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.

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Abstract.Several variational principles are derived for the initial-boundary-value problem of fully coupled linear thermoelasticity for an inhomogeneous, anisotropic continuum. A consistent set of field variables is employed and a method based on the Laplace transform is used to incorporate the initial conditions explicitly into the formulation. These principles lend themselves readily to numerical solutions based on an extended Ritz method.1. Introduction.The application of variational methods for both the unified development of the theory and for the approximate solution of fully coupled initial-boundaryvalue problems in linear thermoelasticity is not new. As a starting point, the work of Biot [1] introduced a variational principle in terms of a pair of vector-valued primary variables, the displacement of a material point and a variable which he termed the entropy displacement. The Euler equations of the principle are the thermoelastic equations of motion and the corrected heat conduction equation. A generalization of Biot's principle [2] incorporated the additional primary variables of the stress tensor and a thermal dis-equilibrium force conjugate to the entropy displacement. Euler equations representing the linear thermoelastic stress-strain relationship, the Fourier heat conduction law, and a relationship between the temperature and the thermal gradient, as well as the aforementioned field equations, are products of that variational principle.The most general variational statement of the coupled thermoelastic problem was made by and later by Ben-Amoz [4], both of whom obtain essentially the same Euler equations and natural boundary conditions. Both prescribe a boundary condition on the entropy displacement vector rather than on the heat flux vector, and Ben-Amoz obtains a sixth Euler equation, representing a relationship between the temperature and the thermal gradient. Impetus for further development of variational principles for the coupled thermoelastic problem is suggested by recent work of Gurtin [5], [6]. In these treatments of linear elastodynamics and transient heat conduction Gurtin, utilizing the operational methods of Mikusinski [7], explicitly introduces the initial conditions appropriate to the problem into the field equations and governing functionals, and derives alternate characterizations of the problems. The following work represents an extension of these concepts to the field theory of linear coupled thermoelasticity, and an attempt to remove

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