Stability of a viscoelastic thread immersed in a confined region∗

In this paper, we consider an infinitely long viscoelastic thread in a tube filled with a Newtonian fluid. We apply Jeffreys model as the rheological relation for the thread. The thread moves due to a constant pressure gradient, but so slowly that the quasi-static creeping flow approximation is applicable. We investigate the effect of the ratio of viscosities, fluid elasticity, confinement and prescribed flow on the stability of the immersed thread. The stability is characterized by the growth rate of a small perturbation. The more viscous the thread is, the more time it takes to break up. A viscoelastic thread breaks up faster than a Newtonian one, and the wave number, which is responsible for the break-up, is smaller for a viscoelastic thread than for a Newtonian one. The latter implies that a viscoelastic thread will break up in larger droplets than a Newtonian one. The thread breaks up slower when the degree of confinement is higher. A critical gap width beyond which the presence of the wall of the tube has no longer an effect on the stability of the thread is found. In case of a Newtonian thread the prescribed flow only causes the thread to be oscillatory unstable with the growth rate equal to the one within a fluid at rest. Moreover, in case of a viscoelastic thread the prescribed flow contributes to both the real and the imaginary part of the growth rate of the perturbation. So, a viscoelastic thread will be oscillatory unstable and will break up faster than the one within a fluid at rest.