Bryan Quaife

High-volume fraction simulations of two-dimensional vesicle suspensions

We consider numerical algorithms for the simulation of the rheology of two-dimensional vesicles suspended in a viscous Stokesian fluid. The vesicle evolution dynamics is governed by hydrodynamic and elastic forces. The elastic forces are due to local inextensibility of the vesicle membrane and resistance to bending. Numerically resolving vesicle flows poses several challenges. For example, we need to resolve moving interfaces, address stiffness due to bending, enforce the inextensibility constraint, and efficiently compute the (non-negligible) long-range hydrodynamic interactions. Our method is based on the work of Rahimian et al. (2010) [33]. It is a boundary integral formulation of the Stokes equations coupled to the interface mass continuity and force balance. We extend the algorithms presented in that paper to increase the robustness of the method and enable simulations with concentrated suspensions. In particular, we propose a scheme in which both intra-vesicle and inter-vesicle interactions are treated semi-implicitly. In addition we use special integration for near-singular integrals and we introduce a spectrally accurate collision detection scheme. We test the proposed methodologies on both unconfined and confined flows for vesicles whose internal fluid may have a viscosity contrast with the bulk medium. Our experiments demonstrate the importance of treating both intra-vesicle and inter-vesicle interactions accurately.

Adaptive time stepping for vesicle suspensions

We present an adaptive high-order accurate time-stepping numerical scheme for the flow of vesicles suspended in Stokesian fluids. Our scheme can be summarized as an approximate implicit spectral deferred correction (SDC) method. Applying a fully-implicit SDC scheme to vesicle flows is prohibitively expensive. For this reason we introduce several approximations. Our scheme is based on a semi-implicit linearized low-order time stepping method. (Our discretization is spectrally accurate in space.) We expect that the accuracy can be arbitrary-order, but our examples suffer from order reduction which limits the observed accuracy to third-order. We also use invariant properties of vesicle flows, constant area and boundary length in two dimensions, to reduce the computational cost of error estimation for adaptive time stepping. We present results in two dimensions for single-vesicle flows, constricted geometry flows, converging flows, and flows in a Couette apparatus. We experimentally demonstrate that the proposed scheme enables automatic selection of the time step size and high-order accuracy.

On preconditioners for the Laplace double-layer in 2D

The discretization of the double-layer potential integral equation for the interior Dirichlet Laplace problem in a domain with smooth boundary results in a linear system that has a bounded condition number. Thus, the number of iterations required for the convergence of a Krylov method is, asymptotically, independent of the discretization size

Second kind integral equation formulation for the modified biharmonic equation and its applications

N

Cell-cell communication enhances bacterial chemotaxis toward external attractants

. Using the Fast Multipole Method (FMM) to accelerate the matrix-vector products, we obtain an optimal

Quantification of mixing in vesicle suspensions using numerical simulations in two dimensions

\mathcal{O}(N)

Integral equation methods for the Yukawa-Beltrami equation on the sphere

solver. In practice, however, when the geometry is complicated, the number of Krylov iterations can be quite large---to the extend that necessitates the use of preconditioning. We summarize the different methodologies that have appeared in the literature (single-grid, multigrid, approximate sparse inverses) and we propose a new class of preconditioners based on an FMM-based spatial decomposition of the double-layer operator. We present an experimental study in which we compare the different approaches and we discuss the merits and shortcomings of our approach. Our method can be easily extended to other second-kind integral equations with non-oscillatory kernels in two and three dimensions.

A boundary-integral framework to simulate viscous erosion of a porous medium

A system of Fredholm second kind integral equations (SKIEs) is constructed for the modified biharmonic equation in two dimensions with gradient boundary conditions. Such boundary value problem arises naturally when the incompressible Navier-Stokes equations are solved via a semi-implicit discretization scheme and the resulting boundary value problem at each time step is then solved using the pure stream-function formulation. The advantages of such an approach (Greengard and Kropinski, 1998) [14] are two fold: first, the velocity is automatically divergence free, and second, complicated (nonlocal) boundary conditions for the vorticity are avoided. Our construction, though similar to that of Farkas (1989) [10] for the biharmonic equation, is modified such that the SKIE formulation has low condition numbers for large values of the parameter. We illustrate the performance of our numerical scheme with several numerical examples. Finally, the scheme can be easily coupled with standard fast algorithms such as FFT, fast multipole methods (Greengard and Rokhlin, 1987) [15], or fast direct solvers (Ho and Greengard, 2012; Martinsson and Rokhlin, 2005) [17,25] to achieve optimal complexity, bringing accurate large-scale long-time fluid simulations within practical reach.

A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points

Bacteria are able to coordinate their movement, growth and biochemical activities through cell-cell communication. While the biophysical mechanism of bacterial chemotaxis has been well understood in individual cells, the role of communication in the chemotaxis of bacterial populations is not clear. Here we report experimental evidence for cell-cell communication that significantly enhances the chemotactic migration of bacterial populations, a finding that we further substantiate using numerical simulations. Using a microfluidic approach, we find that E. coli cells respond to the gradient of chemoattractant not only by biasing their own random-walk swimming pattern through the well-understood intracellular chemotaxis signaling, but also by actively secreting a chemical signal into the extracellular medium, possibly through a hitherto unknown communication signal transduction pathway. This extracellular signaling molecule is a strong chemoattractant that attracts distant cells to the food source. The observed behavior may represent a common evolved solution to accelerate the function of biochemical networks of interacting cells.

A New Family of Regularized Kernels for the Harmonic Oscillator

We study mixing in Stokesian vesicle suspensions in two dimensions on a cylindrical Couette apparatus using numerical simulations. The vesicle flow simulation is done using a boundary integral method, and the advection-diffusion equation for the mixing of the solute is solved using a pseudo-spectral scheme. We study the effect of the area fraction, the viscosity contrast between the inside (the vesicles) and the outside (the bulk) fluid, the initial condition of the solute, and the mixing metric. We compare mixing in the suspension with mixing in the Couette apparatus without vesicles. On the one hand, the presence of vesicles in most cases slightly suppresses mixing. This is because the solute can be only diffused across the vesicle interface and not advected. On the other hand, there exist spatial distributions of the solute for which the unperturbed Couette flow completely fails to mix whereas the presence of vesicles enables mixing. We derive a simple condition that relates the velocity and solute and can be used to characterize the cases in which the presence of vesicles promotes mixing.