We introduce a numerical method based on an integral equation formulation for simulating drops in viscous fluids in the plane. It builds upon the method introduced by Kropinski in 2001, but improves on it by adding an interpolatory quadrature approach for handling near-singular integrals. Such integrals typically arise when drop boundaries come close to one another, and are difficult to compute accurately using standard quadrature rules. Adapting the interpolatory quadrature method introduced by Helsing and Ojala in 2008 to the current application, very general drop configurations can be handled while still maintaining stability and high accuracy. The performance of the new method is demonstrated by some challenging numerical examples.
We present a novel computational framework for simulating suspensions of rigid spherical Janus particles in Stokes flow. We show that long-range Janus particle interactions for a wide array of applications may be resolved using fast, spectrally accurate boundary integral methods tailored to polydisperse suspensions of spherical particles. These are incorporated into our rigid body Stokes platform. Our approach features the use of spherical harmonic expansions for spectrally accurate integral operator evaluation, complementarity-based collision resolution, and optimal O(n) scaling with the numb…
Boundary integral numerical methods are among the most accurate methods for interfacial Stokes flow, and are widely applied. They have the advantage that only the boundary of the domain must be discretized, which reduces the number of discretization points and allows the treatment of complicated interfaces. Despite their popularity, there is no analysis of the convergence of these methods for interfacial Stokes flow. In practice, the stability of discretizations of the boundary integral formulation can depend sensitively on details of the discretization and on the application of numerical filt…
We report on simulations of two-phase flows with deforming interfaces at various density contrasts by solving thermodynamically consistent Cahn-Hilliard Navier-Stokes equations. An (essentially) unconditionally energy-stable Crank-Nicolson-type time integration scheme is used. Detailed proofs of energy stability of the semi-discrete scheme and for the existence of solutions of the advective-diffusive Cahn-Hilliard operator are provided. In space we discretize with a conforming continuous Galerkin finite element method in conjunction with a residual-based variational multi-scale (VMS) approach …
A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integral…
Two-dimensional Stokes flow through a periodic channel is considered. The channel walls need only be Lipschitz continuous, in other words they are allowed to have corners. Boundary integral methods are an attractive tool for numerically solving the Stokes equations, as the partial differential equation can be reformulated into an integral equation that must be solved only over the boundary of the domain. When the boundary is at least C 1 smooth, the boundary integral kernel is a compact operator, and traditional Nyström methods can be used to obtain highly accurate solutions. In the case of Li…
We present a fast, high-order accurate and adaptive boundary integral scheme for solving the Stokes equations in complex---possibly nonsmooth---geometries in two dimensions. The key ingredient is a set of panel quadrature rules capable of evaluating weakly-singular, nearly-singular and hyper-singular integrals to high accuracy. Near-singular integral evaluation, in particular, is done using an extension of the scheme developed in J.~Helsing and R.~Ojala, {\it J. Comput. Phys.} {\bf 227} (2008) 2899--2921. The boundary of the given geometry is ``panelized'' automatically to achieve user-prescri…
Efficient calculation of the integral equation for simulating 2D TE scattering in a homogeneous medium using the Ewald method and a Gabor frame discretization
A highly accurate method for simulating surfactant-covered droplets in two-dimensional Stokes flow with solid boundaries is presented. The method handles both periodic channel flows of arbitrary shape and stationary solid constrictions. A boundary integral method together with a special quadrature scheme is applied to solve the Stokes equations to high accuracy, also for closely interacting droplets. The problem is considered in a periodic setting and Ewald decompositions for the Stokeslet and stresslet are derived. Computations are accelerated using the spectral Ewald method. The time evoluti…
The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary conditions are handled naturally, and the ill-conditioning caused by high order terms in the PDE is preconditioned analytically. Despite these advantages, the adoption of integral equation methods has been slow due to a number of difficulties in their implementation. This work describes a complete integral equation-based flow solver that builds on recently develo…
A framework is developed for a robust and highly accurate numerical solution of the coupled Stokes-Darcy system in three dimensions. The domain decomposition method is based on a Dirichlet-Neumann type splitting of the interface conditions and solving separate Stokes and Darcy problems iteratively. Second kind boundary integral equations are formulated for each problem. The integral equations use a smoothing of the kernels that achieves high accuracy on the boundary, and a straightforward quadrature to discretize the integrals. Numerical results demonstrate the convergence, accuracy, and depen…
Smoothed Particle Hydrodynamics (SPH) is a popular numerical technique developed for simulating complex fluid flows. Among its key ingredients is the use of nonlocal integral relaxations to local differentiations. Mathematical analysis of the corresponding nonlocal models on the continuum level can provide further theoretical understanding of SPH. We present, in this part of a series of works on the mathematics of SPH, a nonlocal relaxation to the conventional linear steady state Stokes system for incompressible viscous flows. The nonlocal continuum model is characterized by a smoothing length…
Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods produce accurate solutions and numerical convergence can be demonstrated using a resolution study, the rigorous convergence proofs are usually limited to particular reformulations and boundary conditions. In this paper, a rigorous error analysis of the marker and cell (MAC) scheme for Stokes interface problems with constant viscosity in the framework of the finite difference method is presented. Without reformulating the problem into elliptic PDEs, t…
A stabilized finite element method is introduced for the simulation of time-periodic creeping flows, such as those found in the cardiorespiratory systems. The new technique, which is formulated in the frequency rather than time domain, strictly uses real arithmetics and permits the use of similar shape functions for pressure and velocity for ease of implementation. It involves the addition of the Laplacian of pressure to the continuity equation with a complex-valued stabilization parameter that is derived systematically from the momentum equation. The numerical experiments show the excellent a…
In transient simulations of particulate Stokes flow, to accurately capture the interaction between the constituent particles and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the particles. Consequently, standard fast direct solvers lose their efficiency since the linear system changes at each time step. This manuscript presents a new computational approach that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, …
The boundary element method (BEM) is an efficient numerical method for simulating harmonic wave propagation. It uses boundary integral formulations of the Helmholtz equation at the interfaces of piecewise homogeneous domains. The discretisation of its weak formulation leads to a dense system of linear equations, which is typically solved with an iterative linear method such as GMRES. The application of BEM to simulating wave propagation through large-scale geometries is only feasible when compression and preconditioning techniques reduce the computational footprint. Furthermore, many different…
Every animal cell is filled with a cytoskeleton, a dynamic gel made of inextensible fibers, such as microtubules, actin fibers, and intermediate filaments, all suspended in a viscous fluid. Numerical simulation of this gel is challenging because the fiber aspect ratios can be as large as $10^4$. We describe a new method for rapidly computing the dynamics of inextensible slender filaments in periodically-sheared Stokes flow. The dynamics of the filaments are governed by a nonlocal slender body theory which we partially reformulate in terms of the Rotne-Prager-Yamakawa hydrodynamic tensor. To en…
Many problems in fluid dynamics are effectively modeled as Stokes flows - slow, viscous flows where the Reynolds number is small. Boundary integral equations are often used to solve these problems, where the fundamental solutions for the fluid velocity are the Stokeslet and stresslet. One of the main challenges in evaluating the boundary integrals is that the kernels become singular on the surface. A regularization method that eliminates the singularities and reduces the numerical error through correction terms for both the Stokeslet and stresslet integrals was developed in Tlupova and Beale, …
Boundary integral methods are highly suited for problems with complicated geometries, but require special quadrature methods to accurately compute the singular and nearly singular layer potentials that appear in them. This paper presents a boundary integral method that can be used to study the motion of rigid particles in three-dimensional periodic Stokes flow with confining walls. A centrepiece of our method is the highly accurate special quadrature method, which is based on a combination of upsampled quadrature and quadrature by expansion (QBX), accelerated using a precomputation scheme. The…
The paper aims at developing low-storage implicit Runge-Kutta methods which are easy to implement and achieve higher-order of convergence for both the velocity and pressure in the finite volume formulation of the incompressible Navier-Stokes equations on a static collocated grid. To this end, the effect of the momentum interpolation, a procedure required by the finite volume method for collocated grids, on the differential-algebraic nature of the spatially-discretized Navier-Stokes equations should be examined first. A new framework for the momentum interpolation is established, based on which…
