Kuzmin, DmitriWegener, Katharina Theresa2024-10-162024-10-162024http://hdl.handle.net/2003/42713http://dx.doi.org/10.17877/DE290R-24548In this work, a Fokker-Planck equation (FPE) is used to approximate the orientation distribution of fibers. FPEs combined with the Navier-Stokes equations (NSE) are widely used to predict the motion of the fibers in fiber suspension flows with low Reynolds numbers. The fibers align in response to the flow and randomize in response to fiber-fiber interactions. A precise formulation takes into account that the flow-fiber interaction is bilateral, so that the suspension rheology also depends on the fiber orientation. Various approaches to model fiber suspensions, including the well-known Folgar- Tucker equation, which relies on orientation tensors, are reviewed. We aim to solve the FPE using the continuous Galerkin method. For each point in the 3d physical space, an equation on the surface of a unit sphere representing the orientation states is solved, while for each point on the sphere an advection equation in the 3d physical space has to be solved. We handle this in the framework of an alternating direction approach including subtime stepping. Algebraic flux correction is performed for each equation to ensure positivity preservation as well as the normalization property of the distribution function. Numerical tests are performed for the individual subproblems. Finally, the velocity field is calculated by the incompressible Navier-Stokes equations (NSE), and benchmark problems for the coupled FPE-NSE system are solved. Thus, the relevance of this two-way coupling across the scales can be validated, and the effect of a different number of fibers is examined.enFiber suspensionsFokker-Planck equationFinite elementsAlternating-direction methodsPositivity preservation510TextFokker-Planck-GleichungNavier-Stokes-GleichungFinite-Elemente-MethodeFaserverstärkter KunststoffRheologie