Lopsided Rigid Dumbbell Rheology from Langevin Equation: A Graduate Tutorial

The modelling 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, which has been detailed elegantly by Bird et al. [Chapter 14 of $Dynamics \ of \ Polymeric \ Liquids$, Vol 2, Ed 2, (1987)]. In this tutorial, a straightforward approach for modelling a lopsided rigid dumbbell particle 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 established results from other methods.