Rotational dynamics of a superhelix towed in a Stokes fluid

Motivated by the intriguing motility of spirochetes (helically-shaped bacteria that screw through viscous fluids due to the action of internal periplasmic flagella), we examine the fundamental fluid dynamics of superhelices translating and rotating in a Stokes fluid. A superhelical structure may be thought of as a helix whose axial centerline is not straight, but also a helix. We examine the particular case where these two superimposed helices have different handedness, and employ a combination of experimental, analytic, and computational methods to determine the rotational velocity of superhelical bodies being towed through a very viscous fluid. We find that the direction and rate of the rotation of the body is a result of competition between the two superimposed helices; for small axial helix amplitude, the body dynamics is controlled by the short-pitched helix, while there is a cross-over at larger amplitude to control by the axial helix. We find far better, and excellent, agreement of our experimental results with numerical computations based upon the method of Regularized Stokeslets than upon the predictions of classical resistive force theory

The orientation of swimming bi-flagellates in shear flows

Biflagellated algae swim in mean directions that are governed by their environments. For example, many algae can swim upward on average (gravitaxis) and toward downwelling fluid (gyrotaxis) via a variety of mechanisms. Accumulations of cells within the fluid can induce hydrodynamic instabilities leading to patterns and flow, termed bioconvection, which may be of particular relevance to algal bioreactors and plankton dynamics. Furthermore, knowledge of the behavior of an individual swimming cell subject to imposed flow is prerequisite to a full understanding of the scaled-up bulk behavior and population dynamics of cells in oceans and lakes; swimming behavior and patchiness will impact opportunities for interactions, which are at the heart of population models. Hence, better estimates of population level parameters necessitate a detailed understanding of cell swimming bias. Using the method of regularized Stokeslets, numerical computations are developed to investigate the swimming behavior of and fluid flow around gyrotactic prolate spheroidal biflagellates with five distinct flagellar beats. In particular, we explore cell reorientation mechanisms associated with bottom-heaviness and sedimentation and find that they are commensurate and complementary. Furthermore, using an experimentally measured flagellar beat for Chlamydomonas reinhardtii, we reveal that the effective cell eccentricity of the swimming cell is much smaller than for the inanimate body alone, suggesting that the cells may be modeled satisfactorily as self-propelled spheres. Finally, we propose a method to estimate the effective cell eccentricity of any biflagellate when flagellar beat images are obtained haphazardly

Sharp quadrature error bounds for the nearest-neighbor discretization of the regularized stokeslet boundary integral equation

The method of regularized stokeslets is a powerful numerical method to solve the Stokes flow equations for problems in biological fluid mechanics. A recent variation of this method incorporates a nearest-neighbor discretization to improve accuracy and efficiency while maintaining the ease-of-implementation of the original meshless method. This method contains three sources of numerical error, the regularization error associated from using the regularized form of the boundary integral equations (with parameter $\varepsilon$), and two sources of discretization error associated with the force and quadrature discretizations (with lengthscales $h_f$ and $h_q$). A key issue to address is the quadrature error: initial work has not fully explained observed numerical convergence phenomena. In the present manuscript we construct sharp quadrature error bounds for the nearest-neighbor discretisation, noting that the error for a single evaluation of the kernel depends on the smallest distance ($\delta$) between these discretization sets. The quadrature error bounds are described for two cases: with disjoint sets ($\delta>0$) being close to linear in $h_q$ and insensitive to $\varepsilon$, and contained sets ($\delta=0$) being quadratic in $h_q$ with inverse dependence on $\varepsilon$. The practical implications of these error bounds are discussed with reference to the condition number of the matrix system for the nearest-neighbor method, with the analysis revealing that the condition number is insensitive to $\varepsilon$ for disjoint sets, and grows linearly with $\varepsilon$ for contained sets. Error bounds for the general case ($\delta\geq 0$) are revealed to be proportional to the sum of the errors for each case.Comment: 12 pages, 6 figure

Efficient Implementation of Elastohydrodynamics via Integral Operators

The dynamics of geometrically non-linear flexible filaments play an important role in a host of biological processes, from flagella-driven cell transport to the polymeric structure of complex fluids. Such problems have historically been computationally expensive due to numerical stiffness associated with the inextensibility constraint, as well as the often non-trivial boundary conditions on the governing high-order PDEs. Formulating the problem for the evolving shape of a filament via an integral equation in the tangent angle has recently been found to greatly alleviate this numerical stiffness. The contribution of the present manuscript is to enable the simulation of non-local interactions of multiple filaments in a computationally efficient manner using the method of regularized stokeslets within this framework. The proposed method is benchmarked against a non-local bead and link model, and recent code utilizing a local drag velocity law. Systems of multiple filaments (1) in a background fluid flow, (2) under a constant body force, and (3) undergoing active self-motility are modeled efficiently. Buckling instabilities are analyzed by examining the evolving filament curvature, as well as by coarse-graining the body frame tangent angles using a Chebyshev approximation for various choices of the relevant non-dimensional parameters. From these experiments, insight is gained into how filament-filament interactions can promote buckling, and further reveal the complex fluid dynamics resulting from arrays of these interacting fibers. By examining active moment-driven filaments, we investigate the speed of worm- and sperm-like swimmers for different governing parameters. The MATLAB(R) implementation is made available as an open-source library, enabling flexible extension for alternate discretizations and different surrounding flows.Comment: 37 pages, 17 figure

Hydrodynamic propulsion of human sperm

The detailed fluid mechanics of sperm propulsion are fundamental to our understanding of reproduction. In this paper, we aim to model a human sperm swimming in a microscope slide chamber. We model the sperm itself by a distribution of regularized stokeslets over an ellipsoidal sperm head and along an infinitesimally thin flagellum. The slide chamber walls are modelled as parallel plates, also discretized by a distribution of regularized stokeslets. The sperm flagellar motion, used in our model, is obtained by digital microscopy of human sperm swimming in slide chambers. We compare the results of our simulation with previous numerical studies of flagellar propulsion, and compare our computations of sperm kinematics with those of the actual sperm measured by digital microscopy. We find that there is an excellent quantitative match of transverse and angular velocities between our simulations and experimental measurements of sperm. We also find a good qualitative match of longitudinal velocities and computed tracks with those measured in our experiment. Our computations of average sperm power consumption fall within the range obtained by other authors. We use the hydrodynamic model, and a prototype flagellar motion derived from experiment, as a predictive tool, and investigate how sperm kinematics are affected by changes to head morphology, as human sperm have large variability in head size and shape. Results are shown which indicate the increase in predicted straight-line velocity of the sperm as the head width is reduced and the increase in lateral movement as the head length is reduced. Predicted power consumption, however, shows a minimum close to the normal head aspect ratio