Abstract
The upper-convected Maxwell model of an elastic liquid is solved by means of the finite element method, using a penalty function approach. It is shown that the algorithm produces oscillations in the stresses in the presence of geometrical discontinuities, for increasing Deborah number De. The cause of this problem is explored and an indication of how to solve this problem is given.
The complete set of equations appears to be of the mixed hyperbolic-elliptic type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
8. References
M.J. Crochet and K. Walters, Numerical methods in non-Newtonian fluid mechanics, Ann. Rev. Fluid Mech. 15 (1983) 241–260.
F. Thomasset, Implementation of finite element methods for Navier-Stokes equations, Berlin, Springer (1981).
R.L. Sani, P.M. Gresho, R.L. Lee and D.F. Griffiths, The cause and cure (?) of the spurious pressures generated by certain FEM solutions of the incompressible Navier-Stokes equations, Part 1, Int. J. Numer. Meth. Fluids 1 (1981) 17–43.
R.L. Sani, P.M. Gresho, R.L. Lee, D.F. Griffiths, and M. Engelman, The cause and cure (!) of the spurious pressures generated by certain FEM solutions of the incompressible Navier-Stokes equations, Part 2, Int. J. Numer. Meth. Fluids 1 (1981) 171–204.
C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Computers & Fluids 1 (1973) 73–100.
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I, R.A.I.R.O. R-3 (1973) 33–76.
D.F. Griffiths, Finite elements for incompressible flow, Math. Meth. Appl. Sci. 1 (1979) 16–31.
M.J. Crochet, The flow of a Maxwell fluid around a sphere. In: R.H. Gallagher, D.H. Norrie, J.T. Oden and O.C. Zienkiewicz, (Eds.), Finite elements in fluids, Vol. IV. New York, Wiley (1982) Chapter 26.
V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Berlin, Springer (1979).
M.A. Mendelson, P.-W. Yeh, R.A. Brown and R.C. Armstrong, Approximation error in finite element calculation of viscoelastic fluid Flows, J. non-Newt. Fluid Mech. 10 (1982) 31–54.
G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963) 27–43.
M. Viriyayuthakorn and B. Caswell, Finite element simulation of viscoelastic flow, J. non-Newt. Fluid Mech. 6 (1980) 245–267.
B. Bernstein, M.K. Kadivar and D.S. Malkus, Steady flow of memory fluids with finite elements: two test problems, Computer Meth. Appl. Mech. & Eng. 27 (1981) 279–302.
B. Caswell, M. Viriyayuthakorn, Finite element simulation of die swell for a Maxwell fluid, J. non-Newt. Fluid Mech. 12 (1983) 13–29.
O. Hassager and C. Bisgaard, A Lagrange finite element method for the simulation of non-Newtonian liquids. J. non-Newt. Fluid Mech. 12 (1983) 153–164.
A. Segal, AFEP, a finite element package, Delft University of Technology, Report NA-14 (1975).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Van Der Zanden, J., Kuiken, G.D.C., Segal, A. et al. Numerical experiments and theoretical analysis of the flow of an elastic liquid of the upper-convected Maxwell type in the presence of geometrical discontinuities. Appl. Sci. Res. 42, 303–318 (1985). https://doi.org/10.1007/BF00384209
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00384209