Slender Body Theory Research Articles

Flow-induced buckling of elastic microfilaments with non-uniform bending stiffness

Buckling plays a critical role in the transport and dynamics of elastic microfilaments in Stokesian fluids. However, previous work has only considered filaments with homogeneous structural properties. Filament backbone stiffness can be non-uniform in many biological systems like microtubules, where the association and disassociation of proteins can lead to spatial and temporal changes into structure. The consequences of such non-uniformities in the configurational stability and transport of these fibers are yet unknown. Here, we use slender-body theory and Euler-Bernoulli elasticity coupled with various non-uniform bending rigidity profiles to quantify this buckling instability using linear stability analysis and Brownian simulations. In shear flows, we observe more pronounced buckling in areas of reduced rigidity in our simulations. These areas of marked deformations give rise to differences in the particle extra stress, indicating a non-trivial rheological response due to the presence of these filaments. The fundamental mode shapes arising from each rigidity profile are consistent with the predictions from our linear stability analysis. Collectively, these results suggest that non-uniform bending rigidity can drastically alter fluid-structure interactions in physiologically relevant settings, providing a foundation to elucidate the complex interplay between hydrodynamics and the structural properties of biopolymers.

Dynamics of flexible filaments in oscillatory shear flows.

The fluid-structure interactions between flexible fibers and viscous flows play an essential role in various biological phenomena, medical problems, and industrial processes. Of particular interest is the case of particles freely transported in time-dependent flows. This work elucidates the dynamics and morphologies of actin filaments under oscillatory shear flows by combining microfluidic experiments, numerical simulations, and theoretical modeling. Our work reveals that, in contrast to steady shear flows, in which small orientational fluctuations from a flow-aligned state initiate tumbling and deformations, the periodic flow reversal allows the filament to explore many different configurations at the beginning of each cycle. Investigation of filament motion during half time periods of oscillation highlights the critical role of the initial filament orientation on the emergent dynamics. This strong coupling between orientation and deformation results in new deformation regimes and novel higher-order buckling modes absent in steady shear flows. The primary outcome of our analysis is the possibility of suppression of buckling instabilities for certain combinations of the oscillation frequency and initial filament orientation, even in very strong flows. We explain this unusual behavior through a weakly nonlinear Landau theory of buckling, in which we treat the filaments as inextensible Brownian Euler-Bernoulli rods whose hydrodynamics are described by local slender-body theory.

Feedback-induced flutter instability of a flexible beam in fluid flow

A pinned-free beam in axial fluid flow, subjected to feedback-based actuation at the pinned end, is investigated. The actuation may be a moment or a prescribed angle and it is proportional to the state (curvature, slope, or displacement) of the beam at some point along its length. All equations and boundary condition terms are non-dimensionalized and the stability of the system is studied over a range of external flow velocity and sensing location. For each combination of flow velocity and sensing location, the critical gain (positive or negative) for the onset of flutter is determined. This process, which is repeated for each combination of actuation and sensing modes, reveals that the closed-loop system exhibits a rich set of stability transitions, each associated with a traveling waveform in the flexible beam at the onset of flutter. With the intent of exploring the use of flexible fluttering beams for underwater propulsion, the efficiency of these waveforms is computed using slender-body theory. Additional insights into the efficiency of the waveforms are obtained through considerations of the smoothness of the traveling waveforms.

Studying multiphase flow around a vehicle moving freely to the water surface

The paper provides an experimental study of a multiphase flow around models in a large high-speed hydrodynamic tunnel with a vertical working section. Experimental dependences of drag coefficients on the cavitation number for vertical cavities are obtained. The results of cavitation flow around bodies of revolution in a vertically descending fluid flow are considered. A methodology for calculating the shape of vertical cavities under the slender-body theory is proposed. Relationships suitable for engineering evaluation are obtained for very long cavities. Numerical simulation of underwater motion is performed with models moving vertically in a nonstationary cavitation flow. The hydrodynamic loads can be evaluated through the determination of the nonstationary boundaries of a gas cavity and a linear load on a water-flown aft. The predicted results are compared with the experimental data. Validation and verification are performed by comparing the analysis results with the experimental data. The applicability of the proposed method to the determination of the hydrodynamic parameters of vehicles under multiphase flow is demonstrated.

Slender body theories for rotating filaments

Slender fibres are ubiquitous in biology, physics and engineering, with prominent examples including bacterial flagella and cytoskeletal fibres. In this setting, slender body theories (SBTs), which give the resistance on the fibre asymptotically in its slenderness $\epsilon$ , are useful tools for both analysis and computations. However, a difficulty arises when accounting for twist and cross-sectional rotation: because the angular velocity of a filament can vary depending on the order of magnitude of the applied torque, asymptotic theories must give accurate results for rotational dynamics over a range of angular velocities. In this paper, we first survey the challenges in applying existing SBTs, which are based on either singularity or full boundary integral representations, to rotating filaments, showing in particular that they fail to consistently treat rotation–translation coupling in curved filaments. We then provide an alternative approach which approximates the three-dimensional dynamics via a one-dimensional line integral of Rotne–Prager–Yamakawa regularized singularities. While unable to accurately resolve the flow field near the filament, this approach gives a grand mobility with symmetric rotation–translation and translation–rotation coupling, making it applicable to a broad range of angular velocities. To restore fidelity to the three-dimensional filament geometry, we use our regularized singularity model to inform a simple empirical equation which relates the mean force and torque along the filament centreline to the translational and rotational velocity of the cross-section. The single unknown coefficient in the model is estimated numerically from three-dimensional boundary integral calculations on a rotating, curved filament.

Mud swimming: Locomotion through a viscoplastic fluid

<h2>Abstract</h2> Different classical models of small, slow (inertialess) swimming are considered when the ambient fluid has a yield stress. A variety of organisms inhabit and have to move through mud, mucus and other biological media, or more generally, soil and sand, all of which can exhibit viscoplastic behaviour. Basic mechanisms for inertialess swimming in ‘simple' viscoplastic (Bingham) fluids are considered from a theoretical and numerical standpoint, with a particular focus on the role of the yield stress, the location of plugged-up regions around the swimmer's body, and the speed and efficiency of locomotion. Taylor's canonical ‘swimming sheet', idealised versions of squirming organisms, and long, thin worm-like motions are all discussed, the latter of which involves a generalisation of classical slender-body theory for viscoplastic fluids.

Hydrodynamic interactions are key in thrust-generation of hairy flagella

The flagellum in many predatory flagellates is equipped with rigid hairs, which reverse the direction of thrust to generate a feeding current. We have discovered a novel mechanism for thrust generation with hairy flagella, which is fundamentally different from the slender body theory governing smooth flagella. We show that the thrust is generated at the crests of the flagellar wave, thanks to the key role of hydrodynamic interactions among hairs, effectively functioning like a sheet. The mechanism suggests that the optimal flagellum should be long and curvy, consistent with the diverse designs of hairy flagella observed in nature.

A semi-analytical solution for energy harvesting via the elastic motion of the circular floater of aquaculture cages attached with piezoelectric

This work's unique research objective is to assess the possibility of energy harvesting by attaching the piezoelectric patches on the high-density polyethylene floater of aquaculture fish cages. Using Euler-Bernoulli's curved beam theory, the vertical elastic motion of the floater in regular linear sea waves was modeled semi-analytically based on slender body theory and matched asymptotic expansions. Regular linear waves refer to the single frequency waves with small amplitudes compared to the wavelength, for which the linear airy wave theory can approximate the wave characteristics. When the floater elastic motions are determined, the resultant strain is implemented to the governing equation of piezoelectric patches to calculate the generated power. The effect of some variables such as wavenumber, floater bending stiffness, relative floater dimension, and the number of patches in the circumference of the floater on the generated power was investigated. The results indicate that by installing the patches in the proper place, the floater can produce the required electrical power supplying some low-powered electric sensors for monitoring sea environmental conditions or other applications. Also, as a suggestion, installing piezoelectric patches throughout the perimeter of a semi-submerged circular elastic floater is a way of energy harvesting from sea waves.

A dynamical study on sea waves interaction with an array of circular elastic floaters of aquaculture fish cages

In this paper, linear oblique wave incident to circular elastic floaters of aquaculture fish cages was studied semi-analytically. This work's unique research objectives include considering the array of elastic floaters where each floater consists of two tori with minor differences in their main radii. Hydrodynamic interaction between the inner and outer of each floater body and hydro-elastic interaction between floaters in the array are the particular features of the problem. Using slender body theory and matched asymptotic expansions, each 3D problem was decomposed into two separate, more straightforward problems named near-field and far-field problems. This approach helps derive fundamental solutions for symmetric and antisymmetric motions. Also, using linear wave theory, the diffracted and radiated waves due to the presence of floaters were expressed in Fourier modes of symmetric and antisymmetric motions. Moreover, Graf's and Gegenbauer's addition theorems for Bessel functions are used to shift between different coordinate systems. This approach is helpful to bridge on calculating related integrals numerically. The developed model verification by BEM-based software demonstrates that the model solves the mathematical formulas with enough accuracy. The effect of various parameters such as floater dimensions, wavenumber, the distances between floaters, and the bending stiffness of floater on the floater's motions was studied. The results demonstrate that as the wavenumber increases, the floater's hydro-elastic interaction becomes essential. Furthermore, the upstream floaters do encounter larger wave loads than a single floater in waves.

A slender body theory for the motion of special Cosserat filaments in Stokes flow

The motion of filament-like structures in fluid media has been a topic of interest since long. In this regard, a well known slender body theory exists, wherein the fluid flow is assumed to be Stokesian while the filament is modeled as a Kirchhoff rod which can bend and twist but remains inextensible and unshearable. In this work, we relax the inextensibility and unshearability constraints on filaments, i.e., the filament is modeled as a special Cosserat rod. Starting with the boundary integral formulation of Stokes flow involving the filament’s surface velocity and fluid traction that acts on the filament surface, the method of matched asymptotic expansion is used to first obtain a leading-order representation of the boundary integral kernels in the filament’s aspect ratio. We then substitute Fourier series expansion (in filament’s circumferential coordinate) of both the filament’s surface velocity and fluid traction in the aforementioned leading-order representation and further linearize it in the rod’s shear strains to reduce the two-dimensional boundary integral over the filament surface into a line integral over the filament’s centerline. Upon further collecting the coefficients of sine and cosine terms, the zeroth-order Fourier mode yields a line integral equation relating the rod’s centerline velocity with the distributed fluid force that acts on the filament. The presence of line integral makes the relation non-local in nature. On the contrary, the first-order Fourier mode yields a simpler local relation between the rod’s angular velocity and the distributed fluid couple. The line integral equation is shown to reduce to the classical slender body theory when shear strains and axial strain are set to zero. The non-dimensional governing equations of the special Cosserat rod are also derived accounting for the distributed fluid force and distributed fluid couple in them which are solved to obtain the filament motion. The presented theory is demonstrated with an example problem of the tumbling of filaments in background shear flow. We show that for relatively shorter filaments where the effect of shear and axial stretch is more dominant, the obtained results deviate from the ones based on the classical slender body theory.

Experimental and theoretical studies of the fluid elasticity on the motion of macroscopic models of active helical swimmers

This work presents experimental and theoretical studies on the locomotion of helical artificial swimmers at low Reynolds number in both Newtonian and viscoelastic ambient liquids. We examine the effect of fluid elasticity on the propulsive force and torque on the body and speed velocity of the swimmer in terms of two physical parameters: Deborah number (De) and Strouhal number (Sh). For this end, some experiments with prototype microorganisms in creeping flow motion are conducted. In the experiments, a macroscopic swimmer that propels itself by mimicking helical flagella are developed and tested. Three swimming models propelled by a helical tail with different wavelengths are investigated, and their motions examined for both cases: when the ambient solvent is a pure Newtonian viscous fluid and when the base fluid is an elastic polymeric solution. In addition, we also apply the slender body theory and the method of regularized Stokeslet in order to calculate theoretically the force and torque, as function of the Strouhal number (Sh), produced by the helical swimmer moving in a Newtonian fluid. The theoretical results are compared with experimental data, and a very good agreement is observed especially for higher values of Sh within the error bars of the experimental data. In the case of a non-Newtonian base fluid, the flow problem of an Oldroyd-B elastic fluid is solved numerically using a computational code based on a finite element method. The helical swimmer propulsive velocity is calculated in terms of the elastic parameter Deborah number and also compared with the experimental observation when the base fluid is non-Newtonian. It is shown experimentally that the swimming speed increases as the elastic effect in the base fluid increases until a critical Deborah number O(1), when the velocity saturates for a constant value within the experimental error bars. The velocity anisotropy measured experimentally by the ratio of the swimmer speed in two different directions is insensitive to the elastic effect in the base fluids. We complete our discussion on the helical swimmers motion in creeping flow by presenting a comparison between predictions of the speed velocity given by finite elements simulations using an Oldroyd-B model for the base elastic fluid and experimental data. The agreement between the two sets of results is very good within the experimental error bars for the elastic parameter varying from 0 to 2. It may be remarked, however, that while the experimental data tend to saturate at larger De, the simulations results seem to have a continuous increase according to the constitutive model used to describe the base elastic liquid.

Aeroelastic modelling of tail fins for small wind turbines

Many small upwind turbines use a tail fin to align the rotor with the wind. Despite the importance of a well-designed fin for efficient operation and in generating ultimate and fatigue loads, the aeroelastic modelling of tail fins is not well developed. This work extends the previous linearized analyses by including nonlinear effects and the difference between the yaw angle and the angle of attack. The analysis is based on unsteady slender body theory, but includes the vortex lift generated at high angles of attack and the effects of vortex bursting. The model predicts with reasonable accuracy the yaw behaviour of delta-shaped tail fins (without a rotor) released at 45° in a wind tunnel, provided allowance is made for the viscous friction in the yaw bearings. The difference in the response frequency between the linear and nonlinear model increases at the higher yaw angle of 80° for which no wind tunnel measurements are available. As a first step towards simplifying the nonlinear model, the difference in yaw angle predictions from the linear model is estimated. The maximum difference is a function of initial yaw angle and is large for yaw angles of magnitude greater than 45°.

On Trapeze Wing Aerodynamics Calculations Based on Improved Vortex Lattice Method

This paper presents, aerodynamics coefficients calculation (Lifting &amp; drag coefficients, pressure central location) of Trapeze wing shape configurations for different aspect ratios (ARs) values by using improved vortex lattice method (VLM), compared with finite-wing and slender body theories. The planar wing was divided into N panels of the size: 6X6 with trapezoid shape panels. As expected, for high ARs the VLM solution for the lifting coefficient is coincided with the finite wing theory whereas for small ARs (&lt;1) it is coincided with the slender body theory (~1). Afterwards, we obtained that the calculated VLM induced drag becomes closer to the finitewing theory as the AR value is increased.

A unified theory of slender wings in asymmetric motion, with an application to a swimming sea snake

Abstract The slender body theory is extended to accommodate cases where a wing, in generally non-uniform motion, sideslips at sufficiently large angle to make the flow separate from one of its long edges but not from any part of the other. In those cases, an approximately triangular wake forms to the side of the wing, rendering the pressure loads on the wing not only asymmetrical, but also dependent on the history of the wing's motion. The theory can have a wide range of possible applications, from estimating lift and drag of a slender wing in pronouncedly asymmetric flight, through analysing an aero-elastic stability of that wing, to estimating thrust, lift and power of a sea snake that swims at an angle to its body axis. Coherence of the theory is demonstrated through numerous numerical simulations.

Tubular-body theory for viscous flows

Cable-like bodies play a key role in many interdisciplinary systems but are hard to simulate. Asymptotic theories, called slender-body theories, are effective but apply in specific regimes and can be hard to extend beyond leading order. In this letter we develop an exact slender-body-like theory for the surface traction of cable-like bodies in viscous flow. This theory expresses the traction as a series of solutions to a well-behaved one-dimensional Fredholm integral equation of the second kind. This process can be simply generalised to other systems.

Locomotion with a wavy cylindrical filament in a yield-stress fluid

A yield stress is added to Taylor's (1952, Proc. Royal Soc. A, 211, 225-239) model of a microscopic organism with a wavy cylindrical tail swimming through a viscous fluid. Viscoplastic slender-body theory is employed for the task, generalising existing results for Bingham fluid to the Herschel–Bulkley constitutive model. Numerical solutions are provided over a range of the two key parameters of the problem: the wave amplitude relative to the wavelength, and a Bingham number which describes the strength of the yield stress. Numerical solutions are supplemented with discussions of various limits of the problem in which analytical progress is possible. If the wave amplitude is sufficiently small, the yield stress of the material inevitably dominates the flow; the resulting ‘plastic locomotion’ results in swimming speeds that depend strongly on the swimming gait, and can, in some cases, even be negative. Conversely, when the yield stress is large, swimming becomes possible at the wave speed, with the swimmer sliding or burrowing along its centreline with a relatively high efficiency.

Techno-Economic Performance of Closed-Loop Geothermal Systems for Heat Production and Electricity Generation

• Closed-loop geothermal systems evaluated for heat and electricity production • Slow heat transfer in rock and limited heat transfer area pose challenges • Targeting high temperatures, existing wells, and multi-laterals beneficial • Relatively high levelized costs for greenfields, unless drilling at low cost Closed-loop geothermal systems, recently referred to as advanced geothermal systems (AGS), have received renewed interest for geothermal heat and power production. These systems consist of a co-axial, U-loop, or other configuration in which the heat transfer or working fluid does not permeate the reservoir but remains within a closed-loop subsurface heat exchanger. Advocates indicate its potential for developing geothermal energy anywhere, independent of site-specific geologic uncertainties, and with limited risk of induced seismicity. Here, we present a technical and economic analysis of closed-loop geothermal systems using a Slender-Body Theory (SBT) model, COMSOL Multiphysics simulator, and the GEOPHIRES analysis tool. We consider a number of different scenarios and evaluate the influence of variations in reservoir temperature (100 to 500℃), well termination depth (2 to 4 km), mass flow rate (10 to 40 kg/s), injection temperature (10 to 40℃), fluid type (liquid water vs. supercritical carbon dioxide), design configuration (co-axial vs. U-loop), and degree of reservoir convection (natural, forced or conduction-only). The resulting average heat production rates range from about 2 to 15 GWh per year for cases considering a co-axial design and from 9 to 67 GWh per year for cases with a U-loop design. Assuming generous economic and operating conditions, estimates of levelized cost of heat range from ∼$20 – $110 per MWh (∼$6 – 32/MMBtu) and ∼$10 – $70 per MWh (∼$3 – $20/MMBtu) for greenfield co-axial and U-loop cases, respectively. In the scenarios in which electricity generation is considered, annual electricity production ranged between 0.12 and 7.5 GWh per year at a levelized cost of electricity from roughly $83 to $2,200 per MWh. In all scenarios, the results exhibit a large rapid drop in production temperature after initiation of operations that levels off to a steady value significantly below the initial reservoir temperature. Operating at lower flow rate increases the production temperature but also lowers the total heat production. The consistently low production temperatures hinder efficient electricity generation in most cases considered. Natural or forced convection can increase thermal output but requires sufficiently high reservoir permeability or formation fluid flow. As expected, overall system costs are heavily dependent on drilling costs; hence, repurposing existing wells could significantly lower capital and levelized costs. In comparison with other types of geothermal systems, our results for closed-loop geothermal systems predict long-term production temperatures considerably below the initial reservoir temperature, and relatively high levelized costs for greenfield closed-loop geothermal systems, particularly for electricity production, unless significant reductions in drilling costs are obtained.

Simulations of dynamically cross-linked actin networks: Morphology, rheology, and hydrodynamic interactions.

Cross-linked actin networks are the primary component of the cell cytoskeleton and have been the subject of numerous experimental and modeling studies. While these studies have demonstrated that the networks are viscoelastic materials, evolving from elastic solids on short timescales to viscous fluids on long ones, questions remain about the duration of each asymptotic regime, the role of the surrounding fluid, and the behavior of the networks on intermediate timescales. Here we perform detailed simulations of passively cross-linked non-Brownian actin networks to quantify the principal timescales involved in the elastoviscous behavior, study the role of nonlocal hydrodynamic interactions, and parameterize continuum models from discrete stochastic simulations. To do this, we extend our recent computational framework for semiflexible filament suspensions, which is based on nonlocal slender body theory, to actin networks with dynamic cross linkers and finite filament lifetime. We introduce a model where the cross linkers are elastic springs with sticky ends stochastically binding to and unbinding from the elastic filaments, which randomly turn over at a characteristic rate. We show that, depending on the parameters, the network evolves to a steady state morphology that is either an isotropic actin mesh or a mesh with embedded actin bundles. For different degrees of bundling, we numerically apply small-amplitude oscillatory shear deformation to extract three timescales from networks of hundreds of filaments and cross linkers. We analyze the dependence of these timescales, which range from the order of hundredths of a second to the actin turnover time of several seconds, on the dynamic nature of the links, solvent viscosity, and filament bending stiffness. We show that the network is mostly elastic on the short time scale, with the elasticity coming mainly from the cross links, and viscous on the long time scale, with the effective viscosity originating primarily from stretching and breaking of the cross links. We show that the influence of nonlocal hydrodynamic interactions depends on the network morphology: for homogeneous meshworks, nonlocal hydrodynamics gives only a small correction to the viscous behavior, but for bundled networks it both hinders the formation of bundles and significantly lowers the resistance to shear once bundles are formed. We use our results to construct three-timescale generalized Maxwell models of the networks.

A single-layer based numerical method for the slender body boundary value problem

Fluid flows containing dilute or dense suspensions of thin fibers are widespread in biological and industrial processes. To describe the motion of a thin immersed fiber, or to describe the forces acting on it, it is convenient to work with one-dimensional fiber centerlines and force densities rather than two-dimensional surfaces and surface tractions. Slender body theories offer ways to model and simulate the motion of immersed fibers using only one-dimensional data. However, standard formulations can break down when the fiber surface comes close to intersecting itself or other fibers. In this paper we introduce a numerical method for a recently derived three-dimensional slender body boundary value problem that can be stated entirely in terms of a one-dimensional distribution of forces on the centerline. The method is based on a new completed single-layer potential formulation of fluid velocity which removes the nullspace associated with the unmodified single layer potential. We discretize the model and present numerical results demonstrating the good conditioning and improved performance of the method in the presence of near-intersections. To avoid the modeling and numerical choices involved with free ends, we consider closed fibers.

Numerical computation of the hydrodynamic coefficients on planing hulls in the six degrees of freedom

In this manuscript a mathematical model to study the forced motions on planing boats is developed, the hydrodynamic coefficients in the six degrees of freedom are computed, the formulation is based on flow potential theory and slender body theory and it allows to evaluate the pressure distribution and the forces in the six degrees of freedom. Virtual towing tank tests of the oscillatory forced motions in the vertical, horizontal, and transverse planes were carried out; the results of the simulations are compared with CFD runs and experimental data by other authors with good agreement, the current solution requires low computational resources in comparison with CFD simulations. The time series of forces in the forced test of heave, pitch and roll motions are high non-linear while the time series of forces in the tests of sway and yawing are linear.