Numerical study of spheres settling in Oldroyd-B fluids

In order to simulate the motion of balls settling in Oldroyd-B fluids, we have generalized to three dimensions the distributed Lagrange multiplier based fictitious domain method we developed in the study by Hao et al. ["A fictitious domain/distributed Lagrange multiplier method for the particulate flow of Oldroyd-B fluids: A positive definiteness preserving approach," J. Non-Newtonian Fluid Mech. 156, 95 (2009)] for viscoelastic particulate flow in two-dimensional channels. The effect of the fluid elastic number and that of the density ratio of the particle and fluid on the chaining of balls have been studied. For the cases of two balls released side-by-side at high elasticity numbers, the two balls attract each other first and then form a chain, and such a chain settles vertically. But at low elasticity numbers, the two balls either stay separate and interact periodically or attract each other, turn, and separate periodically. At high blockage ratios, a stronger wall effect enhances the formation of two ball chains. For the cases of three balls released side-by-side, the ball interaction is slightly more complicated due to its non-symmetrical initial configuration. At high elasticity numbers, either a three ball chain settles vertically or the leading two balls form a chain which leaves the third ball behind. But at low elasticity numbers considered in this article, only the leading two balls form a chain. For the vertical initial configuration, a three ball chain can be obtained at higher elasticity numbers and also the heavier balls can form a vertical chain easily.