Torsional Shear between Parallel Plates

As stated earlier, our goal is to derive general equations that relate these quantities to rheological variables such as shear stress, shear rate, and normal stress differences. Based on the direction of the imposed velocity, the cylindrical symmetry of the flow geometry, and the assumptions that the fluid is incompressible and flow occurs under isothermal conditions, the equations of continuity (Appendix 8.A) yield v = Vg(r,z)eg. Neglecting inertia, the differential linear momentum conservation equations (Appendix 8.B) can be simplified to give 

FIGURE 8.11 Torsional shear flow between parallel plates. 

It is apparent from Equation 8.36 that the shear stress can only be a function of r. Anticipating a relationship between Xe and the shear rate of the general form 

The shear rate is therefore given by y = D.ld)r. The shear stress Xg can be expressed in terms of the measured torque Tby equating the torque exerted by the fluid on the lower plate with the external torque T required to prevent rotation  

Therefore, the radial dependence of the shear stress must be known before the above equation can be exploited to relate the stress to the measured torque. This situation is analogous to the problem we encountered in Section 8.3 we will solve it using a similar approach. First, change variables in Equation 8.40 to introduce the shear rate  

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The velocity field between the cone and the plate is visualized as that of liquid cones described by 0-constant planes, rotating rigidly about the cone axis with an angular velocity that increases from zero at the stationary plate to 0 at the rotating cone surface (23). The resulting flow is a unidirectional shear flow. Moreover, because of the very small i//0 (about 1°—4°), locally (at fixed r) the flow can be considered to be like a torsional flow between parallel plates (i.e., the liquid cones become disks). Thus... 

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). 

Torsional deformation It is where mechanical force is apphed to the sample as a torque. This occurs in rheometers, where the sample is typically sheared between parallel plates. Alternatively, a solid bar can be twisted, resulting in a shear deformation. 

Figure 10,17 Birefringence interference patterns in torsional shearing flow between parallel plates of MBBA, illuminated by monochromatic light and observed between crossed polarizers. The images on the left were obtained at a rotation speed, of 2.09 x 10 sec , while those on the right were obtained at 4.18 x 10 sec >, From top to bottom, the gap is 44, 94, and 194 ixm. (From Wahl and Fischer, reprinted with permission from Mol. Cryst. Liq. Cryst. 22 359, Copyright 1973, Gordon and Breach Publishers.)...

FIG. 5-1. Geometries, coordinates, and dimensions for investigations of viscoelastic liquids, (a) parallel plate simple shear (b) annular pumping (c) rotation between coaxial cylinders (d) torsion between cone and plate (e) torsion between parallel plates (/) axial motion between coaxial cylinders. 

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). 

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