Padé Approximant for Normal Stress Differences in Large-Amplitude Oscillatory Shear Flow
Giacomin, A. Jeffrey
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Analytical solutions for the normal stress differences in large-amplitude oscillatory shear flow (LAOS), for continuum or molecular models, normally take the form of the first few terms of a series expansion in the shear rate amplitude. Here, we improve the accuracy of these truncated expansions by replacing them with rational functions called Padé approximants. Specifically, we examine replacing the truncated expansion for the corotational Jeffreys fluid with its Padé approximants for the normal stress differences. We begin by extending the Goddard integral expansion (GIE) from its third [J Non-Newt Fluid Mech 166, 1081 (2011)] to its fifth order term in the shear rate amplitude. We then explore the Padé approximants for this extension. We uncover the best approximant, the [3,4] approximant, and compare this with both the GIE and the exact solution [Macromol Theory Simul 24, 352 (2015)]. We use Ewoldt grids to show the stunning accuracy of our [3,4] approximant in LAOS. We quantify this accuracy with an objective function and then map its onto Pipkin space. We find the [3,4] approximant to be a simple accurate expression for the normal stress differences in LAOS. Our applications illustrate how to use our new approximant reliably. For this, we use the Spriggs relations to generalize our best approximant to multimode, and then, we compare with measurements on molten high-density polyethylene, and on dissolved polyisobutylene in isobutylene oligomer.