Abstract. This paper presents a new boundary integral equation (BIE) method for simulating particulate and multiphase flows through periodic channels of arbitrary smooth shape in two dimensions. The authors consider a particular system-multiple vesicles suspended in a periodic channel of arbitrary shape-to describe the numerical method and test its performance. Rather than relying on the periodic Green's function as classical BIE methods do, the method combines the free-space Green's function with a small auxiliary basis, and imposes periodicity as an extra linear condition. As a result, we can exploit existing free-space solver libraries, quadratures, and fast algorithms, and handle a large number of vesicles in a geometrically complex channel. Spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures, while a first-order semi-implicit scheme evolves particles by treating the vesicle-channel interactions explicitly. New constraint-correction formulas are introduced that preserve reduced areas of vesicles, independent of the number of time steps taken. By using two types of fast algorithms, (i) the fast multipole method (FMM) for the computation of the vesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) a fast direct solver for the BIE on the fixed channel geometry, the computational cost is reduced to O(N ) per time step where N is the spatial discretization size. Moreover, the direct solver inverts the wall BIE operator at t = 0, stores its compressed representation and applies it at every time step to evolve the vesicle positions, leading to dramatic cost savings compared to classical approaches. Numerical experiments illustrate that a simulation with N = 128,000 can be evolved in less than a minute per time step on a laptop.Key words. Stokes flow, periodic geometry, spectral methods, boundary integral equations, fast direct solvers 1. Introduction. Suspensions of rigid and/or deformable particles in viscous fluids flowing through confined geometries are ubiquitous in natural and engineering systems. Examples include drop, bubble, vesicle, swimmer, and red blood cell (RBC) suspensions. Understanding the spatial distribution of such particles in confined flows is crucial in a wide range of applications including targeted drug delivery [30], enhanced oil recovery [44], and microfluidics for cell sorting and separation [31]. In several of these applications, the long-time behavior of the suspension is sought. For example: What is the optimal size and shape of targeted drug carriers that maximizes their ability to reach the vascular walls escaping from flowing RBCs [30,15]? What is the optimal design of a microfluidic device that differentially separates circulating tumor cells from blood cells [49]? More generally, one is interested in estimating the rheological properties of a given particulate suspension in an applied flow, electric, or magnetic fields. A common mathematical construct that is employed in such a scenario is the periodicity of flow at...

We study the fully nonlinear, nonlocal dynamics of two-dimensional multicomponent vesicles in a shear flow with matched viscosity of the inner and outer fluids. Using a nonstiff, pseudo-spectral boundary integral method, we investigate dynamical patterns induced by inhomogeneous bending for a two phase system. Numerical results reveal that there exist novel phase-treading and tumbling mechanisms that cannot be observed for a homogeneous vesicle. In particular, unlike the well-known steady tank-treading dynamics characterized by a fixed inclination angle, here the phase-treading mechanism leads to unsteady periodic dynamics with an oscillatory inclination angle. When the average phase concentration is around 1/2, we observe tumbling dynamics even for very low shear rate, and the excess length required for tumbling is significantly smaller than the value for the single phase case. We summarize our results in phase diagrams in terms of the excess length, shear rate, and concentration of the soft phase. These findings go beyond the well known dynamical regimes of a homogeneous vesicle and highlight the level of complexity of vesicle dynamics in a fluid due to heterogeneous material properties.

We present a spectrally accurate scheme to turn a boundary integral formulation for an elliptic PDE on a single unit cell geometry into one for the fully periodic problem. The basic idea is to use a small least squares solve to enforce periodic boundary conditions without ever handling periodic Green's functions. We describe fast solvers for the two-dimensional (2D) doubly periodic conduction problem and Stokes nonslip fluid flow problem, where the unit cell contains many inclusions with smooth boundaries. Applications include computing the effective bulk properties of composite media (homogenization) and microfluidic chip design.We split the infinite sum over the lattice of images into a directly summed "near" part plus a small number of auxiliary sources that represent the (smooth) remaining "far" contribution. Applying physical boundary conditions on the unit cell walls gives an expanded linear system, which, after a rank-1 or rank-3 correction and a Schur complement, leaves a well-conditioned square system that can be solved iteratively using fast multipole acceleration plus a low-rank term. We are rather explicit about the consistency and nullspaces of both the continuous and discretized problems. The scheme is simple (no lattice sums, Ewald methods, or particle meshes are required), allows adaptivity, and is essentially dimension-and PDE-independent, so it generalizes without fuss to 3D and to other elliptic problems. In order to handle close-to-touching geometries accurately we incorporate recently developed spectral quadratures. We include eight numerical examples and a software implementation. We validate against highaccuracy results for the square array of discs in Laplace and Stokes cases (improving upon the latter), and show linear scaling for up to 10 4 randomly located inclusions per unit cell.

A multiscale continuum model is constructed for a mechanosensitive (MS) channel gated by tension in a lipid bilayer membrane under stresses due to fluid flows. We illustrate that for typical physiological conditions vesicle hydrodynamics driven by a fluid flow may render the membrane tension sufficiently large to gate a MS channel open. In particular, we focus on the dynamic opening/ closing of a MS channel in a vesicle membrane under a planar shear flow and a pressure-driven flow across a constriction channel. Our modeling and numerical simulation results quantify the critical flow strength or flow channel geometry for intracellular transport through a MS channel. In particular, we determine the percentage of MS channels that are open or closed as a function of the relevant measure of flow strength. The modeling and simulation results imply that for fluid flows that are physiologically relevant and realizable in microfluidic configurations stress-induced intracellular transport across the lipid membrane can be achieved by the gating of reconstituted MS channels, which can be useful for designing drug delivery in medical therapy and understanding complicated mechanotransduction. M echanosensitive (MS) channels are essential to mechanosensation and mechanotransduction in a wide range of cells (1-3). Due to the great diversity of MS channels, the general gating mechanism is found to depend on combinations of the detailed molecular structures (4-7), the gating-associated conformational changes (8-10), and coupling with the lipid bilayer membrane (11)(12)(13)(14). It remains a challenge to elucidate general mechanisms underpinning the gating of MS channels. In this spirit it is useful to have "simple" models to understand the complex response of MS channels and their associated biological functions (15).Stretch-activated (SA) channels are a class of (relatively) simpler MS channels that are stretched open mainly by membrane tension (e.g., due to osmotic shock, stress from fluid flow, or other mechanical sources of tension) for nonselective intracellular transport of ions and macromolecules (16)(17)(18)(19). Their gating mechanisms have been investigated by experiments (18), continuum modeling (20, 21), and molecular dynamics (MD) simulations (22). By exerting an unphysiologically large load onto a membrane patch with a single SA channel in the center, MD simulations show that a SA channel (several nanometers in size) responds to membrane tension within a few nanoseconds (22). Due to computational limitations, MD simulations are restricted to a small lipid patch and a short timescale (approximately microseconds). On the other hand, continuum modeling has been adopted widely in recent studies where the transduction of membrane tension to SA channels is found to depend on the molecular details of lipid binding in the channels (13). For example, the channel gating was shown to depend on the protein surface charge and hydrophobicity (23).Novel technological advancements in microfluidics have made it possible to construct...

Using a rod theory formulation, we derive equations of state for a thin elastic membrane subjected to several different boundary conditions-clamped, simply supported, and periodic. The former is applicable to membranes supported on a softer substrate and subjected to uniaxial compression. We show that a wider family of quasistatic equilibrium shapes exist beyond the previously obtained analytical solutions. In the latter case of periodic membranes, we were able to derive exact solutions in terms of elliptic functions. These equilibria are verified by considering a fluid-structure interaction problem of a periodic, length-preserving bilipid membrane modeled by the Helfrich energy immersed in a viscous fluid. Starting from an arbitrary shape, the membrane dynamics to equilibrium are simulated using a boundary integral method.