STABILITY OF A VISCOUS LIQUID CONTAINED BETWEEN TWO ROTATING CYLINDERS

G.I. TAYLOR, 1923

Notes prepared by Tina Katopodes, Harvard University (1998)

Introduction

The problem of flow between two concentric rotating cylinders lends itself particularly well to a rigorous study of fluid stability due to infinitesimal disturbances. In this classic paper, Taylor presents a mathematical stability analysis and compares the results to thorough laboratory observations.

This problem was first investigated experimentally by Couette (1890) and Mallock (1896). It was observed that the torque needed to rotate the outer cylinder increased linearly with the rotation speed until a critical rotation speed, after which the torque increased more rapidly. This change was due to a transition from stable to unstable flow at the critical rotation speed.

Lord Rayleigh (1916) and von Kármán (1934) carried out analytical studies of the stability of an inviscid fluid moving in concentric layers. Their goal was to determine the condition for which a perturbation resulting from an adverse gradient of angular momentum is unstable. Von Kármán's force argument and Rayleigh's energy argument are outlined by C.S. Yih.

Here we first consider Rayleigh's inviscid flow theory, and then perform a perturbation analysis for the thin-gap limit of the viscous Couette flow examined by Taylor. We consider the problem in circular cylindrical coordinates as shown in the figure. It can be shown that the steady flow field is given by

 

A and B are constants related to the boundary conditions set by the rotation speeds of each cylinder, and , so that

 

where and .







Rayleigh's Inviscid Instability Argument

To study the stability of inviscid circular flow between concentric cylinders, Lord Rayleigh developed an analogy with the stability of a fluid of variable density under the force of gravity. Taylor summarizes this as: ``the varying centrifugal force of the different layers of fluid plays the part of gravity and the resulting condition for stability is that the square of the circulation must increase continuously in passing from the inner to the outer cylinder, just as the density of a fluid under gravity must decrease continuously with height in order that it may be in stable equilibrium.''

To examine this argument, consider a ring of fluid in a flow field with a velocity distribution . The angular momentum per unit volume measured about the central axis is given by . Expressing the velocity in terms of the angular velocity, , gives , and therefore



One can also consider the circulation of the fluid along a circular path,



and notice that the circulation has the dimensions of angular momentum, .

In an inviscid fluid, the angular momentum of a small fluid element is conserved. This can be seen from the Euler equations for inviscid flow,



We assume the motion is axisymmetric, so that the -component of the Euler equations becomes



Multiplication by r leads to



This is simply the material derivative of , so that



and we see that or is constant and angular momentum is conserved.

Next we consider the perturbation resulting when two rings of fluid at different radii are interchanged while conserving angular momentum. The change in angular momentum is



from which it follows that the corresponding change in angular velocity for a displaced fluid ring is

 

We now consider the stability of such a perturbation. The rotating fluid is stable if the displaced fluid ring experiences a force which tends to return it to its original position. Thus, we can imagine displacing a small fluid element from r to such that angular momentum is conserved, and then asking whether the pressure in the surrounding fluid is able to maintain equilibrium. We seek a comparison between the pressure in the equilibrium state, and the pressure in the disturbed state. If the pressure in the undisturbed case is greater than in the disturbed state, it will tend to push the disturbed ring of fluid back to its equilibrium position and maintain stability in the flow. If, on the other hand, the pressure of the disturbed ring of fluid is less than the pressure in the undisturbed fluid surrounding it, instability will set in.

The r-component of the Euler equation gives the centrifugal pressure gradient as



so that is the pressure to within a constant. For fluid elements in the equilibrium state, the angular velocity at r is , and at it is , or to first order. The pressure in the undisturbed flow at is then

 

In the disturbed flow case, the angular velocity at r is and at it is . Since angular momentum is conserved, (from Equation 11), so we then have



The pressure in the disturbed flow at is therefore given by

 

Taking the difference between Eqs. 13 and 15 we obtain



so that



Stability requires that , which implies the following conditions:



If we think of as the circulation of the flow, we come to Rayleigh's conclusion that ``A circulation always increasing outwards ... ensures stability.''

Returning to the case of Couette flow, we have so that . Substituting the value given for A in Equation 2, we require that for stability



Since by definition, we must have for stable flow, or



In -space, the stable region is therefore bounded by the positive -axis and the line . From this we see that if the outer cylinder is fixed ( ), the flow is always unstable.

When and the cylinders are rotating in opposite directions, instability is predicted for all rotation speeds in the absence of viscosity. As explained by Chandrasekhar, however, the instability is only partial. It occurs between the interior cylinder and the nodal surface, defined by , which separates the flow into two parts. The radius of this surface is given by



so there is stability for and instability for . Thus, according to Taylor, ``In the inner region the square of the circulation decreases outwards, so that centrifugal force tends to make the flow unstable. In the outer region the square of the circulation increases so that centrigual force tends to make the flow stable'' (p. 69). It is interesting to note that as , , and the area where the fluid is unstable in the inviscid case becomes infinitesimally small.

When viscous effects are included, there is an added area of stability in -space due to viscous stabilization. Stable flow with , for instance, is possible for less than some critical value. In general, flow is unstable only if



Taylor was the first to investigate this case theoretically and experimentally, and determine the new criterion for the onset of instability. His experiments confirmed his findings, and showed that the marginal state was stationary and showed a steady axisymmetric cellular pattern. He found that when the two cylinders rotate in the same direction, Rayleigh's criterion for inviscid flow still holds, with a small added region of stability. When the cylinders rotate in opposite directions, however, Taylor found a large region of stable flow possible, unlike Rayleigh's prediction that all rotation speeds lead to instability.