Previous studies of slender viscoelastic and Newtonian free surface jets have focused mainly on steady state predictions, guided primarily by industrial textile applications. Dynamical analyses and simulations of fiber flows, even the linearized stability analysis of nontrivial steady states, have lagged behind considerable experimental and textile processing observations and advances. The foundational work of Chang and Lodge [Rheol. Acta., 10 (1971), pp. 448-449] and Petrie [Rheol. Acta., 14 (1975), pp. 955-957], (Elongational Flows, Pitman, London, 1979] has been extended by the axisymmetric modeling and simulation of Beris and Liu [J. Non-Newtonian Fluid Mech., 26 (1988), pp. 363-394], [Comput. Methods Appl. Mech. Engrg., 76 (1989), pp. 179-204], for Maxwell fluids, and Markovich and Renardy [J. Non-Newtonian Fluid Mech., 17 (1985), pp. 13-22] for Johnson-Segalman fluids. The model in [Markovich and Renardy] is a single, nonlinear parabolic equation in which elastic retardation provides dominant smoothing effects. The Maxwell slender jet model that is considered here, similar to the model in [J. Non-Newtonian Fluid Mech., 26 (1988), pp. 341-361], consists of a quasilinear system of four first-order partial differential equations in one space dimension (along the jet axis). The authors focus on hyperbolic behavior in the slender free surface flow dominated by surface tension, inertia, viscosity, elastic relaxation, and gravity. Analyses and numerical computations of the governing system of quasilinear partial differential equations are presented. Results presented consist of the following: a classification of all locally well-posed boundary conditions; boundary conditions relevant for a take-up fiber spinning simulation, computations of the draw ratio (where we define draw ratio as the ratio of take-up speed to initial speed of the filament) as a function of model parameters; classes of exact and numerical steady solutions together with linearized stability analyses, including temporal stability (to superimposed spatial perturbations) and spatial stability (to time-dependent boundary fluctuations); an upwind numerical algorithm for the full initial-boundary value problem of this 4 x 4 quasilinear hyperbolic system; dynamical nonlinear simulations in the neighborhood of the steady states to ascertain spatial and temporal stability, and, furthermore, to observe the onset and development of instabilities. These studies thereby yield robust slender jet structures and identify destabilizing parameter variations that should be realizable in an experiment tailored to these dominant balance effects. Moreover, transient modeling, developed on simplified academic models such as the present one, is a foundation for realistic industrial applications.
