Temporal instability of a viscoelastic liquid thread surrounded by another viscoelastic fluid in presence of insoluble surfactant and inertia

Abstract

In recent years, there has been controversy in the literature regarding the role of viscoelasticity in determining the onset of stability for a viscoelastic thread surrounded by another viscoelastic fluid. Whereas a number of authors have found that viscoelasticity serves to destabilize the system, Patrascu and Balan (2018) obtained very different results. They considered a very simple canonical system subject to axisymmetric perturbations that neglected inertia and did not include the effect of surfactant. Their analysis provided two genuinely startling results: firstly, when both fluids have similar elastic properties, the dispersion relation reduces to the Newtonian limit; and secondly, the presence of elasticity in the external medium decreases the value of the unstable growth rate. In this paper, we investigate the axisymmetric temporal stability of an infinitely long viscoelastic thread surrounded by another immiscible viscoelastic fluid in presence of small amount of insoluble surfactant. Our analysis also includes the effects of inertia. The dispersion relations for cases with and without inertia are carefully derived. For the case of zero inertia and no surfactant, our results directly contradict those of Patrascu and Balan. We show that when both fluids have similar elastic properties that the dispersion relation does not reduce to the Newtonian case. We also show that, when one of the fluids is viscoelastic or both fluids are viscoelastic, the system is more unstable than the case in which both fluids are Newtonian. Moreover, our results also reveal that the system with inertia is less unstable than the system without inertia, and when the inner–outer density ratio becomes larger, the system becomes less unstable. We also carry out a parametric study of the effects of surfactant. Our results show that adding surfactant can decrease the growth rate of the instability and increase the wavenumber of the most unstable mode. We also show that the reduction of the unstable growth rate induced by the surfactant is more significant for larger internal–external total viscosity ratios. Finally, an energy budget analysis is performed to investigate the contribution of each physical factor towards the instability. This reveals the roles that the various physical mechanisms play in determining the temporal growth rate. The energy analysis further confirms that the elasticity in either inner or outer fluid increases the growth rate. • Address the controversy regarding the stability of viscoelastic threads. • Dispersion relation for fluids with same elasticity differs from Newtonian case. • Elasticity in the external medium does not decreases the growth rate. • We examine the role played by inertia and surfactant in determining the instability.