Coaxial Cylinder Couette Flow

In the last decade of the nineteenth century, Maurice Couette invented the concentric cylinder viscometer. This instrument was probably the first rotating device used to measure viscosities. Besides the coaxial cylinders (Couette geometry), other rotating viscometers with cone-plate and plate-plate geometries are used. Most of the viscometers used nowadays to determine apparent viscosities and other important rheological functions as a function of the shear rate are rotating devices. 

The fluid fills the gap between the two cylinders, and the outer cylinder acts as a reservoir for the fluid. The layer in contact with the rotating cylinder rotates with the angular velocity of the cylinder. Drag reduction 

Couette flow has a velocity field whose contravariant components are (2,22,29) 

Let us assume that the coaxial cylinder has a length L and that the radii of the inner and outer cylinders are Ri and i 2, respectively. The shear stress acting on a cyhndrical layer of fluid of radius r is related to the torque M applied to the inner cylinder by the expression 

On the other hand, the angular velocity co of the concentric layers of fluid decreases from the inner to the outer cylinder, and the shear rate can be written as 

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Most rheological measurements measure quantities associated with simple shear shear viscosity, primary and secondary normal stress differences. There are several test geometries and deformation modes, e.g. parallel-plate simple shear, torsion between parallel plates, torsion between a cone and a plate, rotation between two coaxial cylinders (Couette flow), and axial flow through a capillary (Poiseuille flow). The viscosity can be obtained by simultaneous measurement of the angular velocity of the plate (cylinder, cone) and the torque. The measurements can be carried out at different shear rates under steady-state conditions. A transient experiment is another option from which both y q and ]° can be obtained from creep data (constant stress) or stress relaxation experiment which is often measured after cessation of the steady-state flow (Fig. 6.10). 

Among the different possible ways to measure viscosities in rotating viscometers, the coaxial cylinder apparatus is the most commonly used in practice. The measured liquid intersperses the annular gap between the stationary inner cylinder (bob) and the rotating outer cylinder (cup). Therefore a velocity gradient builds between the inner and outer cylinders (Couette flow). The momentum, which is transferred by this downward gradient to the inner cylinder, is directly proportional to the viscosity. Deflection is compensated by a torsion bar and the equilibrium deflection is measured electrically. The measurement of the angular velocity of the cup and the angular deflection of the bob makes it possible to determine the viscosity [4, 11]. 

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. 

Flow birefringence of polymer solutions is, in general, measured with the aid of an apparatus of the Couette type, containing two coaxial cylinders. One of these cylinders is rotated at constant speed, the other is kept in a fixed position. The light beam for the birefringence measurement is directed through the annular gap between these cylinders, in a direction parallel with the axis of the apparatus. In this way, the difference of principal refractive indices An is measured just in the above defined plane of flow (1—2 plane). 

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). 

Example 4.4 Thermomechanical coupling in a circular Couette flow For a circular Couette flow (Figure 4.5), the entropy production rate for an incompressible Newtonian fluid held between two coaxial cylinders is... 

Normal stress differences can be observed in Couette flow, cone-plate and plate-plate geometries, and capillary flow. The only nonzero components of the stress tensor in coaxial cylinders are a. e(r), cSrr(f), CTee(r), and... 

In this case, G is called the gradient of the flow rate or the shear rate. The Couette flow occurs between two parallel moving planes or in the gap between coaxial cylinders rotating at different... 

Couette flow is the shear flow in an annular gap between two coaxial cylinders in relative rotation. 

Couette Mixer - To further investigate the physics of the electrification process in transformer oil/cellulosic systems, a compact Couette flow system of coaxial cylinders for electrification measurements is shown in Fig. 4, where the inner cylinder can rotate at speeds giving laminar and turbulent flows as well as the transition cellular convection regime (Melcher et al., 1986 Lyon, 1987). The cylinder walls on either side of the oil gap can be bare metal or can be covered with oil-impregnated paper used in transformers. 

Show that the shear rate in Couette flow between two coaxial cylinders is given by... 

A continuous highly oriented fiber may be grown from dilute polyethylene solution subjected to shear flow in a Couette apparatus [70,74]. This device consists of a pair of coaxial cylinders, the inner one of which can be rotated rapidly. A dilute polyethylene solution introduced into the gap between the cylinders is subjected to shear flow. When the appropriate conditions of shear, temperature, and concentration are met, the polymer will crystallize on to a seed fiber held in the flow field. As the extended polyethylene fiber grows, it is wound up at a rate equal to its extensional growth rate, the tip of the growing fiber thus remaining at a fixed position. The resulting fiber consists of a bundle of shish... 

For quantitative measurements an apparatus is used which resembles the CoUette viscosimeter. It consists of two coaxial cylinders inside one another the outer or the inner of which can rotate, so that in the liquid between the cylinders a laminar flow is produced for a not too great velocity of the rotating cylinder ... 

COUETTE SHEAR FLOW BETWEEN COAXIAL CYLINDERS... 

FIGURE 8.8 Schematic of a coaxial cylinder or Couette flow rheometer. 

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