : 10.1098 / rsta.1923.0008 . - .