Series expansion for shear stress in large‐amplitude oscillatory shear flow from oldroyd 8‐constant framework

Abstract

When polymeric liquids undergo large-amplitude oscillatory shear flow, the shear stress responds as a Fourier series, the higher harmonics of which are caused by the fluid nonlinearity, and the first harmonic of which is a nonlinear function of both the frequency and the shear rate amplitude. The Oldroyd 8-constant framework for continuum constitutive theory contains a rich diversity of popular special cases for polymeric liquids. The shear stress response for the Oldroyd 8-constant framework has recently yielded to exact analytical solution. However, in its closed form, Bessel functions appear 24 times, each within infinite summations. In this paper, to simplify the exact solution, we expand it in a Taylor series. We truncate the series after its 16th power of the shear rate amplitude. Our main result reduces to the well known expression for the corotational Jeffreys and corotational Maxwell fluids. Whereas these special cases yielded to the Goddard integral expansion (GIE), the more general Oldroyd 8-constant framework does not. We use Ewoldt grids to show our main result to be highly accurate, for the corotational Jeffreys and corotational Maxwell fluids. For these two special cases, our solutions agree closely with the exact solutions so long as Wi=De < 7 2 . We apply our main result for the special case of the Johnson-Segalman fluid to describe the measured frequency sweep and shear rate amplitude sweep responses of molten atactic polystyrene. For this, we use the Spriggs relations to generalize our main result to multimode, which then agrees closely with the measured responses.

The Medium Amplitude Response of Nonlinear Maxwell-Oldroyd Type Models in Simple Shear

“…With the appropriate parameter substitutions for the models defined in Table 1, these series expansions can be used to validate that equation 19 reduces to previously known solutions for the intrinsic nonlinearities in MAOS. Indeed, we find that equations 19 and 50 along with Tables 1 and 2 can be used to derive the MAOS solutions for the Johnson-Segalman, Phan-Thien-Tanner, Larson, stretching and non-stretching Gaussian Rolie-Poly, and Gaussian cDCR-CS models found by Hyun and co-workers [43]; the solution for the Giesekus model found by Gurnon and Wagner [24]; and the solution for the Oldroyd 8constant framework found by Giacomin and co-workers [48]. That all of these solutions, which had previously been derived and represented separately, can be found as realizations of the solution for the cubic Maxwell model further demonstrates both the generality of this framework and its potential utility in model identification.…”

Wire coating and melt elasticity

“…and where the magnitude of the rate of deformation tensor [Eq. (9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27) of Refs. 10 and 11]:…”

Unidirectional large-amplitude oscillatory shear flow of human blood

“…The Pipkin map can be used to indicate the nonlinear regime in experimentalcondition space, (De, Wi) . Lissajous loops can be miniaturized and mapped into cells in Pipkin space to arrive at the Ewoldt grid (see §Ewoldt Grids of [15]), which is now widely used [16,17,18,19] to provide a phenomenal view of how complex fluids reveal their nonlinearity in oscillatory shear flow. These diagrams have been exploited in LAOS to elucidate fluid nonlinearities, but not in udLAOS.…”