In this paper we derive one-space-dimensional, reduced systems of equations (one-dimensional 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 one-dimensional equations based on classical velocity assumptions of von Kármán (1921). Next, we demonstrate that the three-dimensional free-surface boundary conditions overconstrain these truncated systems, so that only a very limited class of solutions exist. We then proceed to derive approximate one-dimensional closure theories through a slender-jet asymptotic scaling, combined with appropriate definitions of velocity, pressure and stress unknowns. Our non-axisymmetric one-dimensional 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 one-dimensional 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. We thereby: (i) derive existing one-dimensional models from the three-dimensional free surface boundary-value problem; (ii) clarify the sense of the one-dimensional approximation; (iii) deduce new one-dimensional closure models for non-axisymmetric viscoelastic free jets.
Derivation of a one-dimensional von Kármán theory for viscoelastic ribbons
