In this paper we derive one space dimensional, reduced systems of equations (1-D closure models) for viscoelastic free jets. We begin with the three-dimensional system of conservation laws and a Maxwell-Jeffreys constitutive law for an incompressible viscoelastic fluid. First, we exhibit exact truncations to a finite, closed system of 1-D equations based on classical velocity assumptions of von Karman. Next, we demonstrate that the 3-D free surface boundary conditions overconstrain these truncated systems, so that only a very limited class of solutions exist. We then proceed to derive approximate 1-D closure theories through a slender jet asymptotic scaling, combined with appropriate definitions of velocity, pressure and stress unknowns. Our nonaxisymmetric 1-D slender jet models incorporate the physical effects of inertia, viscoelasticity (viscosity, relaxation and retardation), gravity, surface tension, and properties of the ambient fluid, and include shear stresses and time dependence. Previous special 1-D slender jet models correspond to the lowest order equations in the present asymptotic theory by an a posteriori suppression to leading order of some of these effects, and a reduction to axisymmetry. Solutions of the lowest order system of equations in this asymptotic analysis are presented: For the special cases of elliptical inviscid and Newtonian free jets, subject to the effects of surface tension and gravity, our model predicts oscillation of the major axis of the free surface elliptical cross section between perpendicular directions with distance down the jet, and drawdown of the cross section, in agreement with observed behavior. 15 refs.
