We experimentally investigate the rotational dynamics of neutrally buoyant axisymmetric particles in a simple shear flow. A custom-built shearing cell and a multi-view shape-reconstruction method are used to obtain direct measurements of the orientation and period of rotation of particles having oblate and prolate shapes (such as spheroids and cylinders) of varying aspect ratios. By systematically changing the viscosity of the fluid, we examine the effect of inertia (which may be originated from either phase) on the dynamical behaviour of these suspended particles up to a particle Reynolds number of approximately one. While no significant effect on the period of rotation is found in this small-inertia regime, a systematic drift among several rotations toward limiting stable orbits is observed. Prolate particles are seen to drift towards the tumbling orbit in the plane of shear, whereas oblate particles are driven either to the tumbling or to the vorticity-aligned spinning orbits, depending on their initial orientation. These results are compared with recent small-inertia asymptotic theories, assessing their range of validity, as well as to numerical simulations in the small-inertia regime for both prolate and oblate particles.

Filaments are ubiquitous within the microscopic world, occurring in biological and industrial environments and displaying a varied dynamics. Their wide range of applications has spurred the development of a branch of asymptotics focused on the behaviour of filaments, called slender-body theory (SBT). Slender-body theories are computationally efficient and focus on the mechanics of an isolated fibre that is slender and not too curved. However, SBTs that work beyond these limits are needed to explore complex systems. Recently, we developed tubular-body theory (TBT), an approach like SBT that allows the hydrodynamic traction on any isolated fibre in a viscous fluid to be determined exactly. This paper extends TBT to model fibres near plane interfaces by performing a similar expansion on the single-layer boundary integrals (BIs) for bodies by a plane interface. This provides a well-behaved SBT inspired approach for fibres by interfaces with a similar versatility to the BIs but without the singular kernels. The derivation of the new theory, called tubular-body theory for interfaces (TBTi), also establishes a criterion for the convergence of the TBTi series representation. The TBTi equations are solved numerically using a approach similar to boundary element methods (BEMs), called TBTi-BEM, to investigate the properties of TBTi empirically. The TBTi-BEM is found to compare favourably with an existing BEM and the lubrication singularity on a sphere, suggesting TBTi is valid for all separations. Finally, we simulate the hydrodynamics of helices beneath a free interface and a plane wall to demonstrate the applicability of the technique.

Many of the cell membrane's vital functions are achieved by the self-organization of the proteins and biopolymers embedded in it. The protein dynamics is in part determined by its drag. A large number of these proteins can polymerize to form filaments. In vitro studies of protein–membrane interactions often involve using rigid beads coated with lipid bilayers, as a model for the cell membrane. Motivated by this, we use slender-body theory to compute the translational and rotational resistance of a single filamentous protein embedded in the outer layer of a supported bilayer membrane and surrounded on the exterior by a Newtonian fluid. We first consider the regime where the two layers are strongly coupled through their inter-leaflet friction. We find that the drag along the parallel direction grows linearly with the filament's length and quadratically with the length for the perpendicular and rotational drag coefficients. These findings are explained using scaling arguments and by analysing the velocity fields around the moving filament. We then present and discuss the qualitative differences between the drag of a filament moving in a freely suspended bilayer and a supported membrane as a function of the membrane's inter-leaflet friction. Finally, we briefly discuss how these findings can be used in experiments to determine membrane rheology. In summary, we present a formulation that allows computation of the effects of membrane properties (its curvature, viscosity and inter-leaflet friction), and the exterior and interior three-dimensional fluids’ depth and viscosity on the drag of a rod-like/filamentous protein, all in a unified theoretical framework.

The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio, $\kappa$. A regular perturbation expansion in the polymer concentration, $c$, a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the $O(c)$ correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon $c\, De$ ($De$ is the imposed shear rate times the polymer relaxation time) and $\kappa$ and quantitatively on $c$. At a small but finite $c\, De$, the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing $\kappa$, the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate $c\, De$, a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller $c\, De$) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing $c\, De$, the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing $c\, De$, the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.

As is known, the presence of surfactants can profoundly influence the dynamics of Newtonian viscous threads. Also, it is known that non-Newtonian viscoelastic threads behave differently from Newtonian ones, particularly in the nonlinear regime. A naturally arising question is how surfactants affect the dynamic behaviour of non-Newtonian viscoelastic threads. To gain some insights into it, we build a one-dimensional model for an Oldroyd-B/finitely extensible nonlinear elastic-Peterlin approximation (FENE-P) viscoelastic liquid thread covered with an insoluble surfactant monolayer based on the slender body theory. A parametric study is performed to examine the effects of the dimensionless numbers related to the surfactant, including the initial concentration, the Marangoni number, the surface Péclet number, the shear Boussinesq number and the dilatational Boussinesq number. It is found that the formation of the beads-on-a-string structure can be greatly delayed by the surfactant. At large values of the surface Péclet number, the exponential thinning of the Oldroyd-B viscoelastic thread is little influenced, but the surfactant may lead to the disappearance of secondary droplets. At moderate values of the surface Péclet number, the surfactant induces the formation of secondary droplets. The primary droplets are axially stretched by the Marangoni or surface viscous stresses and evolve into a prolate or a more singular shape eventually. The surfactant can delay the pinch-off of the FENE-P viscoelastic thread to a great extent, but it affects little the decrease in the minimum thread radius prior to pinch-off when the surface Péclet number is large.

Many microorganisms use cilia to propel themselves in low Reynolds number ($Re$) environments. In this work, we study the dynamics of a composite cilium consisting of an elastic filament and a spherical particle attached at the filament tip driven by an external time-periodic force acting on the particle. The elastic filament is modelled numerically using a slender body theory with hydrodynamic interactions. When tilted at a large angle from the normal direction of the wall, the filament buckles, and the induced velocity field by the cilium shows a large net flux. By varying the tilt angle or the force amplitude, the particle trajectory and the net flux display abrupt changes along with a reversal of the buckling direction. We further demonstrate through a segmental model that the abrupt changes arise from the deviation of the cilium orientation at the start of the recovery stroke from the natural orientation. Our results suggest a simple approach to engineering particle motions and designing artificial cilia for fluid pumping in low $Re$ environments.

In many natural settings, spatial variations in a fluid's viscosity arise due to changes in its local physico-chemical environment. We consider the low-Reynolds-number dynamics of slender bodies in fluids with a linearly varying viscosity. Assuming the spatial change in viscosity to be small compared to the spatial deviation of the body but large relative to the body's aspect ratio, we derive the modifications to resistive-force theory in a fluid due to a constant viscosity gradient. At leading order in the body slenderness, the results are identical to the classical theory in a constant-viscosity fluid but with the viscosity taking everywhere its local, non-constant value. At next order in slenderness, non-local terms arise due to the non-zero viscosity gradient. We use our results to predict the motion of straight and toroidal filaments settling under the action of gravity. We show that viscosity gradients induce rigid-body rotation of the filaments at a rate proportional to the components of the gradients along the filaments. This result contrasts with constant-viscosity fluids where the filaments do not rotate. We demonstrate further that if the viscosity gradient acts in the direction opposite to the gravitational field, then the filaments rotate towards a stable orientation, whose value depends on the ratio between the viscosity gradients parallel and perpendicular to the gravitational field; otherwise, the filaments align in the direction of the gravitational field. Our work shows that viscosity gradients can exert new forces on slender bodies, which could, for example, be used to control their orientation and drift.

The fluid–structure interactions between flexible fibres and viscous flows play an essential role in various biological phenomena, medical problems and industrial processes. Of particular interest is the case of particles transported freely 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 modelling. 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 behaviour through a weakly nonlinear Landau theory of buckling, in which we treat the filaments as inextensible Brownian Euler–Bernoulli rods whose hydrodynamics is described by local slender-body theory.

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.

Earthquakes are a great challenge for the safety of nuclear reactors. To address this challenge, we need to better understand how the reactor core responds to seismic forcing. The reactor core is made of fuel assemblies, which are themselves composed of flexible fuel rods immersed in a strong axial flow. This gives rise to strongly coupled fluid–structure interactions whose accurate modelling generally requires high computational costs. In this paper, we introduce a new model able to capture the mechanical response of the reactor core subjected to seismic forcing with low computational costs. This model is based on potential flow theory for the fluid part, and Euler–Bernoulli beam theory for the structural part, allowing us to predict the response to seismic forcing in presence of axial flow. The linear equations are solved in the Fourier space to decrease computational time. For validation purposes, first we use the proposed model to compute the response of a single cylinder in axial flow. We then implement a multiple-cylinder geometry made of four fuel assemblies, each made of $8\times 8$ cylinders, corresponding to an experimental facility available at CEA. The comparison between numerical results and experiments shows good agreement. The model can predict correctly the added mass. It can also capture qualitatively the coupling between assemblies and the effect of confinement. This shows that a potential flow approach can give insight into the complex fluid–structure interactions within a nuclear reactor and, in particular, be used to predict the response to seismic forcing at low computational cost.

At low Reynolds numbers, axisymmetric ellipsoidal particles immersed in a shear flow undergo periodic tumbling motions known as Jeffery orbits, with the orbit determined by the initial orientation. Understanding this motion is important for predicting the overall dynamics of a suspension. While slender fibres may follow Jeffery orbits, many such particles in nature are neither straight nor rigid. Recent work exploring the dynamics of curved or elastic fibres have found Jeffery-like behaviour along with chaotic orbits, decaying orbital constants and cross-streamline drift. Most work focuses on particles with reflectional symmetry; we instead consider the behaviour of a composite asymmetric slender body made of two straight rods, suspended in a two-dimensional shear flow, to understand the effects of the shape on the dynamics. We find that for certain geometries the particle does not rotate and undergoes persistent drift across streamlines, the magnitude of which is consistent with other previously identified forms of cross-streamline drift. For this class of particles, such geometry-driven cross-streamline motion may be important in giving rise to dispersion in channel flows, thereby potentially enhancing mixing.

The centrifugal spinning method is a recently invented technique to extrude polymer melts/solutions into ultra-fine nanofibres. Here, we present a superior integrated string-based mathematical model, to quantify the nanofibre fabrication performance in the centrifugal spinning process. Our model enables us to analyse the critical flow parameters covering an extensive range, by incorporating the angular momentum equations, the Giesekus viscoelastic constitutive model, the air-to-fibre drag effects and the energy equation into the string model equations. Using the model, we can analyse the dynamic behaviour of polymer melt/solution jets through the dimensionless flow parameters, namely, the Rossby ($Rb$), Reynolds ($Re$), Weissenberg ($Wi$), Weber ($We$), Froude ($Fr$), air Péclet ($Pe^*$) and air Reynolds ($Re^*$) numbers as well as the viscosity ratio ($\delta _s$), corresponding to rotational, inertial, viscous, viscoelastic, surface tension, gravitational, air thermal diffusivity, aerodynamic and viscosity ratio effects. We find that the nonlinear rheology remarkably affects the fibre trajectory, radius and normal stresses. Increasing $Wi$ leads to a thicker fibre, whereas increasing $\delta _s$ shows an opposite trend. In addition, by increasing $Wi$, the fibre curvature is enhanced, causing the fibre to spiral closer to the rotation centre.

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.

In this paper we investigate the statics and the dynamics of large-viscosity ligaments attached to a rod and drawn out of a pure-liquid bath. Following the similar work of films pulling out of a bath (Champougny et al., J. Fluid Mech., vol. 811, 2017, pp. 499–524), a one-dimensional model is applied to describe the ligaments drawn at constant velocities. We focus on the whole drawing dynamics of the ligament up to breakup, for which the breakup height is determined. The breakup height coincides with the maximum static meniscus height for very slow drawing, whose process can be described by quasi-static solutions. We present the numerical results of the static menisci and analytically unravel the mechanism in the low gravity case. Starting from a stable static meniscus, the breakup height of faster drawing depends separately on the rod radius and the drawing velocity, the latter dependency being fully determined by considering the agravic limit. Next, it is shown that the entire lifetime of the ligament drawing can be sequenced into a ductility stage, a capillarity stage and a pinch-off stage, the latter being shown to be almost instantaneous. The ductility and capillarity stages are decorrelated with the help of an approximate solution of the ductility stage, and the transition between the two stages corresponds to the time at which the capillarity-induced contraction velocity exceeds the ductility-induced one. The one-dimensional predictions of the breakup height and the entrained liquid volume attached to the rod quantitatively agree with experimental results of silicone oil ligaments, and the deviations are rationalized in comparison with a two-dimensional model.

We study the behaviour of circular flexible loops sedimenting in a viscous fluid by numerical simulations and linear stability analysis. The numerical model involves a local slender-body theory approximation for the flow coupled to the Euler–Bernoulli elastic forces for an inextensible fibre. Starting from an inclined circle, we simulate the dynamics using truncated Fourier modes to observe three distinct regimes of motion: absolute stability, two- and three-dimensional dynamics, depending on the relative importance of the elastic and gravitational forces. We identify the governing parameter and develop a simple semi-analytic stability criterion, which we verify numerically. In all cases, sedimenting loops converge to stable, planar shape equilibria with one free parameter related to the initial conditions and material properties of the fibre.

The propagation of wave disturbances from a complex multi-fault submarine earthquake of slender rectangular segments in a sea of constant depth is discussed, accounting for both water compressibility and gravity effects. It is found that including gravity effects the modal envelopes of the modified two-dimensional acoustic waves and the tsunami are governed by the Schrödinger equation. An explicit solution is derived using a multi-fault approach that allows capturing the main peak of the tsunami. Moreover, a linear superposition of the solution allows solving complicated multi-fault ruptures, in particular in the absence of dissipation due to large variations in depth. Consequently, the modulations of acoustic waves due to gravity, and of tsunami due to compressibility, are governed simultaneously and accurately, which is essential for practical applications such as tsunami early warning systems. The results are validated numerically against the mild-slope equation for weakly compressible fluids.

The elastohydrodynamics of slender bodies in a viscous fluid have long been the source of theoretical investigation, being pertinent to the microscale world of ciliates and flagellates as well as to biological and engineered active matter more generally. Although recent works have overcome the severe numerical stiffness typically associated with slender elastohydrodynamics, employing both local and non-local couplings to the surrounding fluid, there is no framework of comparable efficiency that rigorously justifies its hydrodynamic accuracy. In this study, we combine developments in filament elastohydrodynamics with a recent regularised slender-body theory, affording algebraic asymptotic accuracy to the commonly imposed no-slip condition on the surface of a slender filament of potentially non-uniform cross-sectional radius. Further, we do this whilst retaining the remarkable practical efficiency of contemporary elastohydrodynamic approaches, having drawn inspiration from the method of regularised Stokeslet segments to yield an efficient and flexible slender-body theory of regularised non-uniform segments.

We present a numerical study of the dynamics of an elastic fibre in a shear flow at low Reynolds number, and seek to understand several aspects of the fibre's motion using the equations for slender-body theory coupled to the elastica. The numerical simulations are performed in the bead-spring framework including hydrodynamic interactions in two theoretical schemes: the generalized Rotne–Prager–Yamakawa model and a multipole expansion corrected for lubrication forces. In general, the two schemes yield similar results, including for the dominant scaling features of the shape that we identify. In particular, we focus on the evolution of an initially straight fibre oriented in the flow direction and show that the time scales of fibre bending, curling and rotation, which depend on its length and stiffness, determine the overall motion and evolution of the shapes. We document several characteristic time scales and curvatures representative of the shape that vary as power laws of the bending stiffness and fibre length. The numerical results are further supported by an interpretation using an elastica model.

The collapse under surface tension of a long axisymmetric capillary, held at both ends and softened by a travelling heater, is used to determine the viscosity or surface tension of silica glasses. Capillary collapse is also used in the manufacture of some optical fibre preforms. Typically, a one-dimensional (1-D) model of the closure of a concentric fluid annulus is used to relate a measure of the change in the cross-sectional geometry, for example the external radius, to the desired information. We here show that a two-dimensional (2-D) asymptotic model developed for drawing of optical fibres, but with a unit draw ratio, may be used and yields analytic formulae involving a single dimensionless parameter, the scaled heater speed $V$, equivalently a capillary number. For a capillary fixed at both ends, this 2-D model agrees with the 1-D model and offers the significant benefit that it enables determination of both the surface tension and viscosity from a single capillary-collapse experiment, provided the pulling tension in the capillary during collapse is measured. The 2-D model also enables our investigation of the situation where both ends of the capillary are not fixed, so that the capillary cannot sustain a pulling tension. Then the collapse of the capillary is markedly different from that predicted by the 1-D model and the ability to determine both surface tension and viscosity is lost.