Simulation of flow-induced fiber orientation with a new closure model using the finite element method

Flow induced alignment of particle suspensions is of practical importance for various industries related to materials processing. The alignment of non-spherical particles influences directly the properties of the products and is linked to the flow of the material while manufacturing. For predicting flow induced alignment, it is possible to use a distribution function. This approach however is computationally intensive since it requires the solution of a complicated partial differential equation. Instead, an approach based on the moments of an orientation distribution function is commonly used. This approach requires a closure model since the equation governing the second moment of the orientation distribution function involves an unclosed fourth order tensor. The further development and validation of a closure model developed at Michigan State University has been pursued in this work. The performance of the model, and some variations of it, has been studied for various specified flow fields. Attempts have been made to identify a constant coefficient that provides satisfactory solutions for these flows. A finite element code has been developed to solve practical problems using this closure model along with other popular closure models. This involved solving an equation for the evolution of the second moment of the distribution function (the second order orientation tensor), coupled with the momentum and continuity equations, as well as with a constitutive model for the non-Newtonian stresses generated by the suspension. The stress developed due to presence of fibers has been accounted in the momentum equation for improving the accuracy of the predicted pressure and stresses in the computational domain. A variety of two dimensional problems are solved, including flow in a sudden compression, flow in an eccentric cylinder, and flow in diverging sections with various angles. The results from the simulations with the closure model are compared with the well-known quadratic and hybrid closure models. The model can predict quantities such as a first normal stress difference, unlike the quadratic closure model. The computational cost of these simulations is quite high, even only for two dimensional problems due to the large number of unknowns. This led to the development of a new method which combines finite element and analytical solutions for the reduction of computational time and memory by using Greens function. The analytical solution, based on Green's functions, can be calculated for a simple domain and combined with the finite element method to obtain a solution in the entire domain. This reduces effectively the computational time for large problems. This method has been applied to two-dimensional heat transfer, and fluid mechanics problems.