Localized stress amplification in inertialess channel flows of viscoelastic fluids

Nonmodal analysis typically uses square-integrated quantities to characterize amplification of disturbances. However, such measures may be misleading in viscoelastic fluids, where polymer stresses can be strongly amplified over a small region. Here, we show that when using a localized measure of disturbance amplification, spanwise-constant polymer-stress fluctuations can be more amplified than streamwise-constant polymer-stress fluctuations, which is the opposite of what is observed when a square-integrated measure of disturbance amplification is used. To demonstrate this, we consider a model problem involving two-dimensional pressure-driven inertialess channel flow of an Oldroyd-B fluid subject to a localized time-periodic body force. Nonmodal analysis of the linearized governing equations is performed using recently developed well-conditioned spectral methods that are suitable for resolving sharp stress gradients. It is found that polymer-stress fluctuations can be amplified by an order of magnitude while there is only negligible amplification of velocity fluctuations. The large stress amplification arises from the continuous spectrum of the linearized problem, and may put the flow into a regime where nonlinear terms are no longer negligible, thereby triggering a transition to elastic turbulence. The results suggest an alternate mechanism that may be useful for understanding recent experimental observations of elastic turbulence in microchannel flows of viscoelastic fluids.