# Spatio temporal linear stability of viscoelastic free shear flows: dilute regime

## TL;DR

This paper conducts a detailed linear stability analysis of viscoelastic free shear flows using the Oldroyd-B model, revealing how viscoelasticity influences flow stability and transition to instability.

## Contribution

It provides the first quantitative linear stability analysis of dilute viscoelastic free shear flows, highlighting the effects of Weissenberg number on flow stability and instability mechanisms.

## Key findings

- Viscoelasticity reduces the maximum growth rate of instabilities.
- The unstable spectrum shifts towards longer waves with increasing Weissenberg number.
- Flow becomes either absolutely or convectively unstable at very low Reynolds numbers.

## 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.