The Taylor Number

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  

---

The results presented here were found by investigations with a special cyUn-der system [45,48]. This system was constructed for an existing Searle viscosimeter (rotation of inner cylinder), such that the gap widths were large in relation to the reference floe diameter of the floccular system used, so that the formation of the floes and their disintegration in the cylinder system are not impaired. For this system, with r2 = 22 mm, rj = 20.04 mm, and Li = 60 mm (r2/ri > 1.098), the following Newton number relationships were determined from the experimental values collected by Reiter [38] for the Taylor number range of 400 < Ta < 3000 used here ... 

The flow domain of TCP can be described by two dimensionless hydrodynamic parameters, corresponding to the rotational speed of the inner cylinder and the imposed axial flow rate the Taylor number, To, and the axial Reynolds number, Re, respectively ... 

Because of the strong effects of plate rotations on the rector performance for both RE and PC electrolyzers, the critical design parameters for these reactors are the Taylor number (a2w/4v)0 5 and the Reynolds number (aVf/v). Here a is the gap width between the plate, w the angular velocity of rotation (in radians per second), v the kinematic viscosity of the fluid, and V the velocity in the feed pipe. Since no asymptotic velocity profile is reached for PC, the length of the cell will be an important design parameter in a pump-cell electrolyzer. Detailed mathematical models for RE and PC electrolyzers are given by Thomas et al. (1988), Jansson (1978), Jansson et al. (1978) and Simek and Rousar (1984). 

According to Taylor (1923), the flow instability is observed when the Taylor number exceeds a critical value where Taylor number is defined by geometrical parameters and the speed of rotation ... 

This can be explained by the fact that the flow in the CCTVFR became closer to plug flow as the Taylor number was dropped closer to. Therefore, the steady-state particle number and the steady-state monomer conversion could be arbitrarily varied by simply varying the rotational speed of the inner cylinder. Moreover, no oscillations were observed, and the rotational speed of the inner cylinder could be kept low, so that the possibility of shear-induced coagulation could be decreased. Therefore, a CCTVFR with these characteristics is considered to be highly suitable as a pre-reactor for a continuous emulsion polymerization process. In the case of the continuous emulsion polymerization of VAc carried out with the same CCTVFR, however, the situation was quite different [365]. Oscillations in monomer conversion were observed, and almost no appreciable increase in steady-state monomer conversion occurred even when the rotational speed of the inner cylinder was decreased to a value close to. Why the kinetic behavior with VAc is so different to that with St cannot be explained at present. 

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] ... 

An empirical correlation which expresses the Sherwood number (Sh) as a function of the Taylor number was proposed by Holeschovsky and Cooney [29] as... 

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). 

Now, following the assumption of Taylor that the principle of exchange of stabilities is satisfied, so that a = 0 at the neutral stability point, the objective of the stability analysis is to obtain nontrivial solutions of (12-147) and (12-148) with a = 0 for various values of a and then determine the minimum value of T as a function of a. This minimum of T is the critical value of the Taylor number for transition to instability. The corresponding value of a is known as the critical wave number and represents the (dimensionless) wave... 

If we substitute in the definition for the Taylor number for this thin-gap approximation, this can be written in the form... 

In the outer gap the velocity of the fluid near the inner cylinder is higher compared to that near the stator. Hence, Taylor vortices caused by centrifugal forces may arise under these circumstances. The criterion for a stable flow is given by the Taylor number Ta... 

The hydrodynamics of the flow between the two cylinders can be characterized by a dimensionless quantity 7h, the Taylor number, named after the mathematician G. I. Taylor. The Taylor number incorporates two dimensionless groups, the Reynolds number and a geometric ratio, and is defined as ... 

THE SOLUTION First we see whether the mass transfer coefficient can be calculated from Eisenberg s Eq. (2.24), by checking the value of the Taylor number, Eq. (2.23), and the value of the Reynolds number. The rotation speed w = 2tc(150/60) = 15.7 radians/s. 

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... 

For low-viscosity liquids flowing in concentric cylinders, with the inner cylinder rotating at high velocities, we see the appearance inertia-driven secondary flow cells called Taylor vortices, see figure 8. The onset of these vorhces is controlled by the Taylor number which is given by... 

Taylor sclassic work deals with the break up of a Newtonian liquid drop in a Newtonian matrix when subjected to a simple shear field. The drop deforms into an ellipse due to the viscous forces upon it and this deformation is resisted by the interfacial tension. The interfacial force depends on y/a, the interfacial tension divided by the drop radius, and the viscous force depends on the matrix viscosity multiplied by the shear rate. The ratio of these is the Taylor number... 

---