Elastoinertial stability analysis and structure formation in viscoelastic subdiffusive pipe flow

The modal temporal stability analysis of viscoelastic, subdiffusive, pressure driven, axisymmetric pipe flow, representing thick polymer solutions, exhibits the presence of temporally stable regions at high fluid inertia. The stress constitutive equation, previously derived for channel flows [Chauhan et al., Phys. Fluids 35(12), 123121 (2023)] is the fractional variant of the upper convected Maxwell equation. The parameters governing the stability are the Reynolds number, Re=ρU0R0η0, the elasticity number, El=λαη0ρR02, and the ratio of the solvent to the polymer solution viscosity, ν=ηsη0, where R0,U0,ρ,λ,α are the pipe radius, the maximum mean flow velocity, density, the polymer relaxation time, and the fractional order of the time derivative, respectively. The neutral curves indicate, in the limit of small elasticity numbers or in the limit when the viscosity ratio approaches unity, El(1−ν)≪1, that the critical Reynold number, Rec diverges as Rec∼[(1−ν)El]−3α/2, while the critical wavenumber, kc increases as kc∼[(1−ν)El]−α/2. Using a novel fractional variant of the pressure correction method as well as a metric in the Riemannian manifold of symmetric positive definite conformation tensors, the direct numerical simulations quantify the formation of spatiotemporally stable macrostructures (or the non-homogeneous regions of high viscosity) at moderate inertia, thereby corroborating the qualitative features of the experimentally observed flow-instability transition of subdiffusive axisymmetric pipe flows.