Couette Shear Flow between Coaxial Cylinders

4 COUETTE SHEAR FLOW BETWEEN COAXIAL CYLINDERS 

Again from the cylindrical symmetry of the flow geometry and the assumptions that (1) the fluid is incompressible, (2) flow occurs under isothermal conditions, and (3) the flow in the gap is nniform (i.e., negligible disturbance from the ends or end effects), we conclude that only spatial derivatives with respect to the radial coordinates are nonzero. Thns, v = Vs(r)es and the shear rate is given by 

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

FIG U RE 8.9 Onset of unstable flow in shear flow between concentric cylinders. (Data from Bird, R. B. et al., Dynamics of Polymeric Liquids. Vol. 1 Fluid Mechanics, 1st edn., Wiley, New York, 1987, pp. 521-523.) 

It follows that only stress gradients with respect to r are nonzero. Neglecting inertia, the differential linear momentum conservation equations in cylindrical coordinates (Appendix 8.B) read 

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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... 

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). 

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

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). 

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... 

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