Exact solutions of the Boeder differential equation for macromolecular orientations in a flowing liquid

The Boeder differential equation is solved in this work over a wide range of $\alpha$, yielding the probability density functions (PDF), that describe the average orientations of rod-like macromolecules in a flowing liquid. The quantity $\alpha$ is the ratio of the hydrodynamic shear rate to the rotational diffusion coefficient. It characterises the coupling of the motion of the macromolecules in the hydrodynamic flow to their thermal diffusion. Previous analytical work is limited to approximate solutions for small values of $\alpha$. Special analytical as well as numerical methods are developed in the present work in order to calculate accurately the PDF for a range of $\alpha$ covering several orders of magnitude, $10^{-6} \le \alpha \le 10^{8}$. The mathematical nature of the differential equation is revealed as a singular perturbation problem when $\alpha$ becomes large. Scaling results are obtained over the differential equation for $\alpha \ge 10^{3}$. Monte Carlo Brownian simulations are also constructed and shown to agree with the numerical solutions of the differential equation in the bulk of the flowing liquid, for an extensive range of $\alpha$. This confirms the robustness of the developed analytical and numerical methods.