Rotational Couette flow

Fig. 15. Flow pattern in rotating Couette flow where and Q2 represent the outer and inner rotational speeds.

Example 6.7. Striation Thickness in Rotational Couette Flow (RCF)... 

FIGURE 6.14 Striation thickness reduction function in rotational Couette flow (RCF) for various ratios of the outside to the inside radius. Newtonian and power-law fluids do not exhibit any significant difference. 

Rotational Couette flow (RCF) is analyzed in the next example. Because the flow is confined inside the Couette flow cell and there is no exit stream, we use the nomenclature G(y) and g(y)dy instead of F(y) mdj y)dy. 

A.12 Rotational Couette Flow for a Power-Law Fluid. Prove that, for a rotational Couette geometry with the inside cylinder rotating, the striation thickness scales inversely proportional to the shear... 

If a fluid is placed between two concentric cylinders, and the inner cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagmed as a substantially enlranced diflfiisivity (typically with effective diflfiision coefficients several orders of magnitude above molecular difhision coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifiircation tlirough a complex sequence of front patterns, see figure A3.14.16. 

Rotating cone viscometers are among the most commonly used rheometry devices. These instruments essentially consist of a steel cone which rotates in a chamber filled with the fluid generating a Couette flow regime. Based on the same fundamental concept various types of single and double cone devices are developed. The schematic diagram of a double cone viscometer is shown in... 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

Laminar Couette flow between concentric oj = rotation rate k = rjr0 = ratio (hi) Sh = 1.0174(ReSc)1 3 + 0.20002L/R, - 0 M13f... 

Various arrangements at the bottom of the inner cylinder are available in Figure 3.2 an indentation is provided so that an air gap is formed and shearing in the sample below the inner cylinder is negligible. Another arrangement is to make the bottom of the inner cylinder a cone. When one of the cylinders is rotated, a Couette flow is generated with fluid particles describing circular paths. The only non-zero velocity component is ve and it varies in the r-direction. In order to minimize secondary flow (Taylor vortices) it is preferable that the outer cylinder be rotated however, in most commercial instruments it is the inner cylinder that rotates. In this case, the fluid s velocity is equal to IXR, at the surface of the inner cylinder and falls to zero at the surface of the outer cylinder. The shear stress is uniform over the curved surface of the inner cylinder and over the outer cylinder (to the bottom of the annular gap). 

The first mode may occur when a droplet is subjected to aerodynamic pressures or viscous stresses in a parallel or rotating flow. A droplet may experience the second type of breakup when exposed to a plane hyperbolic or Couette flow. The third type of breakup may occur when a droplet is in irregular flow patterns. In addition, the actual breakup modes also depend on whether a droplet is subjected to steady acceleration, or suddenly exposed to a high-velocity gas stream.[2701[2751... 

VFF devices consist of a stationary cyhnder, inside which a concentric cyhn-der rotates. The rotating movement of the inner surface of the annular gap creates a Taylor-Couette flow [16], generating Taylor vortices. The filter medium can... 

The Couette rheometer. Another rheometer commonly used in industry is the concentric cylinder or Couette flow rheometer schematically depicted in Fig. 2.48. The torque, T, and rotational speed, 0, can easily be measured. The torque is related to the shear stress that acts on the inner cylinder wall and the rate of deformation in that region is related to the rotational speed. The type of flow present in a Couette device is analyzed in detail in Chapter 5. 

The field of transport phenomena is the basis of modeling in polymer processing. This chapter presents the derivation of the balance equations and combines them with constitutive models to allow modeling of polymer processes. The chapter also presents ways to simplify the complex equations in order to model basic systems such as flow in a tube or Hagen-Poiseulle flow, pressure flow between parallel plates, flow between two rotating concentric cylinders or Couette flow, and many more. These simple systems, or combinations of them, can be used to model actual systems in order to gain insight into the processes, and predict pressures, flow rates, rates of deformation, etc. 

Stresses Generated by CEF Fluids in Various Viscometric Flows What stresses are necessary to maintain a CEF fluid flowing in the following flows (a) parallel-plate drag flow (b) Couette flow with the inner cylinder rotating and (c) parallel-plate pressure flow. Assume the same type of velocity fields that would be expected... 

Dynamic filtration modules are basically of two types rotating disc filter (RDF) and vortex flow filter (VFF). In the latter, the filtration module has a cylindrical shape and has a rotating concentric cylindrical mesh in its interior. The rotational movement of the internal cylinder generates a Taylor-Couette flow in the annular gap (Roth et al., 1997), creating Taylor vortices that minimize concentration polarization and mesh fouling. Continuous perfusion processes based on this type of filter and operating continuously for up to 100 days have been reported (Mercille et al., 1994). 

Indeed, the Biostream Separator [111] operates according to this principle. The device consists of two concentric cylinders the inner one is made to rotate in order to generate a Couette flow. We are no longer dealing with a creeping flow, therefore the liquid density, p, is not negligible. The number of process parameters is increased from two to four. These are (1) the rotational speed, n, of the inner cylinder. (2) In addition, as a result of the possibility of cooling the outer cylinder wall, the temperature difference, AT is an additional process parameter which is freely adjustable. 

The circular Couette flow between concentric cylinders is in the 0-direction only, and satisfies vr = vz = 0, ve = Vfl(r), and T = T(r). The inner cylinder is stationary while the outer cylinder rotates with an angular velocity w. Assuming a steady and laminar flow without end effects, the velocity distribution is... 

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].)... 

Fig. 13. Succession of two-dimensional VEXSY images for cylindrical Couette flow where corresponds to (a) 0.15, (b) 0.50, (c) LOO, and (d) 2.00 rotation cycles for the central rotating cylinder. The theoretical image is shown on the left with the corresponding experimental data on the right. The full width of the image corresponds to 33 cm s . [Reproduced by permission from Callaghan and Manz, 1993.]...

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