The solution of viscous incompressible jet and free-surface flows using finite-element methods

TLDR

The paper develops a finite‑element program to solve incompressible, viscous free‑surface problems in steady axisymmetric or plane flows. The program uses a Galerkin formulation of the governing equations, enabling extension to non‑Newtonian, non‑zero Reynolds, and transient flows. The program successfully predicts a 13 % expansion of Newtonian creeping jets, matching experiments, and also provides solutions for stick‑slip and tube inlet problems with stress singularities.

Abstract

We discuss the creation of a finite-element program suitable for solving incompressible, viscous free-surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. We conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related ‘stick-slip’ problem and the tube inlet problem, both of which also contain stress singularities, are also given.