Elastic instability in a family of rectilinear viscoelastic channel flows devoid of centerline symmetry

We analyze the linear stability of viscoelastic channel flows, with velocity profiles that are asymmetric about the channel centerline, belonging to the one-parameter Couette-Poiseuille family (CPF). These flows are driven by a combination of an imposed pressure gradient and (tangential) wall motion. A particular member of this family, corresponding to a zero net volumetric flux, may be experimentally realized in a shallow lid-driven cavity flow configuration, as well as in the narrow-gap limit of the Taylor-Couette geometry with an obstruction at a fixed azimuthal angle (a narrow gap Taylor-Dean flow). Recent work by Khalid et al. [Phys. Rev. Lett. 127, 134502 (2021)] has shown, using the Oldroyd-B model, that plane Poiseuille flow with a symmetric velocity profile becomes unstable, even in the absence of inertia, to an elastic "center mode" with phase speed close to the base-state maximum. In contrast, viscoelastic plane Couette flow is linearly stable. The objective of this study is to determine parameter regimes where viscoelastic CPF, whose members include the two limiting flows above, is unstable in the inertialess limit. The dimensionless groups that govern stability are the Weissenberg number $W=\lambda U_{\text{avg}}/L$, the parameter $\alpha$ characterizing the relative importance of Couette ($\alpha =0$) and Poiseuille flow ($\alpha =1$) components, and the ratio of solvent to solution viscosities $\beta ={\mu }_s/\mu$. Here, $\lambda$ is the polymer relaxation time, $L$ the channel half-width and $U_{\text{avg}}$ the average speed; $\beta \in [0,1]$, and $\alpha \in (-\infty ,\infty )$ with $\alpha \to \pm \infty$ representing the unidirectional flow in a shallow lid-driven cavity. We show that, similar to plane Poiseuille flow, an elastic center-mode instability does indeed exist for the aforesaid family in the limit of ultra-dilute polymer solutions ($\beta \gtrsim 0.99$); the instability relies on the existence of a base-state maximum, implying its absence for CPF members with $\alpha \in (-0.5,0.25)$. Our results point to the potential relevance of the center-mode instability to viscoelastic Taylor-Dean flows and other curvilinear shear flow configurations.