A Galerkin finite element solution is developed for the flow of fiber suspensions. Primary variables are velocity, pressure, and a second-order tensor describing the fiber orientation. The model treats the orientation as three-dimensioanl, includes fiber-fiber interaction effects, and uses an orthotropic closure approximation. The flow and orientation are strongly coupled through an orientation-dependent constitutive equation. We explore the effects of this coupling on the fluid mechanics of fiber suspensions by studying three flows: an axisymmetric contraction, an axisymmetric expansion, and a center-gated disk. Coupling enhances the corner vortex in the contraction, in quantitative agreement with published experiments and calculations. The expansion results demonstrate that the aligned-fiber approximation is not valid for this flow. In the center-gated disk the effects of coupling are modest and are only noticeable near the center of the disk. This supports the use of decoupled models for injection molding in thin cavities.