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

Published online by Cambridge University Press:  29 March 2006

R. E. Nickell ,
R. I. Tanner  and
B. Caswell
R. E. Nickell
Affiliation:
Division of Engineering, Brown University
R. I. Tanner
Affiliation:
Division of Engineering, Brown University
B. Caswell
Affiliation:
Division of Engineering, Brown University
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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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 65 , Issue 1 , 12 August 1974 , pp. 189 - 206
Copyright
© 1974 Cambridge University Press

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References

Atkinson, B., Brocklebank, M. P., Card, C. C. H. & Smith, J. M. 1969 Low Reynolds number developing flows A.I.Ch.E. J. 15, 548.Google Scholar
Atkinson, B., Card, C. C. & Irons, B. 1970 Application of the finite element method to creeping flow problems. Trans. Inst. Chem. Eng. 48, T 276.Google Scholar
Batchelor, J. & Horsfall, F. 1971 Die swell in elastic and viscous fluids. Rubber and Plastics Res. Assoc. of Great Britain, Res. Rep. 189.Google Scholar
Chan, S. T. K. & Larock, B. E. 1973 Fluid flows from axisymmetric orifices and values Proc. A.S.C.E., J. Hydr. Div. 99, 81.Google Scholar
Cheng, R. T. 1972 Numerical solution of the Navier–Stokes equations by the finite element method Phys. Fluids, 15, 2098.Google Scholar
Felippa, C. A. 1966 Refined finite element analysis of linear and nonlinear two-dimensional structures. University of California at Berkeley, Rep. SESM 66–22.
Finlayson, B. A. 1972 Existence of variational principles for the Navier–Stokes equation Phys. Fluids, 15, 6.Google Scholar
Fox, D. G. & Deardorff, J. W. 1972 Computer methods for simulation of multidimensional nonlinear, subsonic, incompressible flow. A.S.M.E. Paper, 72-HT-61.Google Scholar
Gavis, J. 1964 Contributions of surface tension of expansion and contraction of capillary jets Phys. Fluids, 7, 1097.Google Scholar
Goren, S. L. & Wronski, S. 1966 The shape of low-speed capillary jets of Newtonian liquids J. Fluid Mech. 25, 185.Google Scholar
Herrmann, L. R. 1965 Elasticity equations for incompressible and nearly incompressible materials by a variational theorem A.I.A.A. J. 3, 1896.Google Scholar
Horsfall, F. 1971 A theoretical treatment of die swell in a Newtonian liquid. Rubber and Plastics Res. Assoc. of Great Britain, Res. Rep. no. 192.Google Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid liquid fluid contact line J. Coll. Interface Sci. 35, 85.Google Scholar
Kantorovich, L. V. & Krylov, V. I. 1964 Approximate Methods of Higher Analysis. Wiley.
Lodge, A. S. 1964 Elastic Liquids. Academic.
Michael, D. H. 1958 The separation of a viscous liquid at a straight edge Mathematika, 5, 82.Google Scholar
Middleman, S. & Gavis, J. 1961 Expansion and contraction of capillary jets of Newtonian liquids Phys. Fluids, 4, 355.Google Scholar
Oden, J. T. 1970 Finite element analogue of Navier–Stokes equation Proc. A.S.C.E., J. Engng Mech. Div. 96, 529.Google Scholar
Oden, J. T. & Wellford, L. C. 1972 Analysis of flow of viscous fluids by the finite-element method A.I.A.A. J. 10, 1590.Google Scholar
Olson, M. D. 1972 A variational finite element method for two-dimensional steady viscous flows. Proc. Specialty Conf. on Finite Element Method in Civil Engineering, Engineering Institute of Canada, Montreal, p. 585.
Orszag, S. A. 1971 Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral representation) Studies in Appl. Math. 50, 293.Google Scholar
Pracht, W. E. 1971 A numerical method for calculating transient creep flows J. Comp. Phys. 7, 46.Google Scholar
Richardson, S. 1970a The die-swell phenomenon Rheol. Acta, 9, 193.Google Scholar
Richardson, S. 1970b A ‘stick–slip’ problem related to the motion of a free jet of a low Reynolds numbers. Proc. Camb. Phil. Soc. 67, 477.Google Scholar
Schkade, A. F. 1970 A refined axisymmetric finite element method for the analysis of nearly incompressible solids. Ph.D. dissertation, University of Texas at Austin.
Szabo, B. A. & Lee, G. C. 1969 Derivation of stiffness matrices for problems in plane elasticity by Galerkin's method Int. J. Num. Meth. Eng. 1, 301.Google Scholar
Tanner, R. I. 1963 End effects in falling-ball viscometry J. Fluid Mech. 17, 161.Google Scholar
Taylor, C. & Hood, P. 1973 A numerical solution of the Navier–Stokes equations using the finite element technique Computers and Fluids, 1, 73.Google Scholar
Thompson, E. G., Mack, L. R. & Lin, F. S. 1969 Finite-element method for incompressible slow viscous flow with a free surface Developments in Mechanics, 5, 93.Google Scholar
Waldron, K. J. 1966 M.S. thesis, University of Sydney.
Weissberg, H. L. 1962 End correction for slow viscous flow through long tubes Phys. Fluids, 5, 1033.Google Scholar
Zidan, M. 1969 Zur Rheologie des Spinnprozesses Rheol. Acta, 8, 89.Google Scholar
Zienkiewicz, O. C. 1971 The Finite Element Method in Engineering Science. McGraw-Hill.
Zienkiewicz, O. C. & Taylor, C. 1972 Weighted residual processes in F.E.M. with particular reference to some transient and coupled problems. Lectures for NATO Advanced Study Institute.