Hoop stress and polymer diffusive instabilities in viscoelastic Taylor–Couette flow

The linear stability of inertialess viscoelastic Taylor–Couette flow is analysed using the Oldroyd-B model augmented with stress diffusion. It is well known that this flow is susceptible to a hoop stress-driven ‘purely elastic’ instability (referred henceforth as the ‘hoop stress mode’ (HSM)), that is operative even when fluid inertial effects are negligible. The recent work by Beneitez et al. (Beneitez M, Page J, Kerswell RR 2023 Phys Rev. Fluids 8 , L101901 ( doi:10.1103/PhysRevFluids.8.L101901 )) has shown that the inclusion of stress diffusion in the Oldroyd-B model results in a new type of instability termed the ‘polymer diffusive instability’ (PDI). While certain features of this instability, such as its wavelength approaching the size of the polymer molecule for realistic diffusivity values, raise questions on the feasibility of its experimental realization, the PDI mode is certainly of importance to computational studies of the Oldroyd–B/FENE-P models that often incorporate artificially high values of stress diffusivity for numerical stability. To this end, it is important to identify the parameter regimes where PDI or HSM will be the most critical in the viscoelastic parameter space for Taylor–Couette (TC) flow. This is relevant in discriminating the physical origins of the complex flow structures observed in direct numerical simulation (DNS). In the present work, we analyse the linear stability of the TC flow, using the diffusive Oldroyd-B model, for arbitrary three-dimensional disturbances. Stress diffusion is shown to have opposing effects on HSM and PDI, stabilizing the former while destabilizing the latter. We further show that for the ratio of solvent-to-solution viscosity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>0.8</mml:mn> </mml:math> , it is the PDI that is the more unstable mode compared with HSM, in that the threshold Weissenberg number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> is much lower for the PDI mode. In such cases, it is plausible that the DNS studies of the viscoelastic TC flow with explicit stress diffusion or numerical diffusion (that is likely implicit in such methods) are contaminated, or even dominated, by PDI-induced flow structures. Further, for the special case of the two-dimensional TC flow, the HSM instability is absent, and any complex flow state observed in DNS ought to have a non-HSM origin. Here, we show that the PDI mode likely underlies such complex flow states in the two-dimensional TC flow.