Orientation in Large-Amplitude Oscillatory Shear

Journal of Non-Newtonian Fluid Mechanics

2015

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.

Section: Introductionmentioning

confidence: 99%

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

Schmalzer

Bird

Journal of Non-Newtonian Fluid Mechanics

2015

“…From Figure 19 or Figure 20, we learn that for nonlinear behaviors (be they in nearly time steady shearing flow or the fully nonlinear viscoelastic regime) the orientation distribution is peanut-shaped (called lemniscoidal 40 ). Comparing Figures 4–18 (Multimedia view) with Figures 19–Figure 21, we learn that the lemniscoidal shape of the polymer orientation is imparted by the even harmonics, and specifically by the zeroth and second harmonics, but not by the fourth.…”

Section: Resultsmentioning

confidence: 99%

“…Schmalzer and Giacomin 40–42 recently solved Eq. (6) for large-amplitude oscillatory shear flow, including terms that do not contribute to the rheological responses, using the method of Bird and Armstrong 17 .…”

Section: Methodsmentioning

confidence: 99%

“…(36), (44), (52), and (61) of Ref. 40. In this paper, we undertake the laborious Fourier decomposition of these expressions for

ψ_{1}

through

ψ_{4}

, putting the resulting Fourier coefficients under common denominators.…”

Section: Methodsmentioning

confidence: 99%

“…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). …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Gilbert

Schmalzer

2015

Structural Dynamics

Self Cite

In our previous work, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear flow, a flow experiment that has gained a significant following in recent years. We chose rigid dumbbells since these are the simplest molecular model to give higher harmonics in the components of the stress response. We derived the expression for the dumbbell orientation distribution, and then we used this function to calculate the shear stress response, and normal stress difference responses in large-amplitude oscillatory shear flow. In this paper, we deepen our understanding of the polymer motion underlying large-amplitude oscillatory shear flow by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third, and fourth harmonics). We use three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear steady shear flow (where the Deborah number λω is zero and the Weissenberg number λγ˙0 is above unity), (ii) nonlinear viscoelasticity (where both λω and λγ˙0 exceed unity), and (iii) linear viscoelasticity (where λω exceeds unity and where λγ˙0 approaches zero). We learn that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise tilted and peanut-shaped. We find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting, by the second. The first, third, and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.

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

Jbara

Macro Theory & Simulations

2018

For rigid dumbbells suspended in a Newtonian solvent, the viscoelastic response depends exclusively on the dynamics of dumbbell orientation. The orientation distribution function ψ(θ, φ, t) represents the probability of finding dumbbells within the range (θ, θ + dθ) and (φ, φ + dφ). This function is expressed in terms of a partial differential equation (the diffusion equation), which, for any simple shear flow, is solved by postulating a series expansion in the shear rate magnitude. Each order of this expansion yields a new partial differential equation, for which one must postulate a form for its solution. This paper finds a simple and direct pattern to these solutions. The use of this pattern reduces the amount of work required to determine the coefficients of the power series expansion of the orientation distribution function, ψ i . To demonstrate the usefulness of this new pattern, new expressions for these coefficients are derived up to and including the sixth power of the shear rate magnitude. This work also completes previous findings that ended at the fourth power of the shear rate magnitude.