We carry out linear stability analysis of both inertialess and inertial viscoelastic plane Couette flow using the Giesekus model augmented with stress diffusion. Recent work [Beneitez et al., Phys. Rev. Fluids 8, L101901 (2023)] has shown that plane Couette and Poiseuille flows of an Oldroyd-B fluid are linearly unstable if polymer stress diffusion is included. The unstable mode, termed “polymer diffusive instability” (PDI), however, has wavelengths of the order of the size of the polymer for realistic values of polymer diffusivity, thereby raising a question on the applicability of the Oldroyd-B model, which is valid on length scales large compared to the size of the polymer molecule. To address this issue, in the present study we examine the role of anisotropic mobility of the polymer molecule on the PDI within the framework of the Giesekus model. The stability is governed by the Weissenberg number $\text{Wi}=\lambda V/H$, Reynolds number $\text{Re}$ (or the elasticity number $E=\text{Wi}/\text{Re}$), the anisotropy parameter of the Giesekus model $\alpha$, and a dimensionless polymer diffusivity $\mathcal{D}=\lambda D/H^2$. Here $\lambda$ is the polymer relaxation time, $V$ is the plate velocity, and $H$ is the channel half-width. The conventional (nondiffusive) Giesekus model predicts plane Couette flow to be linearly stable when the constitutive curve is monotonic. For the diffusive Giesekus model, following earlier studies, we employ two different boundary conditions for the polymeric stress tensor: no-flux of the stress tensor and no-diffusion at the walls. Our numerical results reveal that in both the inertialess and inertial regimes, and for both boundary conditions, the PDI is rapidly stabilized as $\alpha$ is increased, resulting in an increase in the critical Weissenberg number (${\text{Wi}}_c$). This is also accompanied by a concomitant shrinking of the neutral curves in the $\text{Wi}\text{−}k$ plane ($k$ is the streamwise wave number), and an eventual disappearance of the unstable region beyond a threshold $\alpha$. The critical Weissenberg number diverges with increasing $\alpha$ for all $\mathcal{D}$'s considered. In the dilute limit, with the solvent-to-solution viscosity ratio $\beta \sim 0.9$, PDI stabilizes rapidly at negligibly small values of $\alpha$ ($\sim {10}^{-5}$). For concentrated polymer solutions ($\beta \sim 0.1$), the PDI is fully stabilized at a slightly higher $\alpha \sim {10}^{-3}$. The suppression of PDI due to anisotropy in the Giesekus model, even for extremely small $\alpha$'s, raises the question whether the PDI will be present when a more rigorous constitutive model (rooted in kinetic theory) is employed, one that accounts for anisotropic diffusion of polymer molecules at higher shear rates.

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• Flow instability
• Non-Newtonian fluids