We study the Doi model for suspensions of rod-like molecules. This model couples a microscopic Fokker-Planck type equation (the Smoluchowski equation) to the macroscopic Stokes equation. The Smoluchowski equation describes the evolution of the distribution of the rod orientation; it comes as a drift-diffusion equation on the sphere at every point in physical space. The macroscopic flow model is coupled to the microscopic description of rod orientations via an elastic stress. The drift term in the microscopic Smoluchowski equation depends on the macroscopic velocity gradient of the flow. Besides the interaction of the rod--like molecules with the flow an interaction between molecules is modeled. The coupled flow problem shows interesting phenomena. We will in particular discuss the spurt phenomena, which describes a sudden increase in the flow rate at a critical stress. Furthermore, we will consider the sedimentation in suspensions of rod-like molecules. We investigate this multiscale model using numerical methods that are based on an accurate approximation of the Smoluchowski equation. The numerical method can be applied more general for the approximation of PDEs on the sphere.