Shear between Cone-and-Plate Fixtures

The cone-and-plate rheometer (Fignre 8.12) is a simple variant of the parallel plate device considered in Section 8.5. In a cone-and-plate rheometer, polymer samples are sandwiched between a blnnt cone and a flat plate. Shear is generated by rotating either the cone or the plate at an angnlar speed Q, while the other fixture is maintained stationary. Typically, the cone angle a is maintained small a 6° to minimize measurement artifacts caused by secondary flow [18]. The main advantage of the cone-and-plate rheometer over the parallel plate device is the homogeneity of the shear field it creates 

FIG U RE 8.12 Torsional shear flow between cone-and-plate fixtnres. 

Based on the direction of the imposed velocity, and the assumptions that the fluid is incompressible and flow occurs under isothermal conditions, the equations of continuity in spherical coordinates (Appendix 8.A) yield v = v (0, r). The shear rate is therefore given by 

Neglecting inertia, the differential linear momentum conservation equations in spherical coordinates (Appendix 8.B) can be written as 

If the shear rate is independent of the radial position in the gap, then the shear stress must also be independent of r, which implies that A(r) = A, where A is a constant. The torque required to maintain the plate stationary can be computed as earlier from a torque balance  

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