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A Lagrangian panel method is presented for vortex sheet motion in threedimensional (3D) flow. The sheet is represented by a set of quadrilateral panels having a tree structure. The panels have active particles that carry circulation and passive particles used for adaptive refinement. The Biot-Savart kernel is regularized and the velocity is evaluated by a treecode. The method is applied to compute the azimuthal instability of a vortex ring, starting from a perturbed circular disc vortex sheet initial condition. Details of the core dynamics are clarified by tracking material lines on the sheet surface. Results are presented showing the following sequence of events: spiral roll-up of the sheet into a ring, wavy deformation of the ring axis, first collapse of the vortex core in each wavelength, second collapse of the vortex core out of phase with the first collapse, formation of loops wrapped around the core and radial ejection of ringlets. The collapse of the vortex core is correlated with converging axial flow.

We demonstrate the ability to simulate complex flows of entangled polymer melts using a high-fidelity slip-link model. Given the strong connections of the underlying molecular model to an atomistic description, nearly ab initio predictions of complex processing are feasible. Moreover, the method retains sufficient information which might allow extraction of detailed polymer chain conformations imposed by the flow. The macroscopic transport equations are solved using smoothed-particle hydrodynamics consistently with the stresses calculated using stochastic simulation of an ensemble of polymer chains in each particle. The polymer model uses only a single adjustable parameter whose value is determined from equilibrium stress relaxation. All other parameters are determined from atomistic simulation. Thereafter, nonlinear rheology predictions are made without any parameter adjustment. As a demonstration, we simulate journal-bearing flow of a moderately entangled polymer. Although the flows considered here are two dimensional, the required computational resources demonstrate that three dimensional flow calculations are accessible.

SUMMARYAccurately characterizing the forces acting on particles in fluids is of fundamental importance for understanding particle dynamics and binding kinetics. Conventional asymptotic solutions may lead to poor accuracy for neighboring particles. In this paper, we develop an accurate boundary integral method to calculate forces exerted on particles for a given velocity field. We focus our study on the fundamental two-bead oscillating problem in an axisymmetric frame. The idea is to exploit a correspondence principle between the unsteady Stokes and linear viscoelasticity in the Fourier domain such that a unifying boundary integral formulation can be established for the resulting Brinkman equation. In addition to the dimension reduction vested in a boundary integral method, our formulation only requires the evaluation of single-layer integrals, which can be carried out efficiently and accurately by a hybrid numerical integration scheme based on kernel decompositions. Comparison with known analytic solutions and existing asymptotic solutions confirms the uniform third-order accuracy in space of our numerical scheme.