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An asymptotic solution to the Giesekus constitutive model of polymeric fluids under homogenous, oscillatory simple shear flow at large Weissenberg number, , and large strain amplitude, , is constructed. Here, , where is the polymerrelaxation time and is the shear rate amplitude, and is a Deborah number, where is the oscillation frequency. Under these conditions, we show that the first normal stress difference is the dominant rheological signal, scaling as , where is the elastic modulus. The shear stress and second normal stress difference are of order . The polymer stress exhibits pronounced nonlinear oscillations, which are partitioned into two temporal regions: (i) A “core region,” on the time scale of , reflecting a balance between anisotropic drag and orientation of polymers in the strong flow; and (ii) a “turning region,” centered around times when the shear flow reverses, whose duration is on the hybrid time scale , in which flow-driven orientation, anisotropic drag, and relaxation are all leading order effects. Our asymptotic solution is verified against numerical computations, and a qualitative comparison with relevant experimental observations is presented. Our results can, in principle, be employed to extract the nonlinearity (anisotropic drag) parameter, , of the Giesekus model from experimental data, without the need to fit the stress waveform over a complete oscillation cycle. Finally, we discuss our findings in relation to recent work on shear banding in oscillatory flows.


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