Spatiotemporal linear stability of viscoelastic free shear flows: Dilute regime

Abstract

We report the temporal and spatio-temporal stability analyses of anti-symmetric, free shear, viscoelastic flows obeying the Oldroyd-B constitutive equation in the limit of low to moderate Reynolds number and Weissenberg number. The resulting fourth order Orr-Sommerfeld equation is reduced to a set of six auxiliary equations which are numerically integrated starting from the rescaled far-field conditions, i. e., via the Compound Matrix Method. Numerical results indicate that with increasing Weissenberg number: (a) the peak of the maximal growth rate (i. e., the maximum value of the imaginary component of the growth rate) is reduced, (b) the entire range of unstable spectrum is shifted towards longer waves (i. e., the entire region of temporal instability is gradually concentrated near zero wavenumber), (c) the vorticity structure contours are dilated and (d) the residual Reynolds stresses are diminished. All these observations suggest that viscoelasticity reduces the temporal instability but does not completely suppress it. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability, absolute and convective instabilties as well as evanescent (false) modes, in a finite range of Reynolds number, Weissenberg number and the viscosity coefficient. The main result is that, unlike Newtonian fluids, the free shear flow of dilute polymeric liquids are either (absolutely/convectively) unstable for all Reynolds number or the transition to instability occurs at very low Reynolds number, a finding attributed to the fact that viscoelasticity aggravates free surface flow instabilities. Although this transitional pathway connecting the temporally stable state to the elastoinertially unstable state have been identified by some in-vitro experiments, but until now, has not been quantified theoretically via linear stability analysis.

Spatiotemporal linear stability of viscoelastic free shear flows: non-affine response regime

“…This instability occurs at short wavelength (i. e., large α r , figure 2i) and low frequency (i. e., small |ω r |, figure 2g). In the dilute regime, the shear flow instabilities were found to arise at zero elasticity number, through a combination of instability via normal stress anisotropy and elasticity [32]. In the non-affine regime, we surmise a similar operative mechanism.…”

“…A positive sign of the temporal growth rate indicates whether absolute instability is possible. The temporal stability analysis for viscoelastic free shear flows in the dilute regime, for low to moderate Re and We, was earlier studied by us [32]. In the limit of large Re and We such that We/Re ∼ O(1), Azaiez conducted the temporal stability analysis through an elastic Rayleigh equation and concluded elasticity as the controlling flow parameter within the dilute flow regime [31].…”

“…where φ (y), ϕ(y) are the transverse perturbations in the streamfunction and the extra stress tensor and α, ω are the complex wavenumber and angular frequency, respectively. We note that equation ( 5) is a solution of the momentum equations for incompressible flow provided there is a dimensionless body force term on the right-hand-side of equation (1b) [32]. We rephrase equations (1)(2)(3)(4) in the streamfunction-vorticity formulation and avail equation ( 6) to arrive at the equation governing the perturbation of the streamfunction, given by the fourth order OSE [31].…”

Implicit-explicit time integration method for fractional advection-reaction-diffusion equations

“…Since we do not wish to study the effects of elasticity, the Weissenberg number is fixed at We = 10.0. The value of µ is fixed at µ = 10 −2 such that the body force term in equation (28a) behaves as a perturbative term for the system which is initially at steady state (29,30). Two different values of ν are considered: ν = 0.3 (elastic stress dominated case) and ν = 0.6 (viscous stress dominated case).…”

Spatiotemporal linear instability of viscoelastic slender jets

“…The first step in the spatiotemporal analysis entails the procedure of finding the most unstable mode for real wavenumber, 𝛼. This mode is the largest positive imaginary component of any root of the dispersion relation, also known as the temporal growth rate, 𝜔 Temp [Sircar & Bansal (2019)]. In other words, this step consists of detecting the admissible saddle points (𝜔 ∈ C, 𝛼 ∈ R) satisfying the equations [Huerre & Monkewitz (1990)],…”