Ricardo Cortez

The method of regularized Stokeslets in three dimensions: Analysis, validation, and application to helical swimming

The method of regularized Stokeslets is a Lagrangian method for computing Stokes flow driven by forces distributed at material points in a fluid. It is based on the superposition of exact solutions of the Stokes equations when forces are given by a cutoff function. We present this method in three dimensions, along with an analysis of its accuracy and performance on the model problems of flow past a sphere and the steady state rotation of rigid helical tubes. Predicted swimming speeds for various helical geometries are compared with experimental data for motile spirochetes. In addition, the regularized Stokeslet method is readily implemented in conjunction with an immersed boundary representation of an elastic helix that incorporates passive elastic properties as well as mechanisms of internal force generation.

The method of images for regularized Stokeslets

The image system for the method of regularized Stokeslets is developed and implemented. The method uses smooth localized functions to approximate a delta distribution in the derivation of the fluid flow due to a concentrated force. In order to satisfy zero-flow boundary conditions at a plane wall, the method of images derived for a standard (singular) Stokeslet is extended to give exact cancellation of the regularized flow at the wall. As the regularization parameter vanishes, the expressions reduce to the known images for singular Stokeslets. The advantage of the regularized method is that it gives bounded velocity fields even for isolated forces or for distributions of forces along curves. These are useful in the simulation of ciliary beats, flagellar motion, and particle suspensions. The expression relating force and velocity can be inverted to find the forces that generate a given velocity boundary condition. The latter is exemplified by modeling a cilium as a filament moving in a three-dimensional flow. The cilium velocity at various times is constructed from known data and used to determine the force field along the filament. Those forces can then reproduce the flow everywhere. The validity of the method is evaluated by computing the drag on a sphere moving near a wall. Comparisons with known expressions for the drag show that the method gives accurate results for spheres even within a distance from the wall equal to the surface discretization size.

Simulation of swimming organisms: coupling internal mechanics with external fluid dynamics

Problems in biological fluid dynamics typically involve the interaction of an elastic structure with its surrounding fluid. A unified computational approach, based on an immersed boundary framework, couples the internal force-generating mechanisms of organisms and cells with an external, viscous, incompressible fluid. Computational simulation, in conjunction with laboratory experiment, can provide valuable insight into complex biological systems that involve the interaction of an elastic structure with a viscous, incompressible fluid. This biological fluid-dynamics setting presents several more challenges than those traditionally faced in computational fluid dynamics - specifically, dynamic flow situations dominate, and capturing time-dependent geometries with large structural deformations is necessary. In addition, the shape of the elastic structures is not preset: fluid dynamics determines it. This article presents our recent progress on coupling the internal molecular motor mechanisms of beating cilia and flagella with an external fluid, as well as the three-dimensional (3D) undulatory swimming of nematodes and leeches. We expect these computational models to provide a testbed for examining different theories of internal force-generation mechanisms.

A Spatial Model of Mosquito Host-Seeking Behavior

Mosquito host-seeking behavior and heterogeneity in host distribution are important factors in predicting the transmission dynamics of mosquito-borne infections such as dengue fever, malaria, chikungunya, and West Nile virus. We develop and analyze a new mathematical model to describe the effect of spatial heterogeneity on the contact rate between mosquito vectors and hosts. The model includes odor plumes generated by spatially distributed hosts, wind velocity, and mosquito behavior based on both the prevailing wind and the odor plume. On a spatial scale of meters and a time scale of minutes, we compare the effectiveness of different plume-finding and plume-tracking strategies that mosquitoes could use to locate a host. The results show that two different models of chemotaxis are capable of producing comparable results given appropriate parameter choices and that host finding is optimized by a strategy of flying across the wind until the odor plume is intercepted. We also assess the impact of changing the level of host aggregation on mosquito host-finding success near the end of the host-seeking flight. When clusters of hosts are more tightly associated on smaller patches, the odor plume is narrower and the biting rate per host is decreased. For two host groups of unequal number but equal spatial density, the biting rate per host is lower in the group with more individuals, indicative of an attack abatement effect of host aggregation. We discuss how this approach could assist parameter choices in compartmental models that do not explicitly model the spatial arrangement of individuals and how the model could address larger spatial scales and other probability models for mosquito behavior, such as Lévy distributions.

Boundary integral solutions of coupled Stokes and Darcy flows

An accurate computational method based on the boundary integral formulation is presented for solving boundary value problems for Stokes and Darcy flows. The method also applies to problems where the equations are coupled across an interface through appropriate boundary conditions. The adopted technique consists of first reformulating the singular integrals for the fluid quantities as single and double layer potentials. Then the layer potentials are regularized and discretized using standard quadratures. As a final step, the leading term in the regularization error is eliminated in order to gain one more order of accuracy. The numerical examples demonstrate the increase of the convergence rate from first to second order and show a decrease in magnitude of the error. The coupled problems require the computation of the gradient of the Stokes velocity at the common interface. This boundary condition is also written as a combination of single and double layer potentials so that the same approach can be used to compute it accurately. Extensive numerical examples show the increased accuracy gained by the correction terms.

Modeling the dynamics of an elastic rod with intrinsic curvature and twist using a regularized Stokes formulation

We develop a Lagrangian numerical algorithm for an elastic rod immersed in a viscous, incompressible fluid at zero Reynolds number. The elasticity of the rod is described by a version of the Kirchhoff rod model, where intrinsic curvature and twist are prescribed, and the fluid is governed by the Stokes equations in R^3. The elastic rod is represented by a space curve corresponding to the centerline of the rod and an orthonormal triad, which encodes the bend and twist of the rod. In this method, the differences between the rod configuration and its intrinsic shape generate force and torque along the centerline. The coupling to the fluid is accomplished by the use of the method of regularized Stokeslets for the force and regularized rotlets for the torque. This technique smooths out the singularity in the fundamental solutions of the Stokes equations for the computation of the velocity of the rod centerline. In addition, the computation of the angular velocity of the rod requires the use of regularized (potential) dipoles. As a benchmark problem, we consider open and closed rods with intrinsic curvature and twist in a viscous fluid. Equilibrium configurations and dynamic instabilities are compared with known results in elastic rod theory. For cases when the exact solution is unknown, the numerical results are compared to those produced by the generalized immersed boundary (gIB) method, where the fluid is governed by the Navier-Stokes equations with small Reynolds number on a finite (periodic) domain. It is shown that the regularization method combined with Kirchhoff rod theory contributes substantially to the reduction of computation time and efficient memory usage in comparison to the gIB method. We also illustrate how the regularized method can be used to model microorganism motility where the organism is propelled by a flagellum propagating sinusoidal waves. The swimming speeds of this flagellum using the regularized Stokes formulation are matched well with classical asymptotic results of Taylors infinite cylinder in terms of frequency and amplitude of the undulation.

The dynamics of sperm detachment from epithelium in a coupled fluid-biochemical model of hyperactivated motility

Hyperactivation in mammalian sperm is characterized by a high-amplitude, asymmetric flagellar waveform. A mechanical advantage of this hyperactivated waveform has been hypothesized to be the promotion of flagellar detachment from oviductal epithelium. In order to investigate the dynamics of a free-swimming sperm׳s binding and escaping from a surface, we present an integrative model that couples flagellar force generation and a viscous, incompressible fluid. The elastic flagellum is actuated by a preferred curvature model that depends upon an evolving calcium profile along its length. In addition, forces that arise due to elastic bonds that form and break between the flagellar head and the surface are accounted for. As in recent laboratory experiments, we find that a hyperactive waveform does result in frequent detaching and binding dynamics that is not observed for symmetric flagellar beats. Moreover, we demonstrate that flagellar behavior depends strongly on the assumptions of the bond model, suggesting the need for more experimental investigation of the biochemistry of epithelial bonding and the shedding of binding proteins on the sperm head.

The action of waving cylindrical rings in a viscous fluid

Dinoflagellates ( Pfisteria piscicida ) are unicellular micro-organisms that swim due to the action of two eucaryotic flagella: a trailing, longitudinal flagellum that propagates planar waves and a transverse flagellum that propagates helical waves. Motivated by the wish to understand the role of the transverse flagellum in dinoflagellate motility, we study the fundamental fluid dynamics of a waving cylindrical tube wrapped into a closed helix. Given an imposed travelling wave on the structure, we determine that the helical ring propels itself in the direction normal to the plane of the circular axis of the helix. The magnitude of this translational velocity is proportional to the square of the helix amplitude. Additionally, the helical ring exhibits rotational motion tangential to its axis. These calculated swimming velocities are consistent when using the method of regularized Stokeslets with prescribed wave kinematics, regularized Stokeslets with dynamic forcing and Lighthills slender-body theory, except in cases where the slenderness parameter is not small. The translational velocity results are nearly indistinguishable using the three approaches, leading to the conjecture that the main contribution to this velocity at a cross-section is the far-field flow generated by the portion on the opposite side of the ring. The largest contribution to the rotational velocity at a cross-section comes from the cross-section itself and others nearby, thus the geometric details of the slender body have a larger effect on the results.

Stokesian peristaltic pumping in a three-dimensional tube with a phase-shifted asymmetry

Many physiological flows are driven by waves of muscular contractions passed along a tubular structure. This peristaltic pumping plays a role in ovum transport in the oviduct and in rapid sperm transport through the uterus. As such, flow due to peristalsis has been a central theme in classical biological fluid dynamics. Analytical approaches and numerical methods have been used to study flow in two-dimensional channels and three-dimensional tubes. In two dimensions, the effect of asymmetry due to a phase shift between the channel walls has been examined. However, in three dimensions, peristalsis in a non-axisymmetric tube has received little attention. Here, we present a computational model of peristaltic pumping of a viscous fluid in three dimensions based upon the method of regularized Stokeslets. In particular, we study the flow structure and mean flow in a three-dimensional tube whose asymmetry is governed by a single phase-shift parameter. We view this as a three-dimensional analog of the phase-shifted two-dimensional channel. We find that the maximum mean flow rate is achieved for the parameter that results in an axisymmetric tube. We also validate this approach by comparing our computational results with classical long-wavelength theory for the three-dimensional axisymmetric tube. This computational framework is easily implemented and may be adapted to more comprehensive physiological models where the kinematics of the tube walls are not specified a priori, but emerge due to the coupling of its passive elastic properties, force generating mechanisms, and the surrounding viscous fluid.

A general system of images for regularized Stokeslets and other elements near a plane wall

We derive a general system of images for regularized sources, Stokeslets, and other related elements starting from an arbitrary regularization kernel (blob) used in the simulation of Stokes flows in three dimensions bounded by a plane. This generalizes previous work in which the image system for a Stokeslet had been derived for one specific blob. The significance of this generalization is that recent work on regularization methods requires the use of blobs designed to satisfy certain properties, such as zero moment conditions and fast decay, and thus it is absolutely necessary to have the system of images starting from an arbitrary blob. The system of images for a regularized element consists of a set of several elements, usually of higher order, that produce a flow that is zero at the bounding plane. In order for the resultant flow to vanish analytically at the wall, two different but related blobs must be used. For any given blob, we provide the formula for the companion blob that accomplishes the cancellation and we derive a systematic way to compute the image system of regularized Stokeslets, sources and dipoles. Other elements can be derived from these. By taking the limit as the regularization parameter approaches zero, the system of images for the corresponding singular elements is found.