On the streamfunction–vorticity formulation in sliding bi-period frames: Application to bulk behavior for polymer blends

The Lees–Edwards description of bi-periodic boundary conditions has been extended to the streamfunction and streamfunction–vorticity formulation in sliding bi-periodic frames. The required compatibility conditions are formulated and uniqueness of the solution is shown. The model has been implemented in a spectral element method context to describe bulk shear behavior far away from walls, where no simple periodic boundary conditions can be used. In the numerical model a Lagrangian multiplier is introduced to couple the shearing boundaries. The proposed method has been validated for a mathematical test problem; convergence is shown and the influence of the order of approximation of the Lagrangian multiplier is studied. Finally, results are presented for drop coalescence across the boundaries of the bi-periodic frame.