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Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids

Non-modal amplification of disturbances in streamwise-constant channel flows of Oldroyd-B fluids is studied from an input-output point of view by analyzing the responses of the velocity components to spatio-temporal body forces. These inputs into the governing equations are assumed to be harmonic in the spanwise direction and stochastic in the wall-normal direction and in time. An explicit Reynolds number scaling of frequency responses from different forcing to different velocity components is developed, showing the same $Re$-dependence as in Newtonian fluids. It is found that some of the frequency response components peak at non-zero temporal frequencies. This is in contrast to Newtonian fluids, where peaks are always observed at zero frequency, suggesting that viscoelastic effects introduce additional timescales and promote development of flow patterns with smaller time constants than in Newtonian fluids. The temporal frequencies, corresponding to the peaks in the components of frequency response, decrease with an increase in viscosity ratio (ratio of solvent viscosity to total viscosity) and show maxima for non-zero elasticity number. Our analysis of the Reynolds-Orr equation demonstrates that the energy-exchange term involving the streamwise/wall-normal polymer stress component $τ_{xy}$ and the wall-normal gradient of the streamwise velocity $\partial_y u$ becomes increasingly important relative to the Reynolds stress term as the elasticity number increases, and is thus the main driving force for amplification in flows with strong viscoelastic effects.