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A numerical case study of a non‐Newtonian flow problem
Author(s) -
Webster M. F.,
Süli E. E.,
Morton K. W.
Publication year - 1988
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620260312
Subject(s) - mathematics , discretization , mathematical analysis , ordinary differential equation , partial differential equation , flow (mathematics) , independent equation , algebraic equation , differential equation , nonlinear system , geometry , physics , quantum mechanics
This paper addresses some of the theoretical aspects involved in the numerical study of non‐Newtonian flow problems. We consider the second‐order Rivlin–Erickson constitutive model due to the simple differential form that emerges for the system of equations that govern the flow when expressed in stream function–vorticity variables. This model describes slightly elastic fluids that exhibit a constant viscosity behaviour. A steady two‐dimensional flow is studied through a planar contraction geometry. An auxiliary variable is introduced into the problem formulation producing a non‐linear system of differential equations comprising two elliptic equations and one hyperbolic equation. This system is discretized by finite difference methods and the resulting system of non‐linear algebraic equations is solved iteratively by successive substitutions. The simple structure of this iteration permits a convergence analysis which is presented in Section 2. This analysis is performed prior to the spatial discretization and establishes the dependence of the iteration upon the material parameters. At the discrete linearized equation level a combination of inner iterations for elliptic equations and direct marching for the hyperbolic equation is used. The stability of the marching scheme is considered in Section 4.3 and a discussion on the results is given in Section 5.