Series Expansion for Shear Stress in Large-Amplitude Oscillatory Shear Flow From Oldroyd 8-Constant Framework
Giacomin, A. Jeffrey
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When polymeric liquids undergo large-amplitude oscillatory shear flow (LAOS), 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 summations to infinity. In this paper, to simplify the exact solution, we expand it in a Taylor series of the dimensionless shear rate amplitude. We truncate the power series expansion after the 16th power of the shear rate amplitude. We find our main result reduces to the well-known expression for the special cases of 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. Our solutions agree closely with the exact solutions so long as Wi/De < 7/2 , for the special cases of the corotational Jeffreys and corotational Maxwell fluids. 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.
URI for this recordhttp://hdl.handle.net/1974/26040
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