Normal stress differences in large-amplitude oscillatory shear flow for dilute rigid dumbbell suspensions

Abstract

Walters leads through his research publications (including his seminal contribution on the complicating role of instrument compliance in large-amplitude oscillatory shear flow measurement [27]), his excellent books on rheology [26,58,59], his educational videography [60-64], his conference organization, his book editorship [65-67], his founding editorships (Journal of Non-Newtonian Fluid Mechanics and Rheology Reviews) and his interesting work on the history and personalities of rheology [61,68]. a b s t r a c tWe examine the simplest relevant molecular model for large-amplitude oscillatory shear (LAOS) flow of a polymeric liquid: the suspension of rigid dumbbells in a Newtonian solvent. We find explicit analytical expressions for the shear rate amplitude and frequency dependences of the zeroth, second and fourth harmonics of the first and second normal stress difference responses. We include a detailed comparison of these predictions with the corresponding results for the simplest relevant continuum model: the corotational Maxwell model. We find that the responses of both models are qualitatively alike. The rigid dumbbell model relies entirely on the dumbbell orientation to explain the viscoelastic response of the polymeric liquid, including the higher harmonics in large-amplitude oscillatory shear flow. Our analysis employs the general method of Bird and Armstrong (1972) for analyzing the behavior of the rigid dumbbell model in any unsteady shear flow.We derive the first three terms of the deviation of the orientational distribution function from the equilibrium state. Then, after getting the ''paren functions,'' we use these for evaluating the normal stress differences for large amplitude oscillatory shear flow. We find the shapes of the first normal stress difference versus shear rate loops predicted to be reasonable [see Fig. 1(a)]. We find that the second normal stress difference is not proportional to the first, and that its shape differs markedly from that of the first [cf. Fig. 1(a) and (b)]. We discover the same remarkable qualitative similarity between the predictions of the rigid dumbbell model and the corotational Maxwell model for the first normal stress difference. We find no qualitative similarities between the dumbbell and the continuum models for any of the predicted coefficients of the second normal stress differences in large-amplitude oscillatory shear flow.

Orientation in Large-Amplitude Oscillatory Shear

“…and of Ref. ) in large‐amplitude oscillatory shear flow, we found that the higher order terms, and thus the higher harmonics, of the orientation distribution function make significant contributions to the rheological properties. From this we learn that even subtle tweaking of the molecular motions, too subtle to observe in our color figures, can matter for the rheology of the polymeric liquid.…”

“…Comparatively little is known about the behavior of the molecules in large‐amplitude oscillatory shear flow. Our previous work on large‐amplitude oscillatory shear derives expressions for the shear stress and for both normal stress difference responses, using both continuum and molecular approaches . For the molecular approach, we use the simplest relevant model of a polymeric liquid, the dilute suspension of rigid dumbbells.…”

“…Our previous work on largeamplitude oscillatory shear derives expressions for the shear stress and for both normal stress difference responses, using both continuum [10,11] and molecular approaches. [12][13][14][15] For the molecular approach, we use the simplest relevant model of a polymeric liquid, the dilute suspension of rigid dumbbells. By simplest relevant, we mean the simplest model that gives the appropriate higher harmonics in the shear stress and normal stress differences.…”

Fourier decomposition of polymer orientation in large-amplitude oscillatory shear flow

“…(13) and (21) of Ref. 39). In our most recent work, we derived the full orientation distribution function, including the terms that make no contribution to the rheological responses 40–42 (see column titled “non-contributing terms” of Table I).…”

Orientation Distribution Function Pattern for Rigid Dumbbell Suspensions in Any Simple Shear Flow

“…; see refs. to explore diffusion equation and application; see also Table I of ref. , Table I of ref.…”