Concentric rotating cylinder problem

A very interesting series of studies of the influence of end effects in the rotating concentric cylinder problem has been published by Mullin and co-workers T. Mullin, Mutations of steady cellular flows in the Taylor experiment,J. Fluid Mech. 121, 207-18 (1982) T. B. Benjamin and T. Mullin, Notes on the multiplicity of flows in the Taylor experiment, J. Fluid Mech. 121, 219-30 (1982) K. A. Cliff and T. Mullin, A numerical and expwerimental study of anomalous modes in the Taylor experiment, J. Fluid Mech. 153, 243-58 (1985) G. Pfister, H. Schmidt, K. A. Cliffe and T. Mullin, Bifurcation phenomena in Taylor-Couette flow in a very short annulus, J. Fluid Mech. 191, 1-18 (1988) K. A. Cliffe, 1.1. Kobine, and T. Mullin, The role of anomalous modes in Taylor-Couette flow, Proc. R. Soc. London Ser. A 439, 341-57 (1992) T. Mullin, Y. Toya, and S. I. Tavener, Symmetry breaking and multiplicity of states in small aspect ratio Taylor-Couette flow, Phys. Fluids 14, 2778-87 (2002). 

As stated, we begin with the special problem of flow between two rotating cylinders whose axes are parallel but offset to produce the eccentric cylinder geometry shown in Fig. 5 1. In the concentric limit, this is the famous Couette flow problem, which was analyzed in Chap. 3. 

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].)... 

Other errors, which could influence the results obtained, are, for example, wall effects ( slipping ), the dissipation of heat, and the increase in temperature due to shear. In a tube, the viscosity of a flowing medium is less near the tube walls compared to the center. This is due to the occurrence of shear stress and wall friction and has to be minimized by the correct choice of the tube diameter. In most cases, an increase in tube diameter reduces the influence of wall slip on the flow rate measured, but for Newtonian materials of low viscosity, a large tube diameter could be the cause of turbulent flow. ° When investigating suspensions with tube viscometers, constrictions can lead to inhomogeneous particle distributions and blockage. Due to the influence of temperature on viscosity (see Section Influence Factors on the Viscosity ), heat dissipated must be removed instantaneously, and temperature increase due to shear must be prevented under all circumstances. This is mainly a constructional problem of rheometers. Technically, the problem is easier to control in tube rheometers than in rotating instruments, in particular, the concentric cylinder viscometers. ... 

To study suspensions, the first choice is a narrow gap, concentric cylinder rheometer. The outer cylinder should rotate to avoid inertia problems. If there are no settling, large particle, or sensitivity limitations, the cone and plate is a good second choice. For either geometry, stress-controlled instruments (see Figure 8.2.10) provide the lowest shear rate data and best measure of yield stress. Most of the stress-controlled instruments can also do sinusoidal oscillations that allow determination of Yc and structure breakdown and recovery measures (see C hapter 10). 

Consider Couette flow between two concentric cylinders, the outer one at Ro fixed and the inner one at Ri rotating with a tangential velocity vg (/ ,). Starting with Equation 16.2 and the equation in Problem 16.5, obtain an expression for the dimensionless tangential velocity profile vg r)hg Rj), for a power-law fluid. Write your expression in terms of the dimensionless quantities r// RglRi, and the flow index n. Plot vg r) vg R versus rlRt for Ro Ri = 2 and n=, V2, and V4. 

The above problem is difficult to avoid when it occurs in filled systems [pastes] because a certain minimum gap is needed to avoid artifticts due to the size of the particles. In general, the g should be a minimum of 10 x particle size, and the remedy for this problem is to reduce the gap to improve the stabilizing effect of surftice tension. However, it is also possible to change the geometry type to a concentric cylinder, with a sufficiently large radius ratio to accommodate the particles. Since the fi ee edge is at the top, and the cause is due to the second normal stress difference in a direction perpendicular to the rotation axis, this effect is almost non-existent. 

---