Motions about a fixed point by hypergeometric functions: new non-complex analytical solutions and integration of the herpolhode

We study four problems in the dynamics of a body moving about a fixed point, providing a non-complex, analytical solution for all of them. For the first two, we will work on the motion first integrals. For the symmetrical heavy body, that is the Lagrange-Poisson case, we compute the second and third Euler angles in explicit and real forms by means of multiple hypergeometric functions (Lauricella, functions). Releasing the weight load but adding the complication of the asymmetry, by means of elliptic integrals of third kind, we provide the precession angle completing some previous treatments of the Euler-Poinsot case. Integrating then the relevant differential equation, we reach the finite polar equation of a special trajectory named the {\it herpolhode}. In the last problem we keep the symmetry of the first problem, but without the weight, and take into account a viscous dissipation. The approach of first integrals is no longer practicable in this situation and the Euler equations are faced directly leading to dumped goniometric functions obtained as particular occurrences of Bessel functions of order $-1/2$.Comment: This is a pre-print of an article published in Celestial Mechanics and Dynamical Astronomy. The final authenticated version is available online at: DOI: 10.1007/s10569-018-9837-

Three-dimensional sedimentation patterns of two interacting disks in a viscous fluid

The sedimentation of two spherical solid objects in a viscous fluid has been extensively investigated and well understood. However, a pair of flat disks (in three dimensions) settling in the fluid shows more complex hydrodynamic behaviors. The present work aims to improve understanding of this phenomenon by performing Direct Numerical Simulations (DNS) and physical experiments. The present results show that the sedimentation processes are significantly influenced by disk shape, characterized by a dimensionless moment of inertia I*, and Reynolds number of the leading disk Re. For the flatter disks with smaller I*, steady falling with enduring contact transits to periodic swinging with intermittent contacts as Re increases. The disks with larger I* tend to fall in a Drafting-Kissing-Tumbling (DKT) mode at low Re and to remain separated at high Re. Based on I* and Re, a phase diagram is created to classify the two-disk falling into ten distinctive patterns. The planar motion or three-dimensional motion of the disks is determined primarily by Re. Turbulent disturbance flows at a high Re contribute to the chaotic three-dimensional rotation of the disks. The chance for the two disks to contact is increased when I* and Re are reduced.Comment: 51 pages, 28 figure

Spin-Orbit Coupling of Europa's Ice Shell and Interior

Europa is an icy ocean world, differentiated into a floating ice shell and solid interior, separated by a global ocean. The classical spin-orbit coupling problem considers a satellite as a single rigid body, but in the case of Europa, the existence of the subsurface ocean enables independent motion of the ice shell and solid interior. This paper explores the spin-orbit coupling problem for Europa from a dynamical perspective, yielding illuminating analytical and numerical results. We determine that the spin behavior of Europa is influenced by processes not captured by the classical single rigid body spin-orbit coupling analysis. The tidal locking process for Europa is governed by the strength of gravity-gradient coupling between the ice shell and solid interior, with qualitatively different behavior depending on the scale of this effect. In this coupled rigid model, the shell can potentially undergo large angular displacements from the solid interior, and the coupling plays an outsize role in the dynamical evolution of the moon, even without incorporating the dissipative effects of shell non-rigidity. We additionally discuss the effects of a realistic viscoelastic shell, and catalogue other torques that we expect to be sub-dominant in Europa's spin dynamics, or whose importance is unknown. Finally, we explore how the choice of tidal model affects the resulting equilibrium spin state.Comment: Revised and updated version, with new figures and additional reference

Study on the Flow Characteristics of a Bluff Body Cut From a Square Cylinder

The objective of this project is to study on the flow characteristics of a bluff body from a square cylinder using numerical method. Flow characteristics for each cutting and rotating angle of the bluff body of the cylinder will be compared and studied using numerical method which is GAMBIT and FLUENT software. A lot of tall buildings are square in horizontal cross sectional shapes. Due to dominant, wind around their area; they will experience dominant wind buildings in certain directions

Improving Swimming Performance and Flow Sensing by Incorporating Passive Mechanisms

As water makes up approximately 70% of the Earth\u27s surface, humans have expanded operations into aquatic environments out of both necessity and a desire to gain potential innate benefits. This expansion into aquatic environments has consequently developed a need for cost-effective and safe underwater monitoring, surveillance, and inspection, which are missions that autonomous underwater vehicles are particularly well suited for. Current autonomous underwater vehicles vastly underperform when compared to biological swimmers, which has prompted researchers to develop robots inspired by natural swimmers. One such robot is designed, built, tested, and numerically simulated in this thesis to gain insight into the benefits of passive mechanisms and the development of reduced-order models. Using a bio-inspired robot with multiple passive tails I demonstrate herein the relationship between maneuverability and passive appendages. I found that the allowable rotation angle, relative to the main body, of the passive tails corresponds to an increase in maneuverability. Using panel method simulations I determined that the increase in maneuverability was directly related to the change in hydrodynamic moment caused by modulating the circulation sign and location of the shed vortex wake. The identification of this hydrodynamic benefit generalizes the results and applies to a wide range of robots that utilize vortex shedding through tail flapping or body undulations to produce locomotion. Passive appendages are a form of embodied control, which manipulates the fluid-robot interaction and analogously such interaction can be sensed from the dynamics of the body. Body manipulation is a direct result of pressure fluctuations inherent in the surrounding fluid flow. These pressure fluctuations are unique to specific flow conditions, which may produce distinguishable time series kinematics of the appendage. Using a bio-inspired foil tethered in a water tunnel I classified different vortex wakes with the foil\u27s kinematic data. This form of embodied feedback could be used for the development of control algorithms dedicated to obstacle avoidance, tracking, and station holding. Mathematical models of autonomous vehicles are necessary to implement advanced control algorithms such as path planning. Models that accurately and efficiently simulate the coupled fluid-body interaction in freely swimming aquatic robots are difficult to determine due, in part, to the complex nature of fluids. My colleagues and I approach this problem by relating the swimming robot to a terrestrial vehicle known as the Chaplygin sleigh. Using our novel technique we determined an analogous Chaplygin sleigh model that accurately represents the steady-state dynamics of our swimming robot. We additionally used the subsequent model for heading and velocity control in panel method simulations. This work was inspired by the similarities in constraints and velocity space limit cycles of the swimmer and the Chaplygin sleigh, which makes this technique universal enough to be extended to other bio-inspired robots

Synchronisation and dynamics of model cilia and flagella

Cilia and flagella are organelles central to fluid transport around tissues, unicellular locomotion, and in early mammalian development. They are observed to undulate, rotate, and beat symmetrically in pairs or even in large numbers via metachronal waves. Inspired by biflagellate swimmers like Chlamydomonas, we analyse the synchronisation and dynamics exhibited by pairs of filaments in a Stokesian flow regime. Using a fully three-dimensional filament model, we study the regions of bistable synchrony exhibited by two filaments tethered to a wall beating via a base-driven, geometric switch model. We establish the existence of two stable and two unstable branches of synchrony, characterising the unstable anti-phase branch using Floquet analysis and characterising the unstable edge state between two basins of attraction via a bisection algorithm and the tracking of the edge behaviour in time. We fully characterise a bifurcation diagram, the nature of the bifurcation points, and model the observed dynamical system with a modified Adler equation. We also find that the two-filament model prefers anti-phase synchrony upon the introduction of small basal-body coupling and a preferred curvature. We further study the effectiveness of oscillator systems in capturing two-filament synchronisation and develop a novel minimum-model which encapsulates the qualitative synchronisation behaviours exhibited in elastic filament dynamics by introducing wall effects to the oscillator system. Finally, extending the base-driven filament model to two filament pairs (four filaments), of varying pairwise separation distances, we study the dynamical behaviours along ciliary rows and between flagellate somatic cells as in the case of microalgae like Gonium or Volvox. We quantify different synchronisation states exhibited by four types of pairs and find that different synchronisation states can coexist simultaneously (a form of bistability). We further characterise the bistable region and its transitions regarding disorder in the dynamical system.Open Acces

The sedimentation of flexible filaments

The dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. Compared to the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic, and gravitational forces in a slender-body theory for zero-Reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. Filaments of both non-uniform and uniform cross-sectional thickness are considered. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations, and shapes. These are shown to match excellently with full numerical simulations. Furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. A linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process. The instability travels as a wave along the filament opposite the direction of gravity as it grows and the predicted growth rates are shown to compare favorably with numerical simulations. The linear eigenmodes of the governing equation are also studied, which agree well with the finite-amplitude buckled shapes arising in simulations

Novel immersed boundary method for direct numerical simulations of solid-fluid flows

Solid-fluid two-phase flows, where the solid volume fraction is large either by geometry or by population (as in slurry flows), are ubiquitous in nature and industry. The interaction between the fluid and the suspended solids, in such flows, are too strongly coupled rendering the assumption of a single-way interaction (flow influences particle motion alone but not vice-versa) invalid and inaccurate. Most commercial flow solvers do not account for twoway interactions between fluid and immersed solids. The current state-of-art is restricted to two-way coupling between spherical particles (of very small diameters, such that the particlediameter to the characteristic flow domain length scale ratio is less than 0.01) and flow. These solvers are not suitable for solving several industrial slurry flow problems such as those of hydrates which is crucial to the oil-gas industry and rheology of slurries, flows in highly constrained geometries like microchannels or sessile drops that are laden with micro-PIV beads at concentrations significant for two-way interactions to become prominent. It is therefore necessary to develop direct numerical simulation flow solvers employing rigorous two-way coupling in order to accurately characterise the flow profiles between large immersed solids and fluid. It is necessary that such a solution takes into account the full 3D governing equations of flow (Navier-Stokes and continuity equations), solid translation (Newton’s second law) and solid rotation (equation of angular momentum) while simultaneously enabling interaction at every time step between the forces in the fluid and solid domains. This thesis concerns with development and rigorous validation of a 3D solid-fluid solver based on a novel variant of immersed-boundary method (IBM). The solver takes into account full two-way fluid-solid interaction with 6 degrees-of-freedom (6DOF). The solid motion solver is seamlessly integrated into the Gerris flow solver hence called Gerris Immersed Solid Solver (GISS). The IBM developed treats both fluid and solid in the manner of “fluid fraction” such that any number of immersed solids of arbitrary geometry can be realised. Our IBM method also allows transient local mesh adaption in the fluid domain around the moving solid boundary, thereby avoiding problems caused by the mesh skewness (as seen in common mesh-adaption algorithms) and significantly improves the simulation efficiency. The solver is rigorously validated at levels of increasing complexity against theory and experiment at low to moderate flow Reynolds number. At low Reynolds numbers (Re 1) these include: the drag force and terminal settling velocities of spherical bodies (validating translational degrees of freedom), Jeffrey’s orbits tracked by elliptical solids under shear flow (validating rotational and translational degrees of freedom) and hydrodynamic interaction between a solid and wall. Studies are also carried out to understand hydrodynamic interaction between multiple solid bodies under shear flow. It is found that initial distance between bodies is crucial towards the nature of hydrodynamic interaction between them: at a distance smaller than a critical value the solid bodies cluster together (hydrodynamic attraction) and at a distance greater than this value the solid bodies travel away from each other (hydrodynamic repulsion). At moderately high flow rates (Re O(100)), the solver is validated against migratory motion of an eccentrically placed solid sphere in Poisuelle flow. Under inviscid conditions (at very high Reynolds number) the solver is validated against chaotic motion of an asymmetric solid body. These validations not only give us confidence but also demonstrate the versatility of the GISS towards tackling complex solid-fluid flows. This work demonstrates the first important step towards ultra-high resolution direct numerical simulations of solid-fluid flows. The GISS will be available as opensource code from February 2015