Temporal stability of a viscoelastic immersed thread 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 fluid and thread move due to a constant pressure gradient, but this highly viscous flow is so slowly that the quasi-static creeping flow approximation is applicable (low-Reynolds number flow). 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 maximum growth rate of a random perturbation. The more viscous the thread is, the more time it takes to break up. A viscoelastic thread breaks up initially faster than a Newtonian one, and with smaller wave number. 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 surrounding 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 flow contributes to both the real and the imaginary parts of the growth rate. So, a viscoelastic thread will be oscillatory unstable. Furthermore, it breaks up faster than the one within a fluid at rest.