Multiple integral constitutive equations in unsteady motions and rheometry D A Siginer Department of Mechanical Engineering, Auburn University, AL 36849 The use of the integral fluid of order three to predict some simple nearly viscometric unsteady flows of viscoelastic liquids, driven by periodic forcing, is discussed. Flow enhancement ef- fects, due to the parallel and orthogonal superposition of oscillatory and simple shear fields, are predicted. It is shown that it is feasible to determine the constitutive constants involved from a series of experiments of rheometry. INTRODUCTION Multiple integral constitutive equations are not in favor with both the theoretician and experimentalist due to the rather large number of constitutive parameters involved at any order larger than two. The analytical difficulties, in particular in complex flows, and the seemingly impossible task of determining the constitutive parameters are discouraging. Of course, the determination of a large num- ber of parameters from any given single experiment of rheometry is out of question. The idea that I would like to develop is that a series of experiments of rheometry may breathe hope into this old query. Multiple integral constitutive structures, in viscoelastic fluids and solids alike, may be needed whenever the effect on the stress of a strain increment may not be considered to be independent of the preceding or following strain increments. If that is the case multiple integral models may not be reduced to a single integral constitutive equation, which is always a possibility in the opposite case. There is strong evidence in the literature, both analytical and experi- mental in nature, which indicates that multiple integral terms may be needed to describe the response of the materi- al. For instance the motion driven by an oscillating vertical rod in a large vat cannot be adequately described without the inclusion of a double nested integral in the constitutive structure. We also would like to make the point that constitutive equations should not be too specialized. We subscribe to the point of view, that it is much better to search for an equa- tion as universal as possible for a limited class of fluids, than to look for equations which are universal only for a re- stricted class of motions of a possibly larger class of fluids. For instance exact universal equations for the prediction of viscometric flows are known, such as the CEF and K-BKZ models. But the predictive powers of these exact stress- strain relationships fail as soon as one considers nearly viscometric flows with possibly large shear rate variations such as pulsating pressure gradient driven flow in a pipe and flow in a cylindrical container with or without a free surface driven by the rotating end caps. It is by no means certain that universality in the sense of Navier-Stokes equa- tions will ever be attainable. Evidence is a plenty that it may even be an impossible, task. Nevertheless, I believe some degree of universality is attainable and is a worth- while goal to strive for. We propose to look at constitutive structures of the fol- lowing type 00 00 F [ G(X,s)] = = JK 1 (s)G(s)ds s=O 0 00 00 + J J K 2 (s 1 ,s 2 )G(s 1 )G(s 2 )ds 1 ds 2 (1) 0 0 00 00 00 + J J jK 3 (s 1 ,s 2 ,s 3 )G(s 1 )G(s 2 )G(s 3 )ds 1 ds 2 ds 3 + ... , 0 0 0 where the even order kernel tensors K; will ultimately de- fine the material functions and therefore characterize the fluid. Mathematically manageable forms of the stress re- sponse functional F can be obtained if F is linearized around some deformation history G 0 . Functional differen- tiability of, say, either Frechet or Gateaux type, may be as- sumed, and the F may be expanded into a series 00 F [ G(X,s)] = F[G 0 ] + OF[G 0 1G 00 ] s=O + 8 2 F[G 0 1G 00 ,G 00 ] + O(IG 00 i3),(2) where 8F and 8 2 F represent functional derivatives at Go • and the history of the motion G(X,s) of the particle X been expressed as a sum of the base state G 0 and the devia- tion G 00 from the base state, Appl Mach Rev vol 44, no 11 , part 2, Nov 1991 8232 © Copyright 1991 American Society of Mechanical Engineers