A fictitious domain/distributed Lagrange multiplier method for the particulate flow of Oldroyd-B fluids: A positive definiteness preserving approach

In this article we present a numerical method for simulating the sedimentation of circular particles in a two-dimensional channel filled with an Oldroyd-B fluid. We have combined a fictitious domain/distributed Lagrange multiplier method with a factorization approach from Lozinski and Owens [J. Non-Newtonian Fluid Mech. 112 (2003) 161] via an operator splitting technique. The new scheme preserves the positive definiteness of the conformation tensor at the discrete level. The method is validated by performing a convergence study which shows that the results are independent of the mesh and time step sizes. Our results show that when the elasticity number (E) is less than a critical value (which depends upon the blockage ratio), two particles will sediment in the channel-like particles in Newtonian fluids; when the elasticity number is greater than the critical value, chains are formed for the case of two particles sedimenting in an Oldroyd-B fluid and the center line is aligned with the falling direction. These results agree with those presented in [P.Y. Huang, H.H. Hu, and D.D. Joseph, J. Fluid Mech. 362 (1998) 297]. For the cases of three and six particles, when the elasticity number is greater than a critical value and the viscoelastic Mach number is less than one, chains are also formed and move to the center of the channel.