Power Series for Normal Stress Differences of Polymeric Liquids in Large-Amplitude Oscillator Shear Flow
Giacomin, A. Jeffrey
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Exact solutions for normal stress differences in polymeric liquids subjected to largeamplitude oscillatory shear flow (LAOS) contain many Bessel functions, each appearing in infinite sums. For the simplest relevant model of a polymeric liquid, the corotational Maxwell fluid in LAOS, Bessel functions appear 38 times in the exact solution. By relevant, we mean that higher harmonics are predicted in LAOS. By contrast, approximate analytical solutions for normal stress differences 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 at all. Perhaps the best example of this, from continuum theory, is the Goddard integral expansion (GIE) that is arrived at laboriously. There is thus practical interest in extending the GIE, to an arbitrary number of terms. However, each term in the GIE requires much more work than its predecessor. For the corotational Maxwell fluid, for instance, the GIE for the normal stress differences has yet to be taken beyond the fifth power of the shear rate amplitude. In this paper, we begin with the exact solution for normal stress difference responses in corotational Maxwell fluids, and then perform an expansion by symbolic computation to confirm up to the fifth power, and to then continue the GIE. In this paper for example, we continue the GIE to the 41st power of the shear rate amplitude. We use Ewoldt grids to show our main result to be highly accurate. We also show, except in its zero-frequency limit, the radius of convergence of the GIE to be infinite. We derive the pattern for the common denominators of the GIE coefficients, and also for every numerator for the zeroth harmonic coefficients. We also find that the numerators of the other harmonics appear to be patternless.