Time-dependent fiber spinning equations. 2. Analysis of the stability of numerical approximations

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Abstract

The truncated time-dependent one-dimensional fiber spinning equations for an Upper Convected Maxwell model, presented in the previous paper [1], are approximated using various numerical methods. The eigenvalues and eigenfunctions of the linearized problem around the steady-state solution are calculated and compared against the analytical results for various values of Deborah and Reynolds numbers.

It is shown that upwind formulations of Finite-Difference and Finite-Element methods produce the most stable results. The most accurate eigenvalues and eigensolutions are those calculated by Pseudospectral methods. However, both Chebyshev and Legendre Pseudospectral methods become unstable at high values of De. Finally, it is shown that the boundary conditions have to be incorporated into the numerical scheme in a suitable way; spectacular instabilities might be generated otherwise.

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