Cylinder shear flow

Fig. 4.5.1 Photograph of the NMRI probe designed to generate a concentric cylinder shear flow. The cylinder axis is parallel to the direction of the static magnetic field B0. The front of the probe is inserted into the magnet...

Viscometric flows used for measurements include well known flows, such as flow in a narrow gap concentric cylinder device and between a small angle cone and a flat plate. In both of these cases the flows established in these devices approximate almost exactly simple shearing flow. There are other viscometric flows in which the shear rate is not constant throughout, these include the wide gap concentric cylinder flow and flow in a circular pipe, discussed above. 

If we neglect the distortion of the segment distribution in the fuzzy cylinder by the shear flow, we can apply Doi s stress expression, Eq. (61), to fuzzy cylinder systems as it stands. The neglect of the distortion may be justified when the shear-rate is low. Equation (61) expresses the contribution of the end-over-end rotation of the chain to asegment distribution is not distorted, the orientational entropy term Sor in the static free energy expression contains only the orientational entropy loss of the entire chain, but not the conformational entropy loss cf. Sect. 2.3. 

For convenience, a mathematically simple arrangement is considered. It consists of a fluid layer of finite constant thickness, confined by two rigid parallel planes of infinite extension. Steady laminar shear flow is created in this layer by fixing one plane in space and moving the other one with constant speed in a direction parallel to both planes. In this way, a truly uniform and time independent shear rate q is created in the liquid. The magnitude of this shear rate is simply given by the ratio of the said speed to the mutual distance of the planes. Experimentally such an arrangement is approximated e.g. by the use of two coaxial cylinders. When the gap between the inner surfaces of these cylinders is made small compared with their radii, the above mentioned situation can be realized to a sufficient extent. 

L-100 (Mw — 1.4 0.1 x 10 , Mw/Mn — 2.2)4 in medicinal white oil as a rather high viscous solvent (1.50 poise at 25° C). In this figure the directly measured shear recovery (s ) (open triangles) is plotted against shear stress pzl of the preceding shear flow. From flow birefringence measurements (in a coaxial cylinder apparatus) and normal thrust measurements (in a cone-and-plate apparatus) values of normal stress difference (pn — p22) were calculated. These values were transformed with the aid of eq. (2.12) into recoverable shears s. The full circles (from... 

For the case that no convection stream is applied, the time, which elapses between the start of shear flow and the establishment of the temperature maximum in the center of the gap has been reported to be useful for the measurement. This measurement could be carried out within 10 sec (202). The possibility of such a procedure, however, follows from the theoretical treatment only for relatively wide gaps of d > 2 mms. For metal cylinders (199) as well as for glass cylinders (203) times of establishment can be calculated which are definitely too short for the application of this method, when smaller gap widths are used. 

Keywords Block copolymers Director Hydrodynamics Layer normal Layered systems Liquid crystals Macroscopic behavior Multilamellar vesicles Onions Shear flow Smectic A Smectic cylinders Undulations... 

Fig. 17 A schematic representation of the multilamellar cylinder in shear flow. The cylinder axis is aligned along the flow...

In what follows, we will explore the first of the two suggested scenarios. Starting with a multilamellar cylinder configuration, we will study its stability under shear flow (Fig. 17). Our aim is to check whether we can find an instability of the cylinder-a secondary instability-that would be responsible for the break-up into onions. 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

As the flow accelerates into the gaps around the cylinder, it possesses a greater relative amount of extension. Ultimately, at distances far downstream from the cylinder, the flow is expected to relax back toward a parabolic profile. In these plots, the symbols represent the measured velocities and the solid curves are the results of a finite element, numerical simulation. The constitutive equation used was a four constant, Phan-Thien-Tanner mod-el[193], which was adjusted to fit steady, simple shear flow shear and first normal stress difference measurements. The fit to the velocity data is very satisfactory. 

A graphic example of the consequences of the existence of in stress in simple steady shear flows is demonstrated by the well-known Weissenberg rod-climbing effect (5). As shown in Fig. 3.3, it involves another simple shear flow, the Couette (6) torsional concentric cylinder flow,3 where x = 6, x2 = r, x3 = z. The flow creates a shear rate y12 y, which in Newtonian fluids generates only one stress component 112-Polyisobutelene molecules in solution used in Fig. 3.3(b) become oriented in the 1 direction, giving rise to the shear stress component in addition to the normal stress component in. 

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. 

If the dimensions of the cylinder and the speed of the ram are known it is possible to calculate the volumetric flow through the die the shear stress and shear rate may be estimated, flow curves constructed, and the material characterized by shear flow over a range of temperature. 

FIG. 15.2 Types of simple shear flow. (A) Couette flow between two coaxial cylinders (B) torsional flow between parallel plates (C) torsional flow between a cone and a plate and (D) Poisseuille flow in a cylindrical tube. After Te Nijenhuis (2007). 

While the experiments under quiescent conditions show the effect of diffusion control, the study by Agarwal and Khakhar [32] of polymerization of PPDT in a coaxial cylinder reactor with a uniform shear flow, most clearly illustrate the role of shearing and orientation on the polymerization. Figure 2 shows the... 

Kato, M. and Launder, B.E. (1993), The modeling of turbulent flow around stationary and vibrating square cylinders. Proceedings of 9th Symposium on Turbulent Shear Flows, Kyoto. 

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). 

Figure 2-9. A number of simple flow geometries, such as concentric cylinder (Couette), cone-and-plate, and parallel disk, are commonly employed as rheometers to subject a liquid to shear flows for measurement of the fluid viscosity (see, e.g., Fig. 3-5). In the present discussion, we approximately represent the flow in these devices as the flow between two plane boundaries as described in the text and sketched in this figure.

We note that, in spite of the superficial similarity to a unidirectional flow, the solution for uo is actually quite different from the linear profile of a simple, planar shear flow. This is not at all surprising for an arbitrary ratio of the cylinder radii. However, we should expect that the approximation to a simple shear flow should improve as the gap width becomes... 

Hence, to achieve the best possible approximation to a linear shear flow, the Couette device must have a very thin gap relative to the cylinder radius. 

The values of ix /fx and "JZ/ix calculated with (3-83) and (3-84) are shown in Table 3-1 for the cases 22 = 0 and 2i / 0 for various values of e. Only in the limit s 1 does ix JZ ix. For e <governing equations (and hence the solutions of these equations) reduce to those for a simple shear flow between parallel plane boundaries. Examining Table 3-1, we see that an error of 10% would be made by using the estimate (3-81) with a gap width that is only 5% of the inner cylinder radius. 

The only changes required in these solutions are due to the fact that a(q2) may be more complex than for uniform streaming flows. For example, a qualitative sketch of the flow structure for a nonrotating cylinder in simple shear flow at low Reynolds number is shown in Fig. 9-16.23 It is evident in this case that there are fom stagnation points on the cylinder smface rather than two, as in the streaming-flow problem. Two of the streamlines that lead to the stagnation points A and C are lines of inflow, and two from B and D are lines of outflow, where we should expect a thin thermal wake. In the limit as these outflow points are approached, we thus expect a breakdown of the similarity solution with g -> 00. At the inflow stagnation points, on the other hand, we require that g be finite. To accommodate... 

Figure 9-16. A qualitative sketch of the streamlines near a nonrotating circular cylinder in a simple 2D shear flow.

Heat Transfer from a Rotating Cylinder in Simple Shear Flow... 

Figure 9-17. The coordinate axes and undisturbed flow for a cylinder in an unbounded simple shear flow.

To begin our analysis of the cylinder problem, it is necessary to consider briefly the velocity field. The fluid mechanics solution for a circular cylinder rotating with an imposed angular velocity il in a simple shear flow was given originally by Bretherton26 for the creeping-flow limit. In this case, the velocity field can be specified in terms of a stream-function (see Chap. 7) such that... 

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