Owing to the spatially local nature of the constitutive equations typically used to model polymeric stresses, the differential operators governing the linearized dynamics of bounded viscoelastic shearing flows have singular points. As a result, the eigenspectra of such shearing flows contain, in addition to discrete eigenvalues, continuous spectra (CS) comprising singular eigenfunctions. On account of the singularity, numerical methods capture these spectra only as extended balloons that slowly converge to the theoretical loci. A clear understanding of the theoretical CS loci is crucial in discriminating physically genuine (discrete) eigenvalues from the poorly approximated numerical CS. This is especially important because, when a parameter such as the Weissenberg number $\left(\text{Wi}\right)$ is varied, discrete eigenvalues can emerge from or disappear into the CS, as the latter are often branch cuts in the complex spectral plane. For rectilinear shear flows of Oldroyd-B fluids, the CS are a pair of line segments, with lengths equal to the base-state range of velocities. In this study, we provide the first comprehensive account of the nature of the CS for both rectilinear and curvilinear shearing flows of the FENE-P fluid. The FENE-P model is one of the most widely used constitutive equations for dilute polymer solutions, and predicts both shear thinning of the polymer viscosity and the first normal stress coefficient. In stark contrast to the CS for the Oldroyd-B fluid mentioned above, we show analytically that there are up to six distinct continuous spectra for shearing flows of FENE-P fluids. Further, owing to shear thinning, and due to the highly coupled nature of the linearized equations, these spectra possess nontrivial structures in the complex plane. When the finite extensibility parameter $L$ is greater than 50, as appropriate for large molecular weight polymers used in experiments, three of the CS are nearly identical, and independent of the solvent-to-solution viscosity ratio $\left(\beta \right)$. The other three CS are $\beta$-dependent, with one of them being the analog of the solvent (viscous) continuous spectrum in the Oldroyd-B fluid. The remaining two $\beta$-dependent CS are novel features of the FENE-P spectrum, and can have phase speeds outside the base range of velocities, including negative ones. For nonlinear shearing flows subjected to streamwise invariant disturbances, the latter two CS have a novel “winglike” structure in the complex plane, symmetric about the imaginary axis. All but two of the theoretical CS are shown to collapse onto master curves for different $(\text{Wi},L)$ pairs, with $\text{Wi}/L$ fixed, a feature not shared by the discrete spectrum. The location and structure of the theoretical CS are broadly in accord with their numerical approximations obtained using the spectral method. The complexity of the CS predicted here, for shearing flows of FENE-P fluids, is expected to carry over to other nonlinear viscoelastic models that exhibit a shear-thinning rheology. Our analytical predictions for the CS will therefore be helpful in identifying physically genuine, discrete eigenvalues in the numerically computed spectra.

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• Flow instability
• Complex fluids
• Non-Newtonian fluids
• Rheological properties