Simple accurate expressions for shear stress in large-amplitude oscillatory shear flow

Abstract

Analytical solutions for shear stress in large-amplitude oscillatory shear flow (LAOS), for continuum or molecular models, often take the form of the first few terms of a power series in the shear rate amplitude. For corotational models, we get this truncated series using the Goddard-Miller integral expansion (GIE). Our previous work shows that the best Padé approximants for this truncated series, and specifically for the corotational Maxwell model in LAOS, can agree closely with the corresponding exact solution. We observe this close agreement, even for the Padé approximant for the series truncated after the fifth shear stress harmonic. In this paper, we begin with the extension of the GIE truncated after the next, seventh, order in the shear rate amplitude [Phys. Fluids 29, 043101 (2017)], and we then explore its Padé approximants. We uncover its best approximant, the [2,4], and compare it with both the GIE and the exact solution [Macromol. Theory Simul. 24, 352 (2015)]. We use Ewoldt grids to show the stunning accuracy of the [2,4] approximant in LAOS. We quantify this accuracy with an objective function and then map this function onto Pipkin space. We find the [2,4] approximant (from the GIE truncated after the seventh order in the shear rate amplitude) to be a simple accurate expression for the shear stress in LAOS. Our worked examples illustrate how researchers can use our new approximant reliably. For this, we use the Spriggs relations to extend the Padé approximant to multimode.