Spatiotemporal linear stability of viscoelastic free shear flows: Nonaffine response regime

Abstract

We provide a detailed comparison of the two-dimensional, temporal, and spatiotemporal linearized analyses of the viscoelastic free shear flows (inhomogeneous flows with mean velocity gradients that develop in the absence of boundaries) in the limit of low to moderate Reynolds number and elasticity number obeying four different types of stress–strain constitutive equations: Oldroyd-B, upper convected Maxwell, Johnson–Segalman (JS), and linear Phan-Thien–Tanner (PTT). The resulting fourth-order Orr–Sommerfeld equation is transformed into a set of six auxiliary equations that are numerically integrated via the compound matrix method. The temporal stability analysis suggests (a) elastic stabilization at higher values of elasticity number {shown previously in the dilute regime [Sircar and Bansal, “Spatiotemporal linear stability of viscoelastic free shear flows: Dilute regime,” Phys. Fluids 31, 084104 (2019)]} and (b) a nonmonotonic instability pattern at low to intermediate values of elasticity number for the JS as well as the PTT model. To comprehend the effect of elasticity, Reynolds number, and viscosity on the temporal stability curves of the PTT model, we consider a fourth parameter, the centerline shear rate, ζc. The “JS behavior” is recovered below a critical value of ζc, and above this critical value, the PTT base stresses (relative to the JS model) are attenuated thereby explaining the stabilizing influence of elasticity. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability and absolute and convective instabilities, as well as evanescent modes, and the results are compared with previously conducted experiments for Newtonian as well as viscoelastic flows past a cylinder. The phase diagrams reveal the two familiar regions of inertial turbulence modified by elasticity and elastic turbulence as well as (a recently substantiated) region of elastoinertial turbulence and the unfamiliar temporally stable region for intermediate values of Reynolds and elasticity number.

Spatiotemporal linear stability of viscoelastic subdiffusive channel flows: a fractional calculus framework

“…and then (among all the possible roots of equation ( 30)) identifying those roots with the largest positive imaginary component of the frequency. Equation ( 30) is solved using a multivariate Newton-Raphson algorithm (refer Author's previously published results [45,55] for a detailed outline of this method).…”

“…The fractional derivative operators account for the complete history to obtain the derivative at an instant. Unlike the classical Maxwell model [45,55] which accounts for only the elastic (or stored) part of the deformation work, the fractional Maxwell model accounts for both forms (stored and dissipated) of energy at any time point. Although Mckinley pointed out that the fractional Maxwell model generally cannot capture polymer shear-thinning [56], the fractional version provides a better fit of the relaxation and creep behavior for a significantly large class of viscoelastic materials using fewer parameters than the classical version [54].…”

“…More significantly, we evaluate the absolute growth rate (or ω Cusp i , details on computing these points are elaborated in Section 2.3) to identify the region of absolute instability, or the region indicating the topological reconfiguration and subsequent pinch-off of the advancing interface [64]. However, evanescent modes are also encountered in our analysis [55]. These modes do not merely depend on the sign of the absolute growth rate and have to be found via the sufficient conditions proposed by Briggs [60] (refer Section 2.3).…”