Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids

Abstract

Amplification of deterministic disturbances in inertialess shear-driven channel flows of viscoelastic fluids is examined by analyzing the frequency responses from spatio-temporal body forces to the velocity and polymer stress fluctuations. In strongly elastic flows, we show that disturbances with large streamwise length scales may be significantly amplified even in the absence of inertia. For fluctuations without streamwise variations, we derive explicit analytical expressions for the dependence of the worst-case amplification (from different forcing to different velocity and polymer stress components) on the Weissenberg number (We), the maximum extensibility of the polymer chains (L), the viscosity ratio, and the spanwise wavenumber. For the Oldroyd-B model, the amplification of the most energetic components of velocity and polymer stress fields scales as We 2 and We 4 . On the other hand, finite extensibility of polymer molecules limits the largest achievable amplification even in flows with infinitely large Weissenberg numbers: in the presence of wall-normal and spanwise forces the amplification of the streamwise velocity and polymer stress fluctuations is bounded by quadratic and quartic functions of L. This high amplification signals low robustness to modeling imperfections of inertialess channel flows of viscoelastic fluids. The underlying physical mechanism involves interactions of polymer stress fluctuations with a base shear, and it represents a close analog of the lift-up mechanism that initiates a bypass transition in inertial flows of Newtonian fluids.

Streak evolution in viscoelastic Couette flow

“…An important question is whether the streamwise vortex would be observed in practice in a real flow. Previous linear analyses of the flow response to body forces in both Newtonian (Bamieh & Dahleh 2001;Jovanović & Bamieh 2005) and weakinertia viscoelastic (Lieu et al 2013) flows highlight a flow sensitivity to streamwise elongated disturbances with O(1) spanwise spacing. This sensitivity underscores the importance of the current results.…”

“…Their analysis provides further evidence of similarities between polymer stretching in highly elastic flows and tilting of mean vorticity in high-Reynolds-number Newtonian flows. Recently, these results have been extended to a more realistic polymer model, where the large energy growth generated by polymer stretch is upper bounded by the maximum extensibility of the polymer chains (Lieu et al 2013). …”

“…Recently, an analogy has been proposed between transient growth in high-Reynoldsnumber Newtonian flows and strongly elastic flows of polymer solutions (Jovanovic & Kumar 2010Lieu, Jovanović & Kumar 2013). This rests on a similarity between the lift-up term in the momentum equation and polymer stretching terms in the constitutive relation.…”

Linear instability of a highly shear-thinning fluid in channel flow

“…Equally, there is the possibility of non-modal growth [13][14][15]. Essentially, if the linear stability problem has close eigenvalues, we may see transient growth of disturbances even if the long-time behaviour (predicted by the sign of the eigenvalues) is stable.…”

The dynamics of spanwise vorticity perturbations in homogeneous viscoelastic shear flow

“…The flow response takes the form of streamwise streaks, similar in appearance to inertial transient growth in Newtonian fluids. Further studies have addressed the influence of weak inertia (Page & Zaki 2014) and finite polymer extensibility (Lieu, Jovanović & Kumar 2013) on this elastic growth. In inertia-dominated flows, Hoda, Jovanović & Kumar (2008, 2009 introduced stochastic forcing in the linear equations for an Oldroyd-B fluid and demonstrated that viscoelasticity enhances the energy density of the response.…”