A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flows

Abstract

A distributed Lagrange multiplier/fictitious domain method (DLM) is developed for simulating the motion of rigid particles suspended in the Oldroyd-B fluid. This method is a generalization of the one described in [R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph, Int. J. Multiphase Flows 25 (1998) 755–794] where the motion of particles suspended in a Newtonian fluid was simulated. In our implementation of the DLM method, the fluid–particle system is treated implicitly using a combined weak formulation in which the forces and moments between the particles and fluid cancel. The governing equations for the Oldroyd-B liquid are solved everywhere, including inside the particles. The flow inside the particles is forced to be a rigid body motion by a distribution of Lagrange multipliers. We use the Marchuk–Yanenko operator-splitting technique to decouple the difficulties associated with the incompressibility constraint, nonlinear convection and viscoelastic terms. The constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor, while the convection term in the constitutive equation is discretized using a third-order upwinding scheme. The nonlinear convection problem is solved using a least square conjugate gradient algorithm, and the Stokes-like problem is solved using a conjugate gradient algorithm. The code is verified performing a convergence study to show that the results are independent of the mesh and time step sizes. Our simulations show that, when particles are dropped in a channel, and the viscoelastic Mach number ( M) is less than 1 and the elasticity number ( E) is greater than 1, the particles chain along the flow direction; this agrees with the results presented in [P.Y. Huang, H.H. Hu, D.D. Joseph, J. Fluid Mech. 362 (1998) 297–325]. In our simulations of the fluidization of 102 particles in a two-dimensional bed, we find that the particles near the channel walls form chains that are parallel to the walls, but the distribution of particles away from the walls is relatively random.