Slender Body Theory Research Articles

Inertial torques and a symmetry breaking orientational transition in the sedimentation of slender fibres

Experimental measurements of the force and torque on freely settling fibres are compared with predictions of the slender-body theory of Khayat & Cox (J. Fluid Mech., vol. 209, 1989, pp. 435–462). Although the flow is viscous dominated at the scale of the fibre diameter, fluid inertia is important on the scale of the fibre length, leading to inertial torques which tend to rotate symmetric fibres toward horizontal orientations. Experimentally, the torque on symmetric fibres is inferred from the measured rate of rotation of the fibres using a quasi-steady torque balance. It is shown theoretically that fibres with an asymmetric radius or mass density distribution undergo a supercritical pitch-fork bifurcation from vertical to oblique settling with increasing Archimedes number, increasing Reynolds number or decreasing asymmetry. This transition is observed in experiments with asymmetric mass density and we find good agreement with the predicted symmetry breaking transition. In these experiments, the steady orientation of the oblique settling fibres provides a means to measure the inertial torque in the absence of transient effects since it is balanced by the known gravitational torque.

Multidisciplinary design optimization of long-range slender guided rockets considering aeroelasticity and subsidiary loads

Due to the insufficient rigidity feature and long-range requirement, it is crucial to consider the aeroelasticity and subsidiary loads caused by the Earth rotation and fuel consumption when designing long-range slender guided rockets (LRSGRs). As a typical multidisciplinary design optimization (MDO) problem, the design optimization of LRSGRs confronts two critical challenges, i.e., accurate multidisciplinary modeling and efficient global optimization. To address the challenges, a novel MDO framework including MDO problem definition, multidisciplinary modeling, and metamodel-based optimizer is developed for LRSGR design. The LRSGR MDO problem is formulated to minimize the total mass subject to a number of practical engineering constraints such as bending mode frequencies, miss distance, and fall angle. Several disciplinary models including structure, aerodynamics, propulsion, mass, aeroelasticity, guidance control, and trajectory are established. To enhance the analysis accuracy, structural finite element analysis (FEA), three-channel autopilot, and high-fidelity trajectory models are adopted. In the aeroelasticity model, the unsteady aerodynamic loads are calculated by slender body theory and aerodynamic derivative method. The subsidiary loads including subsidiary Coriolis force, centrifugal inertial force, Coriolis force, and subsidiary Coriolis moment are incorporated in the trajectory model of LRSGRs. Since structural finite element, aeroelasticity, and trajectory models are computationally expensive (about 1.8 hours for one trial of system analysis on a well-equipped workstation), an adaptive radial basis function metamodel-based optimizer is integrated in the framework to solve the LRSGR MDO problem with moderate computational cost. The total mass of the studied LRSGR is decreased by 88 kg (i.e., 14% of the total mass) after optimization, which demonstrates the effectiveness and practicability of the proposed MDO framework for LRSGRs.

Complex dynamics of long, flexible fibers in shear

The macroscopic properties of polymeric fluids are inherited from the material properties of the fibers embedded in the solvent. The behavior of such passive fibers in flow has been of interest in a wide range of systems, including cellular mechanics, nutrient acquisition by diatom chains in the ocean, and industrial applications such as paper manufacturing. The rotational dynamics and shape evolution of fibers in shear depends upon the slenderness of the fiber and the non-dimensional “elasto-viscous” number that measures the ratio of the fluid’s viscous forces to the fiber’s elastic forces. For a small elasto-viscous number, the nearly-rigid fiber rotates in the shear, but when the elasto-viscous number reaches a threshold, buckling occurs. For even larger elasto-viscous numbers, there is a transition to a “snaking behavior” where the fiber remains aligned with the shear axis, but its ends curl in, in opposite directions. These experimentally-observed behaviors have recently been characterized computationally using slender-body theory and immersed boundary computations. However, classical experiments with nylon fibers and recent experiments with actin filaments have demonstrated that for even larger elasto-viscous numbers, multiple buckling sites and coiling can occur. Using a regularized Stokeslet framework coupled with a kernel independent fast multipole method, we present simulations that capture these complex fiber dynamics.

Deformation and stability of a viscous electrolyte drop in a uniform electric field

We study the deformation and breakup of an axisymmetric electrolyte drop which is freely suspended in an infinite dielectric medium and subjected to an imposed electric field. The electric potential in the drop phase is assumed small, so that its governing equation is approximated by a linearized Poisson-Boltzmann or modified Helmholtz equation (the Debye-H\"{u}ckel regime). An accurate and efficient boundary integral method is developed to solve the low-Reynolds-number flow problem for the time-dependent drop deformation, in the case of arbitrary Debye layer thickness. Extensive numerical results are presented for the case when the viscosity of the drop and surrounding medium are comparable. Qualitative similarities are found between the evolution of a drop with a thick Debye layer (characterized by the parameter $\chi\ll 1$, which is an inverse dimensionless Debye layer thickness) and a perfect dielectric drop in an insulating medium. In this limit, a highly elongated steady state is obtained for sufficiently large imposed electric field, and the field inside the drop is found to be well approximated using slender body theory. In the opposite limit $\chi\gg 1$, when the Debye layer is thin, the drop behaves as a highly conducting drop, even for moderate permittivity ratio $Q=\epsilon_1/\epsilon_2$, where $\epsilon_1, \epsilon_2$ is the dielectric permittivity of drop interior and exterior, respectively. For parameter values at which steady solutions no longer exist, we find three distinct types of unsteady solution or breakup modes. These are termed conical end formation, end splashing, and open end stretching. The second breakup mode, end splashing, resembles the breakup solution presented in a recent paper [R. B. Karyappa et al., J. Fluid Mech. 754, 550-589 (2014)]. We compute a phase diagram which illustrates the regions in parameter space in which the different breakup modes occur.

Coupled CFD-FEM simulation of hydrodynamic responses of a CALM buoy

In this paper, the Finite-Analytic Navier-Stokes (FANS) code is coupled with an in-house finite-element code to study the dynamic interaction between a floating buoy and its mooring system. Hydrodynamic loads on the buoy are predicted with the FANS module, in which Large Eddy Simulation (LES) is used as the turbulence model. The mooring lines are modeled based on a slender body theory. Their dynamic responses are simulated with a nonlinear finite element module, MOORING3D. The two modules are coupled by transferring the forces and displacements of the buoy and its mooring system at their connections through an interface module. A free-decay model test was used to calibrate the coupled method. In addition, to investigate the capability of the present coupled method, numerical simulations of two degree-of-freedom vortex-induced motion of a CALM buoy in uniform currents were performed. With the study it can be verified that accurate predictions of the motion responses and tension responses of the CALM buoy system can be made with the coupling CFD-FEM method.

Swimming of Spermatozoa in a Maxwell Fluid.

It has been suggested that the swimming mechanism used by spermatozoa could be adopted for self-propelled micro-robots in small environments and potentially applied to biomedical engineering. Mammalian sperm cells must swim through a viscoelastic mucus layer to find the egg cell. Thus, understanding how sperm cells swim through viscoelastic liquids is significant not only for physiology, but also for the design of micro-robots. In this paper, we developed a numerical model of a sperm cell in a linear Maxwell fluid based on the boundary element slender-body theory coupling method. The viscoelastic properties were characterized by the Deborah number (De), and we found that, under the prescribed waveform, the swimming speed decayed with the Deborah number in the small-De regime (De < 1.0). The swimming efficiency was independent of the Deborah number, and the decrease in the swimming speed was not significantly affected by the wave pattern.

Dropping slender-body theory into the mud

The equations describing classical viscous fluid flow are notoriously challenging to solve, even approximately, when the flow is host to one or many immersed bodies. When an immersed body is slender, the smallness of its aspect ratio can sometimes be used as a basis for a ‘slender-body theory’ describing its interaction with the surrounding environment. If the fluid is complex, however, such theories are generally invalid and efforts to understand the dynamics of immersed bodies are almost entirely numerical in nature. In a valiant effort, Hewitt &amp; Balmforth (J. Fluid Mech., vol. 856, 2018, pp. 870–897) have unearthed a theory to describe the motion of slender bodies in a viscoplastic fluid, ‘fluids’ such as mud or toothpaste which can be coaxed to flow, but only with a sufficiently large amount of forcing. Mathematical theories for some tremendously complicated physical systems may now be within reach.

Scale-dependent alignment, tumbling and stretching of slender rods in isotropic turbulence

We examine the dynamics of slender, rigid rods in direct numerical simulation of isotropic turbulence. The focus is on the statistics of three quantities and how they vary as rod length increases from the dissipation range to the inertial range. These quantities are (i) the steady-state rod alignment with respect to the perceived velocity gradients in the surrounding flow, (ii) the rate of rod reorientation (tumbling) and (iii) the rate at which the rod end points move apart (stretching). Under the approximations of slender-body theory, the rod inertia is neglected and rods are modelled as passive particles in the flow that do not affect the fluid velocity field. We find that the average rod alignment changes qualitatively as rod length increases from the dissipation range to the inertial range. While rods in the dissipation range align most strongly with fluid vorticity, rods in the inertial range align most strongly with the most extensional eigenvector of the perceived strain-rate tensor. For rods in the inertial range, we find that the variance of rod stretching and the variance of rod tumbling both scale as $l^{-4/3}$, where $l$ is the rod length. However, when rod dynamics are compared to two-point fluid velocity statistics (structure functions), we see non-monotonic behaviour in the variance of rod tumbling due to the influence of small-scale fluid motions. Additionally, we find that the skewness of rod stretching does not show scale invariance in the inertial range, in contrast to the skewness of longitudinal fluid velocity increments as predicted by Kolmogorov’s $4/5$ law. Finally, we examine the power-law scaling exponents of higher-order moments of rod tumbling and rod stretching for rods with lengths in the inertial range and find that they show anomalous scaling. We compare these scaling exponents to predictions from Kolmogorov’s refined similarity hypotheses.

On nonlinear rotating electrified non-Newtonian jets

We consider a nonlinear non-Newtonian rotating jet flow whose centerline is curved and is elongated by an electric force due to an imposed electric field. The jet is driven both by the imposed electric and rotational forces. We introduce a non-Newtonian viscosity model, which, in particular, takes into account both extension thinning and thickening of the jet. From the governing equations and the boundary conditions of such jet flow, we construct a modeling system based on the slender-body theory for such nonlinear jet and calculate numerically the expressions for the nonlinear steady solutions for the jet quantities such as radius, speed, stretching rate, induced electric field and surface charge versus arc length. We determine these quantities for different values of the parameters that represent effects due to rotation, electric field, electric conductivity, viscosity and relaxation time. We find, in particular, that the jet speeds up, stretches up, and its diameter reduces significantly with increasing the imposed electric force, rotational forces and the jet relaxation time. The notable jet radius reduction that is due to strong electric and rotational effects is found to be for jet dominated by negative surface charge whose magnitude enhances with the rotation rate and the electrical conductivity.

Viscoplastic slender-body theory

The theory of slow viscous flow around a slender body is generalized to the situation where the ambient fluid has a yield stress. The local flow around a cylinder that is moving along or perpendicular to its axis, and rotating, provides a first step in this theory. Unlike for a Newtonian fluid, the nonlinearity associated with the viscoplastic constitutive law precludes one from linearly superposing solutions corresponding to each independent component of motion, and instead demands a full numerical approach to the problem. This is accomplished for the case of a Bingham fluid, along with a consideration of some asymptotic limits in which analytical progress is possible. Since the yield stress of the fluid strongly localizes the flow around the body, the leading-order slender-body approximation is rendered significantly more accurate than the equivalent Newtonian problem. The theory is applied to the sedimentation of inclined cylinders, bent rods and helices, and compared with some experimental data. Finally, the theory is applied to the locomotion of a cylindrical filament driven by helical waves through a viscoplastic fluid.

3D motion dynamics of axisymmetric bodies falling through water

A new simplified method for analysis of 3D motion dynamics of axisymmetric slender bodies falling from air through water is presented. Slender-body theory for potential flow of incompressible water is combined with 2D + t theory to account for viscous cross-flow separation in the submerged phase. The effect of flow separation from the upstream end of a pipe plays an important role. Furthermore, asymmetric vortex shedding triggering important 3D motions is considered. The water impact loads are based on strip theory and empirical slamming coefficients together with added mass and buoyancy loads. The theory is validated by comparing numerical analyses with experimental drop tests of pipes performed by Aanesland (1987). Error analyses of the theoretical method are performed and compared with tests to explore the sensitivity to input variations.

Violent water entry of spheres

Water entry is an impulsive two-phase flow often met in nature and in industrial processes. Violent vertical entries are observed when the projectile is heavy and when the Froude number is high enough: they form a distinct category characterized by an elongated cavity trailed by the projectile. The guiding idea in the present approach is to regard the union of the deep cavity caped by the sphere as a composite deformable body relevant to the theory of flows past slender bodies. This process allows linearization, making the problem analytically tractable when air behaves as a passive component, a case occurring very often due to the contrast in density between air and water. Asymptotic analysis shows that the length of the slender body is of the order of RF 2∕3, with F=U(gR)−12 the Froude number, R being the radius, U the speed at entry and g the gravity. That gives rise to a parameter scaling the importance of the air flow inside the cavity, D=(ρa∕ρ)F4∕3, with ρ and ρa the water and air densities. In what follows we adopt the mathematical symbols o and O , with D=o(1) when D≪1 and D=O(1) when D is of the order of 1 but cannot become ≪1. As long as water and air are considered as inviscid, the dimensionless parameters governing the flow are F, D and ρb∕ρ, with ρb the projectile density. In the present contribution we introduce the concept of ideal deep cavity, which does not take into account some transitory perturbations occurring just after impact. Access is obtained to the water velocity potential and to the equations governing the shape of the cavity and the air flow filling the cavity. The problem is solved for D=o(1), which is met in most cases owing to the smallness of ρa /ρ, even if F is very high. Water velocity and air pressure are expressed in terms of the sectional area S of the slender body. Still for D=o(1), a full solution including friction, turbulence and separation is obtained afterward provided the characteristic Reynolds number Re is high. However, transient unsteadiness instigated by a sudden impingement, or transition from laminar to turbulent regimes cannot be characterized by quantitative criteria put in advance, making the task difficult. One of significant features revealed by the present contribution is the important role of friction, which has been generally ignored in previous investigations. The cavity seal appears to be closely connected to friction and instability through the way how separation occurs. The theoretical model is consistent with separation taking place above the equator. A typical property arises: violent cavity sealing occurs in the first third of the distance between atmosphere and the sphere. Present results come complete past entry models elaborated at moderate Froude number. Finally, validation is based on results obtained almost seventy years ago at the Naval Ordnance Laboratory, USA.

A higher-order slender-body theory for axisymmetric flow past a particle at moderate Reynolds number

Slender-body theory is utilized to derive an asymptotic approximation to the hydrodynamic drag on an axisymmetric particle that is held fixed in an otherwise uniform stream of an incompressible Newtonian fluid at moderate Reynolds number. The Reynolds number,$Re$, is based on the length of the particle. The axis of rotational symmetry of the particle is collinear with the uniform stream. The drag is expressed as a series in powers of$1/\text{ln}(1/\unicode[STIX]{x1D716})$, where$\unicode[STIX]{x1D716}$is the small ratio of the characteristic width to length of the particle; the series is asymptotic for$Re\ll O(1/\unicode[STIX]{x1D716})$. The drag is calculated through terms of$O[1/\text{ln}^{3}(1/\unicode[STIX]{x1D716})]$, thereby extending the work of Khayat &amp; Cox (J. Fluid Mech., vol. 209, 1989, pp. 435–462) who determined the drag through$O[1/\text{ln}^{2}(1/\unicode[STIX]{x1D716})]$. The calculation of the$O[1/\text{ln}^{3}(1/\unicode[STIX]{x1D716})]$term is accomplished via the generalized reciprocal theorem (Lovalenti &amp; Brady,J. Fluid Mech., vol. 256, 1993, pp. 561–605). The first dependence of the inertial contribution to the drag on the cross-sectional profile of the particle is at$O[1/\text{ln}^{3}(1/\unicode[STIX]{x1D716})]$. Notably, the drag is insensitive to the direction of travel at this order. The asymptotic results are compared to a numerical solution of the Navier–Stokes equations for the case of a prolate spheroid. Good agreement between the two is observed at moderately small values of$\unicode[STIX]{x1D716}$, which is surprising given the logarithmic error associated with the asymptotic expansion.

Biological fluid mechanics under the microscope: a tribute to John Blake

John Blake (1947–2016) was a leader in fluid mechanics, his two principal areas of expertise being biological fluid mechanics on microscopic scales and bubble dynamics. He produced leading research and mentored others in both Australia, his home country, and the UK, his adopted home. This article reviews John Blake’s contributions in biological fluid mechanics, as well as gives the author’s personal viewpoint as one of the many graduate students and researchers who benefitted from his supervision, guidance and inspiration. The key topics from biological mechanics discussed are: “squirmer” models of protozoa, the method of images in Stokes flow and the “blakelet” solution, discrete cilia modelling via slender body theory, physiological flows in respiration and reproduction, blinking stokeslets in microorganism feeding, human sperm motility and embryonic nodal cilia. doi:10.1017/S1446181118000020

Supervised Learning to Predict Sperm Sorting by Magnetophoresis

Machine learning is gaining popularity in the commercial world, but its benefits are yet to be well-utilised by many in the microfluidics community. There is immense potential in bridging the gap between applied engineering and artificial intelligence as well as statistics. We illustrate this by a case study investigating the sorting of sperm cells for assisted reproduction. Slender body theory (SBT) is applied to compute the behavior of sperm subjected to magnetophoresis, with due consideration given to statistical variations. By performing computations on a small subset of the generated data, we train an ensemble of four supervised learning algorithms and use it to make predictions on the velocity of each sperm. Our results suggest that magnetophoresis can magnify the difference between normal and abnormal cells, such that a sorted sample has over twice the proportion of desirable cells. In addition, we demonstrated that the predictions from machine learning gave comparable results with significantly lower computational costs.

The boundary integral formulation of Stokes flows includes slender-body theory

The incompressible Stokes equations can classically be recast in a boundary integral (BI) representation, which provides a general method to solve low-Reynolds-number problems analytically and computationally. Alternatively, one can solve the Stokes equations by using an appropriate distribution of flow singularities of the right strength within the boundary, a method that is particularly useful to describe the dynamics of long slender objects for which the numerical implementation of the BI representation becomes cumbersome. While the BI approach is a mathematical consequence of the Stokes equations, the singularity method involves making judicious guesses that can only be justified a posteriori. In this paper, we use matched asymptotic expansions to derive an algebraically accurate slender-body theory directly from the BI representation able to handle arbitrary surface velocities and surface tractions. This expansion procedure leads to sets of uncoupled linear equations and to a single one-dimensional integral equation identical to that derived by Keller &amp; Rubinow (J. Fluid Mech., vol. 75, 1976, p. 705) and Johnson (J. Fluid Mech., vol. 99, 1979, p. 411) using the singularity method. Hence, we show that it is a mathematical consequence of the BI approach that the leading-order flow around a slender body can be represented using a distribution of singularities along its centreline. Furthermore, when derived from either the single-layer or the double-layer modified BI representation, general slender solutions are only possible in certain types of flow, in accordance with the limitations of these representations.

Hull form optimization in the conceptual design stage considering operational efficiency in waves

This study focuses on the optimization of ship dimensions by considering hydrodynamic performance in waves. In actual seaways, a ship experiences speed loss due to environmental loads by waves and wind. Therefore, along with calm water resistance, speed loss in waves should be considered in the hull form design in order to improve operational efficiency in waves. However, a trade-off may be needed between total resistance on the ship and the speed loss in waves. To address this problem, Non-dominated Sorting Genetic Algorithm II, which is a multi-objective optimization method, is used to minimize the total resistance on a ship in seaways and the speed loss by additional resistance. In the optimization process, added resistance is predicted using a numerical method based on slender-body theory, Maruo’s far-field formulation, and an empirical formula for added resistance in short waves. The speed loss in waves, which can be expressed by a weather factor ( fw), is estimated using power–speed curves. This article introduces some examples of the sensitivity analysis of added resistance and speed loss in waves to the variations of ship dimensions. Finally, the optimization solutions on a Pareto front set are compared to a basis ship in terms of hull form, and the corresponding hydrodynamic performances are evaluated.

A study on supercavitation in axisymmetric subsonic liquid flow past slender conical body

A novel theoretical model for solving the cavities in axisymmetric supercavitating flow past a slender conical body in subsonic fluid has been established in the present paper based on the slender body theory. The fluid compressibility has been taken into consideration in the present model. The nonlinear integral differential equation is derived for solving subsonic supercavitating flow. The numerical discrete method and the iterative process to solve the equation are presented in this paper. The critical Mach number are obtained to describe the subsonic flow. The results of supercavity shapes and the hydrodynamic coefficients obtained by the present theoretical model are compared with the results of other literatures, which verifies the present model have theoretical accuracy and broad application. Finally we discuss the compressibility effects on cavity shape, surface pressure distribution and drag coefficient in the subsonic liquid flow.

Spheroidal and conical shapes of ferrofluid-filled capsules in magnetic fields

We investigate the deformation of soft spherical elastic capsules filled with a ferrofluid in external uniform magnetic fields at fixed volume by a combination of numerical and analytical approaches. We develop a numerical iterative solution strategy based on nonlinear elastic shape equations to calculate the stretched capsule shape numerically and a coupled finite element and boundary element method to solve the corresponding magnetostatic problem, and employ analytical linear response theory, approximative energy minimization, and slender-body theory. The observed deformation behavior is qualitatively similar to the deformation of ferrofluid droplets in uniform magnetic fields. Homogeneous magnetic fields elongate the capsule, and a discontinuous shape transition from a spheroidal shape to a conical shape takes place at a critical field strength. We investigate how capsule elasticity modifies this hysteretic shape transition. We show that conical capsule shapes are possible but involve diverging stretch factors at the tips, which gives rise to rupture for real capsule materials. In a slender-body approximation we find that the critical susceptibility above which conical shapes occur for ferrofluid capsules is the same as for droplets. At small fields capsules remain spheroidal, and we characterize the deformation of spheroidal capsules both analytically and numerically. Finally, we determine whether wrinkling of a spheroidal capsule occurs during elongation in a magnetic field and how it modifies the stretching behavior. We find the nontrivial dependence between the extent of the wrinkled region and capsule elongation. Our results can be helpful in order to quantitatively determine capsule or ferrofluid material properties from magnetic deformation experiments. All results also apply to elastic capsules filled with a dielectric liquid in an external uniform electric field.

Reduced-order model for inertial locomotion of a slender swimmer.

The inertial locomotion of an elongated model swimmer in a Newtonian fluid is quantified, wherein self-propulsion is achieved via steady tangential surface treadmilling. The swimmer has a length 2l and a circular cross section of longitudinal profile aR(z), where a is the characteristic width of the cross section, R(z) is a dimensionless shape function, and z is a dimensionless coordinate, normalized by l, along the centerline of the body. It is assumed that the swimmer is slender, ε=a/l≪1. Hence, we utilize slender-body theory to analyze the Navier-Stokes equations that describe the flow around the swimmer. Therefrom, we compute an asymptotic approximation to the swimming speed, U, as U/u_{s}=1-β[V(Re)-1/2∫_{-1}^{1}zlnR(z)dz]/ln(1/ε)+O[1/ln^{2}(1/ε)], where u_{s} is the characteristic speed of the surface treadmilling, Re is the Reynolds number based on the body length, and β is a dimensionless parameter that differentiates between "pusher" (propelled from the rear, β<0) and "puller" (propelled from the front, β>0) -type swimmers. The function V(Re) increases monotonically with increasing Re; hence, fluid inertia causes an increase (decrease) in the swimming speed of a pusher (puller). Next, we demonstrate that the power expenditure of the swimmer increases monotonically with increasing Re. Further, the power expenditures of a puller and pusher with the same value of |β| are equal. Therefore, pushers are superior in inertial locomotion as compared to pullers, in that they achieve a faster swimming speed for the same power expended. Finally, it is demonstrated that the flow structure predicted from our reduced-order model is consistent with that from direct numerical simulation of swimmers at intermediate Re.