# On the numerical modeling of an oldroyd-type constitutive equation

- Authors
- Journal
- Journal of Computational Physics 0021-9991
- Publisher
- Elsevier
- Publication Date
- Volume
- 19
- Issue
- 4
- Identifiers
- DOI: 10.1016/0021-9991(75)90069-8

## Abstract

Abstract A one-dimensional coupled set of equations consisting of appropriate forms of an equation of motion and a three-constant Oldroyd constitutive equation, used in the description of certain non-Newtonian flows, has been modeled using a finite-difference technique. These one-dimensional equations may be applicable along the centerline of a symmetric almost parallel flow field, where extensive molecular stretching occurs, with the restrictions of zero pressure gradient and zero shear. Mainly, this article differs from previous numerical work on these equations, such as Townsend's work on a four-constant Oldroyd equation, in that the nonlinear convection terms, which did not appear in this earlier work due to the parallel flow being considered, are now present. In comparison with the full set of equations, it is seen that the structure of the motion and constitutive equation used retain the essential characteristics of the complete set. In the equations, the time and spatial derivatives of both velocity and stress were centered so that in the Newtonian limit of zero relaxation and retardation times, the motion equation would reduce to Burgers' equation with the diffusion term time lagged. Stability of the differencing scheme for the constitutive equation dictated that the convective derivative of the deformation rate be time lagged with respect to the convective derivative of the stress. A von Neumann stability analysis was performed on the model equation yielding restrictions on Δt based on the usual viscous condition and, in addition, on the convective inertial and elastic propagation velocities, and also on the time scale of the straining motion. As a test for the proposed scheme an initial-boundary-value problem was formulated. An analytic steady-state solution to the problem can be obtained in Lagranian coordinates if a constant velocity and a linear stress-deformation rate condition are assumed at the inflow to the flow field. This analytic solution, expressed in Eulerian coordinates, was then used as a check for the iterated steady-state solution.

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