Couette flow Taylor number

Taylor vortices are toroidal in form and constitute a secondary laminar flow superimposed on the basic Oouette flow. For many years it has been suggested that the sequence of flows occurring as the Taylor number is increased is Couette flow, Taylor vortex flow, wavy Taylor vortex flow with travelling azimuthal waves superimposed on the... 

When the shear rate reaches a critical value, secondary flows occur. In the concentric cylinder, a stable secondary flow is set up with a rotational axis perpendicular to both the shear gradient direction and the vorticity axis, i.e. a rotation occurs around a streamline. Thus a series of rolling toroidal flow patterns occur in the annulus of the Couette. This of course enhances the energy dissipation and we see an increase in the stress over what we might expect. The critical value of the angular velocity of the moving cylinder, Qc, gives the Taylor number ... 

Smith, G. P., and A. A. Townsend. 1982. Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mechanics 123 187-217. 

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. 

Many studies have been devoted to the Taylor-Couette problem (flow between two concentric cylinders with radii R and R2, Ri < R2, of infinite length, and rotating with angular velocities fij and 02 repectively). For instance Zielinska and Demay [74] consider the general Maxwell models with —1 <0 < 1. They show that the axisymmetric steady flow (the Couette flow) does not exist for 2dl values of parameters where the steady state exists moreover all models, except for a very close to —1, predict stabilization of the Couette flow in the spectral sense, for small enough values of the Weissenberg number. (See also [55].)... 

When the Taylor number exceeds a critical value (Tac), a transition from stable Couette flow to vortical Taylor-Couette flow occurs. The critical Taylor number can be calculated by [24] ... 

Next we turn to the stability of Couette flow for parallel rotating cylinders. This is an important flow for various applications, and, though it is a shear flow, the stability is dominated by the centrifugal forces that arise because of centripetal acceleration. This problem is also an important contrast with the first two examples because it is a case in which the flow can actually be stabilized by viscous effects. We first consider the classic case of an inviscid fluid, which leads to the well-known criteria of Rayleigh for the stability of an inviscid fluid. We then analyze the role of viscosity for the case of a narrow gap in which analytic results can be obtained. We show that the flow is stabilized by viscous diffusion effects up to a critical value of the Reynolds number for the problem (here known as the Taylor number). 

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... 

Figure 3-8. Stability diagram for Couette flow with 0 < . 2 / -2 i < 1, plotted as Taylor number T versus ih/ The flow is stable for T < Tc and unstable for T > Tc.

In both cases, the increase in rotation speed in the range of 1000-3000 rpm, which corresponds to arange of Ta from about 2000-6000 for the test device, resulted in a substantial increase in flux with a transition point observed at a rotation speed of 2000 rpm—this corresponds approximately to a Taylor number of 3500, where turbulent Couette flow can be assumed. 

The Couette-Taylor flow reactor consists of two concentric cylinders in which the outer one is fixed and jacketed, while the inner one rotates. Under some particular conditions, a flow pattern characterized by counter-rotating toroidal vortices is formed. This Couette-Taylor flow makes the RTD in this reactor similar to that of a train of CSTRs [74]. However, because viscosity may change substantially as polymerization proceeds (along the reactor), it is difficult to maintain the required Taylor number in the whole reactor. The use of a conical outer cylinder may counteract the viscosity increase [75]. However, no example of the production of a commercial-like latex (i.e., high solids content) has been reported. 

The critical Taylor number T for the onset of Taylor vortices can be predicted by examining the stability of snail amplitude disturbances when superimposed on the basic Couette flow. The use of this linear stability analysis for concentric cylinders has been extensively reviewed by Chandrasekhar (1) and Stuart (2). All such analyses assune that the cylinders are infinitely long. In addition to T they predict an initial Taylor vortex celf axial length, . ... 

The Taylor vortices described above are an example of stable secondary flows. At high shear rates the secondary flows become chaotic and turbulent flow occurs. This happens when the inertial forces exceed the viscous forces in the liquid. The Reynolds number gives the value of this ratio and in general is written in terms of the linear liquid velocity, u, the dimension of the shear gradient direction (the gap in a Couette or the radius of a pipe), the liquid density and the viscosity. For a Couette we have ... 

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. 

Fig. 4. The computation of the dimension of an attractor is illustrated using velocity data obtained fof a weakly turbulent flow in the Couette-Taylor system at R/R =16.0, where R is the Reynolds number for the onset of time-independent Taylor vortex flow. The different curves correspond to different embedding dimensions m. (a) The dependence of N(e), the average number of points within a ball of radius e, on e. (b) The slope of the curves shown in (a). Regions A,...

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