The Differentiation Method in Rheology: II. Characteristic Derivatives of Ideal Models in Poiseuille Flow
- J.G. Savins (Socony Mobil Oil Co.) | G.C. Wallick (Socony Mobil Oil Co.) | W.R. Foster (Socony Mobil Oil Co.)
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- December 1962
- Document Type
- Journal Paper
- 309 - 316
- 1962. Original copyright American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Copyright has expired.
- 5.3.1 Flow in porous media
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Inasmuch as the differentiation and integration methods represent different modes of rheological analyses, a dual scheme of analysis using both methods should lead to a generalized method of data analysis. A dual differentiation.integration method of analysis is applied here to the Poiseuille flow of a variety of ideal Generalized Newtonian and viscoplastic models. Using machine processing techniques, the result is a spectrum of response patterns which are expressed in terms of certain derivative functions. It is shown that these characteristic functions form the basis of a highly-sensitive analytic technique for optimizing the selection of the most appropriate functional relationship between shear rate and shearing stress.
In the first paper of this series, Savins, Wallick and Foster presented an historical review of the salient features of the differentiation method of rheological analysis in Poiseuille flow, and also indicated how the method could be applied to problems involving plane Poiseuille flow. It was shown that the differentiation and integration methods, although basically not incompatible, do represent different modes of rheological analysis. This suggests that valuable background information regarding the probable response characteristics of real data obtained with the differentiation method can be obtained from an integration method-differentiation method analysis of the response of a variety of ideal rheological models. The present paper describes how this dual method of analysis has been applied to suites of idealized models representing a wide variety of Generalized Newtonian and viscoplastic behavior which have received attention at various places in the literature.
For a liquid of constant viscosity
By substituting Eq. 1 in Eq. 9 of Ref. 1 and integrating, it is easily shown that for Poiseuille flow
GENERALIZED NEWTONIAN SYSTEMS
This model is of the form
Note that it represents a Maclaurin-type series expansion, based on the Newtonian model, which is restricted to the odd powers of the stress.
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