Taylor-Couette instability

In this section, we study the stability of Couette flow between a pair of concentric rotating cylinders. The steady flow was considered earlier in Section C of Chap. 3. 

The general problem of the stability of various types of shear flow has occupied a great deal of research for nearly a century. There are two basic classes of problem. First, is the instability of a parallel shear flow such as simple shear flow or either 2D or axisymmetric Poiseuille flow, in which the problem of stability is a very subtle balance of viscous and inertia effects. Although the basic flow is very simple in these cases, the analysis of instability is difficult and involved, so much so that complete books have been written on this subject alone.21 At the end of this chapter, we will return to a very brief discussion of this class of problems. 

In the case of Couette flow, the base flow whose stability we wish to study, was shown in Chap. 3, Section C, to be 

The variables in (12-114) including the coefficients C[ and C 2 are all dimensional, as indicated by the fact that they are primed. We see that the fluid moves in circular paths and thus the fluid particles undergo a centripetal acceleration. However, the centrifugal force associated with this centripetal acceleration does not produce a secondary flow, because it is exactly balanced by a radial pressure gradient, 

As noted already in Chap. 3, the balance between centrifugal force and radial pressure gradient produces a potentially unstable arrangement that can be thought of as resulting from a gradient of angular momentum. 

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On the other hand, Muller et al. [75-78] have reported on a purely elastic Taylor-Couette instability for models with or without Newtonian contribution (Jeffreys or Maxwell). The conclusion of their studies is that negative second normal stresses are stabilizing, especially for very small gap ratios, and that the Newtonian relative contribution has a stabilizing influence. 

E.S.G. Shaqfeh, S.J. Muller and R.G. Larson, The effects of gap width and dilute solution properties on the viscoelastic Taylor-Couette instability, J. Fluid Mech., 235 (1992) 285-317. 

An important problem is to analyze the stability of fluid flows. With the exception of the Taylor-Couette and Saffman Taylor problems, this chapter has focused on stability questions when the base state of the system was one with no motion (or rigid-body motion), so that instability addresses the conditions for spontaneous onset of flow. An equally valid question is whether a particular flow, such as Poiseuille flow in a pipe (or any of the other flows that we have analyzed in previous chapters of this book), is stable, especially to infinitesimal perturbations as linear instability determines whether the particular flow is actually realizable in experiments. This question was first mentioned back in Chapter 3 when we analyzed simple unidirectional flow problems and noted that solutions such as Poiseuille s solution for flow through a tube was a valid solution of the Navier-Stokes equations for all Reynolds numbers, even though common experience tells us that beyond some critical Reynolds number there is a transition to turbulent flow in the tube. 

Problem 12-7. The Dean Problem. A problem related to the Taylor-Couette problem is the instability of pressure-gradient-driven flow in a curved channel. An example of this is the flow that is due to a pressure gradient in the azimuthal direction between curved walls that are sections of a pair of nonrotating concentric cylinders of radius R and R2, respectively. 

Vaezi, V., and R. C. Aldredge. 2000. Laminar-flame instabilities in a Taylor-Couette combustor. Combustion Flame 121 356. 

Viscoelastic phenomena may be described through three aspects, namely stress relaxation, creep and recovery. Stress relaxation is the decline in stress with time in response to a constant applied strain, at a constant temperature. Creep is the increase in strain with time in response to a constant applied stress, at a constant temperature. Recovery is the tendency of the material to return partially to its previous state upon removal of an applied load. The material is said to have memory as if it remembers where it came from. Because of the memory effect, in transient flows the behavior of viscoelastic fluids wUl be dramatically different from that of Newtonian fluids. Viseoelastie fluids are fiiU of instabilities. Some examples inelude instabilities in Taylor-Couette flow, in eone-and-plate and plate-and-plate flows (Larson 1992). The extrudate distortion, commonly called melt fraeture, is a notorious example of viscoelastic instability in polymer processing. The viseoelastie instability in injection molding can result in specific surface defects such as tiger stripes (Bogaerds et al. 2004). 

MOBBS, F.R. "Marginal instability in Taylor-Couette flows at very high Taylor nutiber." J. Fluid Mech. 1979, 94,... 

However, several flow transition regimes have been identified between laminar and fully turbulent flow. The cessation of laminar Couette flow is marked by the appearance of Taylor vortices in the gap between the two cylinders. For the case of stationary outer cylinder, the critical angular velocity, C0crit> of inner cylinder at which these flow instabilities first appear can be estimated by using the following equations [102] ... 

Reactors which generate vortex flows (VFs) are common in both planktonic cellular and biofilm reactor applications due to the mixing provided by the VF. The generation of Taylor vortices in Couette cells has been studied by MRM to characterize the dynamics of hydrodynamic instabilities [56], The presence of the coherent flow structures renders the mass transfer coefficient approaches of limited utility, as in the biofilm capillary reactor, due to the inability to incorporate microscale details of the advection field into the mass transfer coefficient model. 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

Kataoka, K. Taylor vortices and instabilities in circular Couette flows, in N.P. Cheremisinoff (Ed.), Encyclopedia of Eluid Mechanics , vol. 1. Gulf Publishing, Houston (1986), p. 236. 

For Taylor numbers exceeding Tc, the flow develops a secondary flow pattern in which ur and uz are both nonozero. A sketch of the stability criteria given by (3-86) is shown in Fig. 3 8. The reader who is interested in a detailed description of the stability analysis that leads to the criterion (3-86) is encouraged to consult Chap. 12 or one of the standard textbooks on hydrodynamic stability theory (see Chandrashekhar [1992] for a particularly lucid discussion of the instability of Couette flows).12... 

Couette flow is a laminar circular flow occurring between a rotating (inner) cylinder and a static one, and the extension via increased speed of rotation to centrifugally-driven instabilities leads to laminar Taylor vortex flow, tending to turbulent flow as speed increases. Poiseuille flow is axial. 

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