A weak-coupling expansion applied to the motion of a set- tling rod in a viscoelastic fluid

Viscoelastic fluids are fluids that, in addition to having the property of viscosity, also have the property of elasticity. This is often the result of the introduction of a polymeric material to a non-viscoelastic fluid— the presence of these materials greatly affects the behavior of the fluid by introducing additional internal stresses. An example of this— and the focus of this paper— is the dynamics of a settling rod in a viscoelastic fluid. In a Newtonian fluid, a rod settling under gravity adopts a horizontal orientation; whereas, in a viscoelastic fluid, the same rod would adopt a vertical orientation. In this paper, we discuss a number of theoretical and computational models toward the goal of demonstrating the stability of the vertical orientation of the rod in a viscoelastic fluid. The most important of these techniques is the weakcoupling expansion. In this technique, we consider the elastic stresses and Newtonian stresses in the fluid to be coupled by a parameter, β. Considering this to be small, we expand the equations of motion for the rod in terms of β, allowing for an efficient solution of these equations.