The Differentiation Method in Rheology: III. Couette Flow

The theory of the differentiation method for the Couette flow experiment is reviewed. Particular attention is given to the requirements on data analyses in the case of the class of non-Newtonian materials described as viscoplastics, i. e., materials characterized by a yield point or yield stress. Here, changes in boundary conditions arise when the shearing stress attains a critical value with the result that the form of the basic integral equation for Couette flow is determined by the flow conditions existing during the measurement. Introduction In the preceding papers in this series, the salient features of the differentiation method of rheological analysis in Poiseuille-type flow were discussed. It was shown that a dual differentiation- integration method analysis of the Poiseuille flow of idealized generalized Newtonian and visco-plastic models could be used to develop a spectrum of highly sensitive response patterns in terms of certain characteristic derivative functions. These functions were shown to optimize the selection of the most appropriate functional relationship between f(p) and p from the Poiseuille flow experiment. The present paper reviews the theory of the differentiation method as applied to the equally important Couette flow experiment. We will also discuss the range of variables over which the basic integral equation for Couette flow is applicable when the non-Newtonian material is of the viscoplastic type, i.e., characterized by a yield point or yield stress. THEORY Having described the application of the differentiation method to Poiseuille-type flow in the preceding papers, we now proceed to the case where the test liquid is confined to the annular space between coaxial cylinders of length L, one of which is in motion, i.e., Couette flow, formulating the basic integral equation after the method of Mooney. The observed kinematical and dynamical quantities are the angular velocityand the torque T. Here, the one nonvanishing component of the shear-rate tensor is ........................(1) and the corresponding component of the shearing-stress tensor at any point r is given by ..........................(2) The shearing stresses at the inner surface of radius R(1) and the outer surface of radius R(2) are related by .................(3) Combining Eqs. 1, 2 and 3, letting = 0 at p = p1 and = at p = p2 and integrating yield .........................(4) Note that the definite integral has a finite lower limit. Differentiating Eq. 4 with respect to p1, following the rule of Leibnitz (i.e., in Eq. 11 of Ref. 1), gives a difference equation in the desired function ..................(5) This result was initially obtained by Mooney who used it as a starting point for an approximate solution. Several other approximate solutions of the difference equation have been described, the principal results of which are described in the succeeding sections. The interested reader is referred to the original papers for the details. SPEJ P. 14^