Christina Papenfuss†, Maedeh Ranjbar‡
Abstract: During the mold-filling process, the fiber orientation in a flowing fiber suspension plays a crucial role in determining the material properties of the resulting fiber composite. A common modelling approach introduces the second-order orientation tensor governed by the Folgar–Tucker equation, which serves as the equation of motion, combined with an appropriate closure relation. In the present work, analytical solutions in three dimensions of the quadratically closed Folgar–Tucker equation are presented for a planar flow. The analysis reveals that the solution does not reduce to a purely two-dimensional orientation tensor. However, in the long-term limit, only two components of the orientation tensor remain independent—corresponding to the same number as in the two-dimensional case. A reconstruction of the orientation distribution function (ODF) further highlights the distinction between the fully three-dimensional orientation state and the effectively two- dimensional assumption in steady flow conditions. The analytical solution also shows that the steady-state orientation is highly sensitive to the shear rate. At very low shear rates the solution is nearly isotropic orientation tensor, whereas at very high shear rates it approaches a configuration that is strongly aligned in the velocity direction.
Keywords: Folgar-Tucker equation, analytical solution, planar flow, fiber suspension, orientation tensor, Couette flow.
MSC: 76T20, 34A05.
DOI 10.56082/annalsarscimath.2025.3.293
† Christina.Papenfuss@HTW-Berlin.de, Hochschule fur Technik und Wirtschaft Berlin, Wilhelminenhofstraße 75A, 12459 Berlin, Germany
‡ Maedeh.Ranjbar@htw-berlin.de, Hochschule fur Technik und Wirtschaft Berlin, 12459 Berlin, Germany
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 17 no 3, 2025
