Power Series for Shear Stress of Polymeric Liquid in Large-amplitude Oscillatory Shear Flow

Exact solutions for shear stress in a polymeric liquid subjected to large-amplitude oscillatory shear flow (LAOS) contain many Bessel functions. Approximate analytical solutions for shear stress in LAOS often take the form of the first few terms of a power series in the shear rate amplitude, and without any Bessel functions. There is thus interest in extending the Goddard integral expansion (GIE), to an arbitrary number of terms. In continuum theory, these truncated series are arrived laboriously using GIE. However, each term in the GIE requires much more work than its predecessor. In this paper, we begin with the exact solution for shear stress responses in corotational Maxwell fluids, and then perform an expansion by symbolic computation to confirm up to the sixth power, and to then continue the GIE. In this paper for example, we continue the GIE to the 40th power of the shear rate amplitude. We use Ewoldt grids to show our main result to be highly accurate. We also show the radius of convergence of the GIE to be an infinite.