Sedimentation of a sphere near a vertical wall in an Oldroyd-B fluid

A code based on the distributed Lagrange multiplier/fictitious domain method (DLM) is used to study the motion of a sphere sedimenting in a viscoelastic liquid near a vertical wall. The viscoelastic liquid is assumed to be shear thinning and modeled by a shear-thinning Oldroyd-B model. Our simulations show that when the Deborah number based on the sphere velocity is O(1) and its initial position is sufficiently close to the wall, it moves towards the wall. This tendency of a sedimenting sphere to move closer to the vertical wall is enhanced by shear thinning, and also by an increase in the Deborah number. In a Newtonian liquid, on the other hand, the sphere moves away from the vertical wall and attains a steady position between the channel center and the wall. The sense of rotation of a sedimenting sphere when it is close to the vertical wall, for both Newtonian and viscoelastic liquids, is anomalous, i.e. the sphere rotates as if rolling up the wall. However, when the sphere is away from the wall the direction of rotation reverses. These results are in agreement with the experimental data reported in [D.D. Joseph et al., J. Non-Newtonian Fluid Mech. 54 (1994) 45-86; Y.J. Liu et al., J. Non-Newtonian Fluid Mech. 50 (1993) 305-329; D.D. Joseph et al., Theoretical and Applied Rheology, Elsevier, Amsterdam, 1992, pp. 60-65; D.L.E. Becker et al., J. Non-Newtonian Fluid Mech. 63 (1996) 45-86]. In two dimensions, on the other hand, simulations show that a sedimenting cylinder moves away from the wall in both Newtonian and viscoelastic liquids. These numerical results prove that the attraction between a wall and a particle sedimenting in a viscoelastic liquid is a three-dimensional effect, i.e. exists for a sphere but not for a cylinder, and it is enhanced by shear thinning.