Normal Stress Differences from Oldroyd 8-Constant Framework: Exact Analytical Solution for Large-Amplitude Oscillatory Shear Flow
Giacomin, A. Jeffrey
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The Oldroyd 8-constant framework for continuum constitutive theory contains a rich diversity of popular special cases for polymeric liquids. In this paper, we use part of our exact solution for shear stress to arrive at unique exact analytical solutions for the normal stress difference responses to large amplitude oscillatory shear flow (LAOS). The nonlinearity of the polymeric liquids, triggered by LAOS, causes these responses at even multiples of the test frequency. We call responses at frequency higher than twice the test frequency higher harmonics. We find the new exact analytical solutions to be compact and intrinsically beautiful. These solutions reduce to those of our previous work on the special case of the corotational Maxwell fluid. Our solutions also agree with our new truncated Goddard integral expansion for the special case of the corotational Jeffreys fluid. The limiting behaviors of our exact solutions for the Oldroyd 8-constant framework yield new explicit expressions for the normal stress difference responses in small-amplitude amplitude oscillatory shear flow (SAOS). Finally, we use our exact solutions to see how η∞ affects the normal stress differences in LAOS.