Lopsided rigid dumbbell rheology from Langevin equation: A graduate tutorial

The modeling of symmetric rigid dumbbell particles suspended in a Newtonian fluid, as a model of a rigid-rod polymeric solution, has been accomplished exclusively through the diffusion equation, detailed elegantly by Bird et al. [Dynamics of Polymeric Liquids: Kinetic Theory, 2nd ed. (Wiley, NY, 1987), Vol. 2]. In this Tutorial, a straightforward approach for modeling a lopsided rigid dumbbell particle in suspension is presented by the Langevin analysis. The connector force between the dumbbell beads is obtained through the rigidity constraint of the center-to-center vector of the dumbbell using its Langevin equation. By directly averaging via the Langevin equation, the evolution of the center-to-center vector and the configuration tensor are derived. The stress expressions for the dumbbell from the Langevin equation and the diffusion equation for the orientation distribution function of the center-to-center vector of the dumbbell are also derived, and the final expressions agree with the established results from other methods.