Benchmarking solutions of the Folgar–Tucker-Equation and its reduction to a linear problem for non-linear closure forms

Abstract

The Folger-Tucker-Equation (FTE) arises from a modification of the Jeffery equation and describes the orientation of elongated particles in a flow by means of the orientation tensor A . Though the FTE represents the most widespread and commercially used relaxation equation for A in simulations of injection molding of short-fiber reinforced fluids, analytical solutions of the formulation applied in industry are hardly investigated. Previous work focused on the solution of the underlying (modified) Jeffery’s equation and its integration to the orientation tensor rather than a direct analytical solution of the FTE. The present paper firstly introduces a lemma that reduces nonlinear formulations of the FTE to linear problems. Its solution is numerically computational less intensive than the one of the original problems. Secondly, this paper presents analytical benchmarking solutions of the FTE to validate simulation algorithms. Thirdly, a closer look at two dimensional solutions enables a deeper understanding of the mathematical behaviour of this popular differential equation. The influence of rotational components in the velocity field on the orientation process is analysed. Finally an extension to the three dimensional case is discussed shortly. • We present analytical solutions to the Folgar-Tucker-Equation applied in industry. • The solutions provide an effective benchmark to validate numerical algorithms. • A lemma is proposed reducing nonlinear formulations of the FTE to linear problems. • The lemma reduces computational effort for its solution.