Linear stability analysis of a three-dimensional viscoelastic liquid jet surrounded by a swirling air stream

A theoretical model is established to investigate the instability of a viscoelastic liquid jet with axisymmetric and non-axisymmetric disturbances, which is moving in a swirling air stream. The dispersion relation is derived by a temporal linear stability analysis. Results show that the three-dimensional viscoelastic liquid jet is more unstable than its Newtonian counterpart when considering the air swirl. The effects of air swirl strength, jet velocity, surface tension, liquid viscosity and gas density on the instability of viscoelastic jet surrounded by a swirling gas are analogous with the example of the Newtonian jet. Note that air swirl is also a stabilizing factor on the instability of the viscoelastic jet. The axisymmetric mode can prevail over the non-axisymmetric when the swirl strength is strong enough, while the non-axisymmetric mode is dominant in a liquid jet with a high liquid Weber number and a low liquid Reynolds number. It is also found that the maximum unstable growth rate of a viscoelastic liquid jet increases as liquid elasticity increases or the time constant ratio decreases. Furthermore, when air swirl velocity is introduced, it is the gas-to-liquid relative axial velocity that governs jet instability in axisymmetric disturbances, while the absolute gas axial velocity is another influencing factor in the non-axisymmetric mode. Finally, the competition between the gas rotating and axial velocities on jet instability is examined. In the case of a relatively smaller resultant gas velocity, the effect of the gas axial velocity can prevail over that of the air swirl when the gas velocity ratio is larger in the non-axisymmetric mode, whereas it can always exceed the effect of the air swirl in the axisymmetric mode. For a larger resultant gas velocity, the effect of the gas axial velocity is predominant only at a large gas velocity ratio in the axisymmetric mode.