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20 Years of Regularized Stokeslets: Applications, Computation, and Theory

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A special issue of Fluids (ISSN 2311-5521). This special issue belongs to the section "Mathematical and Computational Fluid Mechanics".

Deadline for manuscript submissions: closed (25 August 2022) | Viewed by 24569

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Prof. Dr. Ricardo Cortez

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Guest Editor

Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
Interests: computational fluid dynamics; numerical methods and scientific computing; biological fluid flow applications; regularized Stokeslets; Lagrangian vortex/impulse method

Prof. Dr. Sarah D. Olson

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Guest Editor

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA
Interests: micro-swimmers; fluid-structure interactions; computational fluid dynamics; mathematical biology; regularized Stokeslets; regularized Brinkmanlets

Special Issue Information

Dear Colleagues,

The Method of Regularized Stokeslets, first introduced in 2001, has had profound and broad impacts on the field of fluid dynamics, particularly at microscopic length scales. This Special Issue of Fluids will highlight current perspectives of the Method of Regularized Stokeslets, its contributions to numerical analysis, and how it is leveraged to investigate applications such as the study of microswimmer locomotion strategies and fluid mixing. We invite submissions ranging from review articles to mathematical analysis, algorithmic advances, and computational issues related to the method of Regularized Stokeslets, as well as studies that utilize these methods to investigate the fluid dynamics of applications at the microscale.

Prof. Dr. Ricardo Cortez
Prof. Dr. Sarah D. Olson
Guest Editors

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Keywords

method of regularized Stokeslets
computational fluid dynamics
fluid-structure interaction
cilia
flagella
micro-swimmers
fluid mixing

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Published Papers (10 papers)

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Research

24 pages, 661 KiB

Open AccessArticle

A Framework for Generating Radial and Surface-Oriented Regularized Stokeslets

by Nicholas G. Chisholm and Sarah D. Olson

Fluids 2022, 7(11), 351; https://doi.org/10.3390/fluids7110351 - 14 Nov 2022

Cited by 1 | Viewed by 1881

Abstract

Error in the method of regularized Stokeslets is highly dependent on the choice of the blob or regularization function that is utilized to handle singularities in the flow. In this work, we develop a general framework to choose regularizations at the level of the vector potential via smoothing factors. We detail the derivation for radial smoothing factors and specify properties which ensure that the solution is a regularized flow satisfying the incompressible Stokes equations. Error analysis is completed for both the far-field flow (away from the location of the forces) as well as at the location of the forces, relating our newly derived smoothing factors to commonly used blob functions and moment conditions. When forces are on a surface, we extend the radial smoothing factor case to the case of non-radial regularizations that are surface-oriented. We illustrate the utility of this framework by computing the forward and inverse problems of a translating sphere using radial and surface-oriented regularizations. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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20 pages, 4620 KiB

Open AccessArticle

Performance of a Helical Microswimmer Traversing a Discrete Viscoelastic Network with Dynamic Remodeling

by Rudi Schuech, Ricardo Cortez and Lisa Fauci

Fluids 2022, 7(8), 257; https://doi.org/10.3390/fluids7080257 - 29 Jul 2022

Cited by 5 | Viewed by 2230

Abstract

Microorganisms often navigate a complex environment composed of a viscous fluid with suspended microstructures such as elastic polymers and filamentous networks. These microstructures can have similar length scales to the microorganisms, leading to complex swimming dynamics. Some microorganisms secrete enzymes that dynamically change the elastic properties of the viscoelastic networks through which they move. In addition to biological organisms, microrobots have been engineered with the goals of mucin gel penetration or dissolving blood clots. In order to gain insight into the coupling between swimming performance and network remodeling, we used a regularized Stokeslet boundary element method to compute the motion of a microswimmer consisting of a rotating spherical body and counter-rotating helical flagellum. The viscoelastic network is represented by a network of points connected by virtual elastic linkages immersed in a viscous fluid. Here, we model the enzymatic dissolution of the network by bacteria or microrobots by dynamically breaking elastic linkages when the cell body of the swimmer falls within a given distance from the link. We investigate the swimming performance of the microbes as they penetrate and move through networks of different material properties, and also examine the effect of network remodeling. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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17 pages, 7732 KiB

Open AccessFeature PaperArticle

Actin Turnover Required for Adhesion-Independent Bleb Migration

by Calina Copos and Wanda Strychalski

Fluids 2022, 7(5), 173; https://doi.org/10.3390/fluids7050173 - 18 May 2022

Viewed by 2363

Abstract

Cell migration is critical for many vital processes, such as wound healing, as well as harmful processes, such as cancer metastasis. Experiments have highlighted the diversity in migration strategies employed by cells in physiologically relevant environments. In 3D fibrous matrices and confinement between two surfaces, some cells migrate using round membrane protrusions, called blebs. In bleb-based migration, the role of substrate adhesion is thought to be minimal, and it remains unclear if a cell can migrate without any adhesion complexes. We present a 2D computational fluid-structure model of a cell using cycles of bleb expansion and retraction in a channel with several geometries. The cell model consists of a plasma membrane, an underlying actin cortex, and viscous cytoplasm. Cellular structures are immersed in viscous fluid which permeates them, and the fluid equations are solved using the method of regularized Stokeslets. Simulations show that the cell cannot effectively migrate when the actin cortex is modeled as a purely elastic material. We find that cells do migrate in rigid channels if actin turnover is included with a viscoelastic description for the cortex. Our study highlights the non-trivial relationship between cell rheology and its external environment during migration with cytoplasmic streaming. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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20 pages, 10645 KiB

Open AccessArticle

Optimal Design of Bacterial Carpets for Fluid Pumping

by Minghao W. Rostami, Weifan Liu, Amy Buchmann, Eva Strawbridge and Longhua Zhao

Fluids 2022, 7(1), 25; https://doi.org/10.3390/fluids7010025 - 5 Jan 2022

Cited by 2 | Viewed by 1839

Abstract

In this work, we outline a methodology for determining optimal helical flagella placement and phase shift that maximize fluid pumping through a rectangular flow meter above a simulated bacterial carpet. This method uses a Genetic Algorithm (GA) combined with a gradient-based method, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, to solve the optimization problem and the Method of Regularized Stokeslets (MRS) to simulate the fluid flow. This method is able to produce placements and phase shifts for small carpets and could be adapted for implementation in larger carpets and various fluid tasks. Our results show that given identical helices, optimal pumping configurations are influenced by the size of the flow meter. We also show that intuitive designs, such as uniform placement, do not always lead to a high-performance carpet. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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30 pages, 11798 KiB

Open AccessArticle

The Role of the Double-Layer Potential in Regularised Stokeslet Models of Self-Propulsion

by David J. Smith, Meurig T. Gallagher, Rudi Schuech and Thomas D. Montenegro-Johnson

Fluids 2021, 6(11), 411; https://doi.org/10.3390/fluids6110411 - 13 Nov 2021

Cited by 1 | Viewed by 2702

Abstract

The method of regularised stokeslets is widely used to model microscale biological propulsion. The method is usually implemented with only the single-layer potential, the double-layer potential being neglected, despite this formulation often not being justified a priori due to nonrigid surface deformation. We describe a meshless approach enabling the inclusion of the double layer which is applied to several Stokes flow problems in which neglect of the double layer is not strictly valid: the drag on a spherical droplet with partial-slip boundary condition, swimming velocity and rate of working of a force-free spherical squirmer, and trajectory, swimmer-generated flow and rate of working of undulatory swimmers of varying slenderness. The resistance problem is solved accurately with modest discretisation on a notebook computer with the inclusion of the double layer ranging from no-slip to free-slip limits; the neglect of the double-layer potential results in up to 24% error, confirming the importance of the double layer in applications such as nanofluidics, in which partial slip may occur. The squirming swimmer problem is also solved for both velocity and rate of working to within a small percent error when the double-layer potential is included, but the error in the rate of working is above 250% when the double layer is neglected. The undulating swimmer problem by contrast produces a very similar value of the velocity and rate of working for both slender and nonslender swimmers, whether or not the double layer is included, which may be due to the deformation’s ‘locally rigid body’ nature, providing empirical evidence that its neglect may be reasonable in many problems of interest. The inclusion of the double layer enables us to confirm robustly that slenderness provides major advantages in efficient motility despite minimal qualitative changes to the flow field and force distribution. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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24 pages, 15167 KiB

Open AccessArticle

Using Experimentally Calibrated Regularized Stokeslets to Assess Bacterial Flagellar Motility Near a Surface

by Orrin Shindell, Hoa Nguyen, Nicholas Coltharp, Frank Healy and Bruce Rodenborn

Fluids 2021, 6(11), 387; https://doi.org/10.3390/fluids6110387 - 29 Oct 2021

Cited by 2 | Viewed by 1720

Abstract

The presence of a nearby boundary is likely to be important in the life cycle and evolution of motile flagellate bacteria. This has led many authors to employ numerical simulations to model near-surface bacterial motion and compute hydrodynamic boundary effects. A common choice has been the method of images for regularized Stokeslets (MIRS); however, the method requires discretization sizes and regularization parameters that are not specified by any theory. To determine appropriate regularization parameters for given discretization choices in MIRS, we conducted dynamically similar macroscopic experiments and fit the simulations to the data. In the experiments, we measured the torque on cylinders and helices of different wavelengths as they rotated in a viscous fluid at various distances to a boundary. We found that differences between experiments and optimized simulations were less than 5% when using surface discretizations for cylinders and centerline discretizations for helices. Having determined optimal regularization parameters, we used MIRS to simulate an idealized free-swimming bacterium constructed of a cylindrical cell body and a helical flagellum moving near a boundary. We assessed the swimming performance of many bacterial morphologies by computing swimming speed, motor rotation rate, Purcell’s propulsive efficiency, energy cost per swimming distance, and a new metabolic energy cost defined to be the energy cost per body mass per swimming distance. All five measures predicted that the optimal flagellar wavelength is eight times the helical radius independently of body size and surface proximity. Although the measures disagreed on the optimal body size, they all predicted that body size is an important factor in the energy cost of bacterial motility near and far from a surface. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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20 pages, 1773 KiB

Open AccessArticle

A Model for Stokes Flow in Domains with Permeable Boundaries

by Ricardo Cortez, Marian Hernandez-Viera and Owen Richfield

Fluids 2021, 6(11), 381; https://doi.org/10.3390/fluids6110381 - 23 Oct 2021

Cited by 2 | Viewed by 2147

Abstract

We derive a new computational model for the simulation of viscous incompressible flows bounded by a thin, flexible, porous membrane. Our approach is grid-free and models the boundary forces with regularized Stokeslets. The flow across the porous membranes is modeled with regularized source doublets based on the notion that the flux velocity across the boundary can be viewed as the flow induced by a fluid source/sink pair with the sink on the high-pressure side of the boundary and magnitude proportional to the pressure difference across the membrane. Several validation examples are presented that illustrate how to calibrate the parameters in the model. We present an example consisting of flow in a closed domain that loses volume due to the fluid flux across the permeable boundary. We also present applications of the method to flow inside a channel of fixed geometry where sections of the boundary are permeable. The final example is a biological application of flow in a capillary with porous walls and a protein concentration advected and diffused in the fluid. In this case, the protein concentration modifies the pressure in the flow, producing dynamic changes to the flux across the walls. For this example, the proposed method is combined with finite differences for the concentration field. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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31 pages, 6945 KiB

Open AccessArticle

Flagellar Cooperativity and Collective Motion in Sperm

by Julie Simons and Alexandra Rosenberger

Fluids 2021, 6(10), 353; https://doi.org/10.3390/fluids6100353 - 8 Oct 2021

Cited by 3 | Viewed by 3561

Abstract

Sperm have thin structures known as flagella whose motion must be regulated in order to reach the egg for fertilization. Large numbers of sperm are typically needed in this process and some species have sperm that exhibit collective or aggregate motion when swimming in groups. The purpose of this study is to model planar motion of flagella in groups to explore how collective motion may arise in three-dimensional fluid environments. We use the method of regularized Stokeslets and a three-dimensional preferred curvature model to simulate groups of undulating flagella, where flagellar waveforms are modulated via hydrodynamic coupling with other flagella and surfaces. We find that collective motion of free-swimming flagella is an unstable phenomenon in long-term simulations unless there is an external mechanism to keep flagella near each other. However, there is evidence that collective swimming can result in significant gains in velocity and efficiency. With the addition of an ability for sperm to attach and swim together as a group, velocities and efficiencies can be increased even further, which may indicate why some species have evolved mechanisms that enable collective swimming and cooperative behavior in sperm. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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17 pages, 1178 KiB

Open AccessArticle

Regularized Stokeslets Lines Suitable for Slender Bodies in Viscous Flow

by Boan Zhao and Lyndon Koens

Fluids 2021, 6(9), 335; https://doi.org/10.3390/fluids6090335 - 21 Sep 2021

Cited by 1 | Viewed by 2150

Abstract

Slender-body approximations have been successfully used to explain many phenomena in low-Reynolds number fluid mechanics. These approximations typically use a line of singularity solutions to represent flow. These singularities can be difficult to implement numerically because they diverge at their origin. Hence, people have regularized these singularities to overcome this issue. This regularization blurs the force over a small blob and thereby removing divergent behaviour. However, it is unclear how best to regularize the singularities to minimize errors. In this paper, we investigate if a line of regularized Stokeslets can describe the flow around a slender body. This is achieved by comparing the asymptotic behaviour of the flow from the line of regularized Stokeslets with the results from slender-body theory. We find that the flow far from the body can be captured if the regularization parameter is proportional to the radius of the slender body. This is consistent with what is assumed in numerical simulations and provides a choice for the proportionality constant. However, more stringent requirements must be placed on the regularization blob to capture the near field flow outside a slender body. This inability to replicate the local behaviour indicates that many regularizations cannot satisfy the no-slip boundary conditions on the body’s surface to leading order, with one of the most commonly used blobs showing an angular dependency of velocity along any cross section. This problem can be overcome with compactly supported blobs, and we construct one such example blob, which can be effectively used to simulate the flow around a slender body. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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24 pages, 1190 KiB

Open AccessArticle

Remarks on Regularized Stokeslets in Slender Body Theory

by Laurel Ohm

Fluids 2021, 6(8), 283; https://doi.org/10.3390/fluids6080283 - 14 Aug 2021

Cited by 4 | Viewed by 2175

Abstract

We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius

ϵ > 0

. Denoting the regularization parameter by

δ

, we consider regularized SBT based on the most [...] Read more.

We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius

ϵ > 0

. Denoting the regularization parameter by

δ

, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive

L^{\infty}

bounds for the difference between regularized SBT and its classical counterpart in terms of

δ

ϵ

, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to

log (δ / ϵ)

; in particular,

δ = ϵ

is necessary to avoid an

O (1)

discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to

log (1 + δ^{2} / ϵ^{2})

, and any choice of

δ \propto ϵ

will result in an

O (1)

discrepancy as

ϵ \to 0

. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed slender body PDE which classical SBT is known to approximate. Numerics verify this

O (1)

discrepancy but also indicate that the difference may have little impact in practice. Full article

(This article belongs to the Special Issue 20 Years of Regularized Stokeslets: Applications, Computation, and Theory)

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