Slender Body Theory Research Articles

Phoretic self-propulsion of helical active particles

Chemically active colloids self-propel by catalysing the decomposition of molecular ‘fuel’ available in the surrounding solution. If the various molecular species involved in the reaction have distinct interactions with the colloid surface, and if the colloid has some intrinsic asymmetry in its surface chemistry or geometry, there will be phoretic flows in an interfacial layer surrounding the particle, leading to directed motion. Most studies of chemically active colloids have focused on spherical, axisymmetric ‘Janus’ particles, which (in the bulk, and in absence of fluctuations) simply move in a straight line. For particles with a complex (non-spherical and non-axisymmetric) geometry, the dynamics can be much richer. Here, we consider chemically active helices. Via numerical calculations and slender body theory, we study how the translational and rotational velocities of the particle depend on geometry and the distribution of catalytic activity over the particle surface. We confirm the recent finding of Katsamba et al. (J. Fluid Mech., vol. 898, 2020, p. A24) that both tangential and circumferential concentration gradients contribute to the particle velocity. The relative importance of these contributions has a strong impact on the motion of the particle. We show that, by a judicious choice of the particle design parameters, one can suppress components of angular velocity that are perpendicular to the screw axis, or even select for purely ‘sideways’ translation of the helix.

Regularized Stokeslets Lines Suitable for Slender Bodies in Viscous Flow

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.

Dynamics of gas bubbles in fiber suspensions

The collapse/expansion of a spherical bubble in a Newtonian fluid filled with non-Brownian fibers is numerically investigated. An experimental test was conducted to observe that the spherical geometry of the bubble is maintained during the foaming process. The constitutive equation for the fiber suspension arises from the slender body theory and is coupled with an evolution equation describing the fiber dynamics. First solutions are proposed using steady fiber orientation distributions and the effect of the coupling coefficient on the bubble radius is examined. Later the condition of fixed microstructure is relaxed and the fibers are able to orient along with bubble expansion. Starting with a 3D random orientation, fibers close to the interface orient randomly in planes orthogonal to the radial direction, depending on the level of fiber–fiber interactions. For interaction coefficients encountered in polymer processing, fibers exhibit a closely 2D random planar orientation in a plane tangent to the cavity surface which prevents the bubble to collapse, as observed experimentally.

Remarks on Regularized Stokeslets in Slender Body Theory

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∞ 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+δ2/ϵ2), and any choice of δ∝ϵ will result in an O(1) discrepancy as ϵ→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.

Hydrodynamic impulse generated by slender bodies undergoing unsteady motion in viscous flows

The forces and moments on a UUV hull undergoing unsteady sway motions are analyzed using both traditional methods which integrate the pressure and shear stress at the wall, and hydrodynamic impulse based methods which analyze the vortical structure of the flow around the body. Computational fluid dynamics simulations have been conducted of a body with a slenderness ratio of 8.5, at a Reynolds number of 3.4 million, undergoing a variety of sway maneuvers. These include steady translations at incidence angles from 0∘ to 20∘, impulsive acceleration with varied initial incidence, and sinusoidally oscillating sway over a range of frequencies and amplitudes. The impulse based method is based on a body-fixed, non-inertial frame used to analyze the vortical structures near the body. It is demonstrated that this impulse based analysis provides a sensible decomposition of unsteady forces into those due to added mass and those due to unsteady circulation and motion history. Motion history effects are seen to be primarily described by the Lamb vector distribution around the body. These effects decay towards the quasi-steady limit at large times for impulsive motion and at low frequencies for sinusoidal motion. This is demonstrated with the analysis of the impulse based force components and with visualizations of the vortex structures in the flow field. A simplified form of the impulse based equations is derived and validated, based on approximations from slender body theory.

Asymptotic theory of hydrodynamic interactions between slender filaments

Hydrodynamic interactions (HIs) are important in biophysics research because they influence both the collective and the individual behaviour of microorganisms and self-propelled particles. For instance, HIs at the micro-swimmer level determine the attraction or repulsion between individuals, and hence their collective behaviour. Meanwhile, HIs between swimming appendages (e.g. cilia and flagella) influence the emergence of swimming gaits, synchronised bundles and metachronal waves. In this study, we address the issue of HIs between slender filaments separated by a distance larger than their contour length (d>L) by means of asymptotic calculations and numerical simulations. We first derive analytical expressions for the extended resistance matrix of two arbitrarily-shaped rigid filaments as a series expansion in inverse powers of d/L>1. The coefficients in our asymptotic series expansion are then evaluated using two well-established methods for slender filaments, resistive-force theory (RFT) and slender-body theory (SBT), and our asymptotic theory is verified using numerical simulations based on SBT for the case of two parallel helices. The theory captures the qualitative features of the interactions in the regime d/L>1, which opens the path to a deeper physical understanding of hydrodynamically governed phenomena such as inter-filament synchronisation and multiflagellar propulsion. To demonstrate the usefulness of our results, we next apply our theory to the case of two helices rotating side-by-side, where we quantify the dependence of all forces and torques on the distance and phase difference between them. Using our understanding of pairwise HIs, we then provide physical intuition for the case of a circular array of rotating helices. Our theoretical results will be useful for the study of HIs between bacterial flagella, nodal cilia, and slender microswimmers.

Study on the multi-objective optimization method of trimaran outrigger layout

ABSTRACT The selection of the outrigger layout plays a significant role in trimaran design as it will has effect on various hydrodynamic performance of the trimaran. In this paper, the high-speed slender body theory (2.5D method) is applied to the problem of trimaran swaying motion and added resistance prediction. Furthermore, in order to optimise the resistance and swaying motion of the trimaran, this paper propose a trimaran outrigger layout optimisation method with resistance and swaying motion are considered comprehensively based on NSGA-II. By analysing the convergence of the results, the optimisation method proposed in this paper is verified. In addition, this paper gives a suggested outrigger layout for a trimaran sailing in oblique waves with wavelength to ship length ratio equals 1.0 and . The optimisation result is also compared with the other six layouts in the experiment to verify the outrigger layout optimisation method proposed in this paper.

Stability of sedimenting flexible loops

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Self-propelled slender objects can measure flow signals net of self-motion

The perception of hydrodynamic signals by self-propelled objects is a problem of paramount importance ranging from the field of bio-medical engineering to bio-inspired intelligent navigation. By means of a state-of-the-art fully resolved immersed boundary method, we propose different models for fully coupled self-propelled objects (swimmers, in short), behaving either as “pusher” or as “puller.” The proposed models have been tested against known analytical results in the limit of Stokes flow, finding excellent agreement. Once tested, our more realistic model has been exploited in a chaotic flow field up to a flow Reynolds number of 10, a swimming number ranging between zero (i.e., the swimmer is freely moving under the action of the underlying flow in the absence of propulsion) and one (i.e., the swimmer has a relative velocity with respect to the underlying flow velocity of the same order of magnitude as the underlying flow), and different swimmer inertia measured in terms of a suitable definition of the swimmer Stokes number. Our results show the following: (i) pusher and puller reach different swimming velocities for the same, given, propulsive force: while for pusher swimmers, an effective slender body theory captures the relationship between swimming velocity and propulsive force, this is not for puller swimmers. (ii) While swimming, pusher and puller swimmers possess a different distribution of the vorticity within the wake. (iii) For a wide range of flow/swimmer Reynolds numbers, both pusher and puller swimmers are able to sense hydrodynamic signals with good accuracy.

The numerical study on active control strategies of T-foil

Objectives An active control T-foil can effectively improve the longitudinal anti-rolling motion of the ship, but there is still no conclusion on what control strategy has the best anti-rolling effect. So the paper has studied and analyzed the active control strategies of the T-foil. Methods The paper uses the slender body theory to calculate the longitudinal motion of Wigley III in regular waves of different wavelengths. Then, the damping effect of the ship with the passive and active T-foil is calculated and analyzed under the same speed and wave conditions. Results The results show that the passive control T-foil has a good anti-rolling effect in the resonance region, however, the anti-rolling effect in the long and short wave region is relatively limited. The control with a pitch velocity signal is very effective in reducing the pitch motion and the acceleration of bow motion, but it increases the heave motion in the longwave area. The anti-rolling effect of bow motion velocity signal on-pitch motion and acceleration of bow motion is 10%-20% higher than that of a pitch velocity signal, and there is no reduction effect on heave motion. Conclusions For the active control strategies of T-foil, the bow motion velocity signal can achieve a better anti-rolling effect by comprehensively considering both pitch and heave motions.

Dynamics of a confined pipe aspirating fluid and concurrently subjected to external axial flow: A theoretical investigation

A linear model is developed for a hanging pipe aspirating fluid with internal flow velocity Ui, flowing up the pipe from its lower end, and also subjected to an external flow of velocity Uo flowing down from the upper end of the annulus formed between the pipe and a relatively shorter rigid outer tube; both the tube and pipe are cantilevered from their top ends. In this model, the pipe is modelled as an Euler–Bernoulli beam, with the internal flow modelled as a plug flow and the external one by means of slender-body theory; flow over the unconfined portion of the pipe is also taken into account. This model is discretized and solved using the Galerkin method to obtain the eigenfrequencies of the pipe for a number of flow velocity ratios Uo/Ui. In all cases, the pipe loses stability, mostly via first-mode flutter, at sufficiently high values of Ui. Subsequently, the theoretical results are compared with earlier theoretical results and also with all available experimental results. The present theoretical results are in reasonably good agreement with the experimental results.The system under consideration represents an idealized mode of operation of ‘salt-mined caverns’, which are large underground storage spaces created to store hydrocarbons, such as natural gas and oil.

An integral model based on slender body theory, with applications to curved rigid fibers

We propose a novel integral model describing the motion of both flexible and rigid slender fibers in viscous flow and develop a numerical method for simulating dynamics of curved rigid fibers. The model is derived from nonlocal slender body theory (SBT), which approximates flow near the fiber using singular solutions of the Stokes equations integrated along the fiber centerline. In contrast to other models based on (singular) SBT, our model yields a smooth integral kernel which incorporates the (possibly varying) fiber radius naturally. The integral operator is provably negative definite in a nonphysical idealized geometry, as expected from the partial differential equation theory. This is numerically verified in physically relevant geometries. We discuss the convergence and stability of a numerical method for solving the integral equation. The accuracy of the model and method is verified against known models for ellipsoids. Finally, we develop an algorithm for computing dynamics of rigid fibers with complex geometries in the case where the fiber density is much greater than that of the fluid, for example, in turbulent gas-fiber suspensions.

Accuracy of slender body theory in approximating force exerted by thin fiber on viscous fluid

AbstractWe consider the mapping properties of the integral operator arising in nonlocal slender body theory (SBT) for the model geometry of a straight, periodic filament. It is well known that the classical singular SBT integral operator suffers from high wavenumber instabilities, making it unsuitable for approximating the slender body inverse problem, where the fiber velocity is prescribed and the integral operator must be inverted to find the force density along the fiber. Regularizations of the integral operator must therefore be used instead. Here, we consider two regularization methods: spectral truncation and the ‐regularization of Tornberg and Shelley (2004). We compare the mapping properties of these approximations to the underlying partial differential equation (PDE) solution, which for the inverse problem is simply the Stokes Dirichlet problem with data constrained to be constant on cross sections. For the straight‐but‐periodic fiber with constant radius , we explicitly calculate the spectrum of the operator mapping fiber velocity to force for both the PDE and the approximations. We prove that the spectrum of the original SBT operator agrees closely with the PDE operator at low wavenumbers but differs at high frequencies, allowing us to define a truncated approximation with a wavenumber cutoff . For both the truncated and ‐regularized approximations, we obtain rigorous ‐based convergence to the PDE solution as : A fiber velocity with regularity gives convergence, while a fiber velocity with at least regularity yields convergence. Moreover, we determine the dependence of the ‐regularized error estimate on the regularization parameter .

Regularised non-uniform segments and efficient no-slip elastohydrodynamics

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Universal features of the shape of elastic fibres in shear flow

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Control of Uniflagellar Soft Robots at Low Reynolds Number Using Buckling Instability

Abstract In this paper, we analyze the inverse dynamics and control of a bacteria-inspired uniflagellar robot in a fluid medium at low Reynolds number. Inspired by the mechanism behind the locomotion of flagellated bacteria, we consider a robot comprising a flagellum—a flexible helical filament—connected to a spherical head. The flagellum rotates about the head at a controlled angular velocity and generates a propulsive force that moves the robot forward. When the angular velocity exceeds a threshold value, the hydrodynamic force exerted by the fluid can cause the soft flagellum to buckle, characterized by a dramatic change in its shape. In this computational study, a fluid–structure interaction model that combines Discrete Elastic Rods algorithm with Lighthill's Slender Body Theory is employed to simulate the locomotion and deformation of the robot. We demonstrate that the robot can follow a prescribed path in three-dimensional space by exploiting buckling of the flagellum. The control scheme involves only a single (binary) scalar input—the angular velocity of the flagellum. By triggering the buckling instability at the right moment, the robot can follow the path in three-dimensional space. We also show that the complexity of the dynamics of the helical filament can be captured using a deep neural network, from which we identify the input–output functional relationship between the control input and the trajectory of the robot. Furthermore, our study underscores the potential role of buckling in the locomotion of natural bacteria.

Integral-based spectral method for inextensible slender fibers in Stokes flow

Every animal cell is filled with a cytoskeleton, a dynamic gel made of inextensible fibers, such as microtubules, actin fibers, and intermediate filaments, all suspended in a viscous fluid. Numerical simulation of this gel is challenging because the fiber aspect ratios can be as large as $10^4$. We describe a new method for rapidly computing the dynamics of inextensible slender filaments in periodically-sheared Stokes flow. The dynamics of the filaments are governed by a nonlocal slender body theory which we partially reformulate in terms of the Rotne-Prager-Yamakawa hydrodynamic tensor. To enforce inextensibility, we parameterize the space of inextensible fiber motions and strictly confine the dynamics to the manifold of inextensible configurations. To do this, we introduce a set of Lagrange multipliers for the tensile force densities on the filaments and impose the constraint of no virtual work in an $L^2$ weak sense. We augment this approach with a spectral discretization of the local and nonlocal slender body theory operators which is linear in the number of unknowns and gives improved spatial accuracy over approaches based on solving a line tension equation. For dynamics, we develop a second-order semi-implicit temporal integrator which requires at most a few evaluations of nonlocal hydrodynamics and a few block diagonal linear solves per time step. After demonstrating the improved accuracy and robustness of our approach through numerical examples, we apply our formulation to a permanently cross-linked actin mesh in a background oscillatory shear flow. We observe a characteristic frequency at which the network transitions from quasi-static, primarily elastic behavior to dynamic, primarily viscous behavior. We find that nonlocal hydrodynamics increases the viscous modulus by as much as 25%, even for semi-dilute fiber suspensions.

Hydrodynamic Modeling and Design of Robotic Fish using Slender Body Theory

Design of bio-inspired robotic fish has been an important area of research for applications such as water quality monitoring, aquatic animal behavioral study and underwater exploration. The modeling of robotic fish requires high dimensional analysis, which increases the design complexity. To address this complexity, approximate models are preferred and frequently used in the design of robotic fishes. The accuracy of these models mainly relies on various hydrodynamic parameter identification of which drag coefficient estimation is highly challenging. This paper utilizes a simple deceleration motion model to estimate the drag coefficient effectively so that the linearization error is reduced. The non-linearity in the deceleration model is simplified and mapped as a linear model to predict the drag coefficient. The estimated drag coefficient is used for dynamic modeling of robotic fish using slender body theory. The dynamic model is validated through experimental setup using accelerometer and visual sensor data. The maximum swimming speed of 0.32 ms-1 is achieved at 1.2 Hz caudal fin oscillations.

Fastest Fish Shapes and Optimal Supercavitating and Hypersonic Bodies of Revolution

Background. The best swimmers have a streamlined shape that provides a flow pattern without boundary layer separation and delays the laminar-to-turbulent transition. Their shape itself could be the reason of small drag and high locomotion velocity. The fastest fish, e.g., sailfish, swordfish, black marlin, etc. have another feature of their shape – a very sharp nose – rostrum, the purpose of which remains unclear. Popular belief that the rostrum is used by these predators to pierce their prey is often disputed. Objective. In this study, we analyze the hydrodynamic aspects of the rostrum presence and the possible use of similar hulls for supercavitating underwater vehicles and fast penetration into water. We illustrate that shapes with the very sharp nose could be useful for hypersonic motion in order to eliminate overheating of the vehicle fuselage. Methods. We use the known exact solutions of the Euler equations for the incompressible fluid to simulate the pressure distribution on the bodies of revolution with a sharp nose. The slender body theory is used to simulate the supercavitation and the axisymmetric air flows at high Mach numbers. Results. Bodies of revolution with a rostrum similar to trunks of the fastest fish (sailfish, swordfish, black marlin) and corresponding pressure ant temperature coefficients were calculated. The proposed shapes ensure no stagnation points and no high pressures and temperatures on their noses at sub- and supersonic speeds both in water and air. The drag on such bodies of revolution was estimated for attached, supercavitating and supersonic flow patterns. Conclusions. A method of calculation of axisymmetric bodies without stagnation points on their surface was proposed. This peculiarity of the shape allows diminishing the maximum pressure and temperature on the nose without a significant increase in drag. Such shapes with the sharp concave nose could be recommended for high-speed attached and supercavitating bodies of revolution and for the hypersonic motion.

Semi-analytical study on regular sea wave interaction with circular elastic floaters of aquaculture fish cages

In this paper, a semi-analytical model was developed to find the vertical motions and wavefield around elastic circular floaters of aquaculture fish cages in regular waves. The fluid is assumed inviscid, incompressible, and irrotational flow. A regular train of deep water waves was incident to a floater, which consist of two tori of high-density polyethylene (HDPE) pipes. The problem was analyzed with the assumption of the relatively large incident wavelength in comparison to the cross-section of tori. Slender body theory and matched asymptotic expansions were used with the inner solution close to the body section composed of suitable radiation solution at large distance of floater by distributing source and doublet along centerlines of floater constituent tori. In this approach, the fundamental solutions were developed for symmetric and anti-symmetric motions. Then, the governing equations on diffraction and incident waves were decomposed to Fourier modes of these solutions. The main influence parameters on vertical accelerations such as floater dimensions, wave number, the distances between tori, and bending stiffness of floater were studied. The model results demonstrate good agreement with those of the BEM. Furthermore, the Energy Transmission Index (ETI) was introduced for describing the effect of the floater on the downstream environment.