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

Abstract : The authors 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. The authors 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. (Modified author abstract)

[1]  Roger I. Tanner,et al.  End effects in falling-ball viscometry , 1963, Journal of Fluid Mechanics.

[2]  H. L. Weissberg.  End Correction for Slow Viscous Flow through Long Tubes , 1962 .

[3]  J. Smith,et al.  Low Reynolds number developing flows , 1969 .

[4]  George C. Lee,et al.  Derivation of stiffness matrices for problems in plane elasticity by Galerkin's method , 1969 .

[5]  R. T. Cheng,et al.  Numerical Solution of the Navier‐Stokes Equations by the Finite Element Method , 1972 .

[6]  J. Gavis.  Contribution of Surface Tension to Expansion and Contraction of Capillary Jets , 1964 .

[7]  M. Zidan,et al.  Zur Rheologie des Spinnprozesses , 1969 .

[8]  Computer Methods for Simulation of Multidimensional, Nonlinear, Subsonic, Incompressible Flow , 1972 .

[9]  S. Richardson,et al.  A ‘stick-slip’ problem related to the motion of a free jet at low Reynolds numbers , 1970, Mathematical Proceedings of the Cambridge Philosophical Society.

[10]  Simon L. Goren,et al.  The shape of low-speed capillary jets of Newtonian liquids , 1966, Journal of Fluid Mechanics.

[11]  S. Richardson.  The die swell phenomenon , 1970 .

[12]  S. Middleman,et al.  Expansion and Contraction of Capillary Jets of Newtonian Liquids , 1961 .

[13]  J. Batchelor,et al.  Die swell in elastic and viscous fluids , 1973 .

[14]  P. Hood,et al.  A numerical solution of the Navier-Stokes equations using the finite element technique , 1973 .

[15]  O. Zienkiewicz.  The Finite Element Method In Engineering Science , 1971 .

[16]  William E. Pracht,et al.  A numerical method for calculating transient creep flows , 1971 .

[17]  S. Orszag.  Numerical Simulation of Incompressible Flows Within Simple Boundaries. I. Galerkin (Spectral) Representations , 1971 .

[18]  A theoretical treatment of die swell in a Newtonian liquid , 1973 .

[19]  L. Herrmann.  Elasticity Equations for Incompressible and Nearly Incompressible Materials by a Variational Theorem , 1965 .

[20]  B. Finlayson.  Existence of Variational Principles for the Navier‐Stokes Equation , 1972 .

[21]  L. Kantorovich,et al.  Approximate methods of higher analysis , 1960 .

[22]  L. Scriven,et al.  Hydrodynamic Model of Steady Movement of a Solid / Liquid / Fluid Contact Line , 1971 .

[23]  D. Michael.  The separation of a viscous liquid at a straight edge , 1958 .

[24]  J. Tinsley Oden,et al.  Analysis of Flow of Viscous Fluids by the Finite-Element Method , 1972 .

[25]  J. Oden.  Finite-Element Analogue of Navier-Stokes Equation , 1970 .