Dynamics Of Rods Research Articles

Nonlinear dynamics of a flexible rod partially sliding in a rigid sleeve under the action of gravity and configurational force

We investigate various methods of analyzing systems with moving boundaries, using as an example a flexible rod sliding in an ideal frictionless sleeve in the field of gravity. Special attention is paid to the configurational force acting on the rod at the sleeve opening and thus determining the rod’s dynamics. The non-material kinematic description used in simulations is based on the re-parametrization of the Lagrangian arc length coordinate. The variational formulation uses the energy expressions written for the entire rod, comprising the free segment and the one inside the sleeve. A novel finite element scheme is efficient for highly flexible rods, which may undergo complete ejection. A simplified two degrees of freedom model, which accelerates simulations, shows a good agreement as the bending stiffness increases. An analytical study using Hamiltonian mechanics exploits the separation of variables into fast oscillations and slow axial motion. The adiabatic invariant approach leads to approximate closed-form solutions for the slow dynamics and yields the maximum injection depth of the rod into the sleeve.

Transport and clogging dynamics of flexible rods in pore constrictions.

The transport and clogging behavior of flexible particles in confined flows is a complex interplay between elastic and hydrodynamic forces and wall interactions. While the motion of non-spherical particles in unbounded flows is well understood, their behavior in confined spaces remains less explored. This study introduces a coupled computational fluid dynamics-discrete element method (CFD-DEM) approach to investigate the transport and clogging dynamics of flexible rod-shaped particles in confined pore constrictions. The spatio-temporal analysis reveals the influence of the rod's initial conditions and flexibility on its transport dynamics through a pore constriction. The simulation results demonstrate an increase in the lateral drift of the rod upon exiting the pore that can be scaled with channel height confinement. The clogging dynamics are explored based on hydrodynamic and mechanical forces, unveiling conditions for mechanical clogging through sieving. The developed method allows for the deconvolution of the forces that contribute to particle trajectories in confined flow, which is highly relevant in particle separation processes, fibrous-shaped virus filtration, biological flows, and related applications. The method is embedded into the open-source CFDEM framework, facilitating future extensions to explore multiple particle dynamics, intermolecular forces, external influences, and complex geometries.

Numerical simulation of fluid-vegetation coupled dynamic using a promoted semi-resolved coupling model

Flexible vegetation is an important component of riverine and coastal ecosystems. The numerical simulation is difficult due to the complex motions of flexible vegetation and fluid. In this work, a semi-resolved numerical model coupling the computational fluid dynamics (CFD) and flexible rod dynamics (FRD) using a promoted two-way domain expansion method was proposed with the accuracy of the resolved model and the efficiency of the unresolved model. The governing equations of the flexible rod dynamics are solved by the finite element method (FEM). The effect of vegetation is added as source terms to the Reynolds-averaged Navier-Stokes equations, which are discretized and solved by the finite volume method (FVM). The anisotropic Gaussian kernel function is adopted to calculate the interactive forces between the fluid and the flexible rod with an anisotropic cross-section. The model was first validated by simulating the flow passing through a single flexible rod with a rectangular cross-section. Meanwhile, the main factors were calibrated. The engineering application of the model was further validated by simulating the flow passing through a finite patch of flexible vegetation. These results indicate that the proposed model could well simulate the open channel flows with the flexible vegetation.

Jet space extensions of infinite-dimensional Hamiltonian systems

We analyze infinite-dimensional Hamiltonian systems corresponding to partial Differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and non-quadratic Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes-Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

Soft confinement of self-propelled rods: simulation and theory.

We present an analytical framework for evolving the dynamics of active rods under any periodic external potential, including confining channels and arrays of harmonic traps. As a proof of concept, we analyze the structure and dispersion of self-propelled rods under a soft, periodic one-dimensional (1D) confinement potential and under a two-dimensional (2D) periodic radial harmonic trap. While passive rods and polymers nematically order under 1D confinement, their diffusive transport along the director is limited by thermal diffusion. In contrast, self-propelled rods can generate large convective fluxes when combined with nematic ordering, producing a strong dispersion along the director. Combining theory and simulation, we demonstrate that nematic alignment and self-propulsion generates an exponential enhancement in active diffusivity along the director, in contrast to passive rods that experience at most a 2-fold increase.

Boundary value problems of the dynamics of thermoelastic rods and their solutions

We consider spatially one-dimensional boundary value problems of uncoupled thermoelasticity, which can be used to study various rod structures under thermal heating conditions. As is known, rod structures are connecting and transmission links of various parts of the machines and mechanisms. Here we propose a unified technique for solving various boundary value problems typical for practical applications. The problems of determining the thermally stressed state of a thermoelastic rod under various boundary conditions at its ends and acting power and heat sources along the entire length of the rod are considered. Based on the method of generalized functions, generalized solutions of non-stationary and stationary direct and semi-inverse boundary value problems under the action of power and heat sources of various types, including stationary sources of periodic oscillations, are constructed. Acting sources can also be specified by singular generalized functions, under various boundary conditions at the ends of the rod. Shock elastic waves that arise in such structures under the action of shock loads are considered. Regular integral representations of generalized solutions are obtained, which provide an analytical solution to the posed boundary value problems. The peculiarity of the constructed solutions makes them convenient for studying network thermoelastic systems that can be modeled by thermoelastic graphs.

Spatially-dependent model for rods and cones in the retina

We develop a mathematical model for photoreceptors in the retina. We focus on rod and cone outer segment dynamics and interactions with a nutrient source associated with the retinal pigment epithelium cells. Rod and cone densities (number per unit area of retinal surface) are known to have significant spatial dependence in the retina with cones located primarily near the fovea and the rods located primarily away from the fovea. Our model accounts for this spatial dependence of the rod and cone photoreceptor density as well as for the possibility of nutrient diffusion. We present equilibrium and dynamic solutions, discuss their relation to existing models, and estimate model parameters through comparisons with available experimental measurements of both spatial and temporal photoreceptor characteristics. Our model compares well with existing data on spatially-dependent regrowth of photoreceptor outer segments in the macular region of Rhesus Monkeys. Our predictions are also consistent with existing data on the spatial dependence of photoreceptor outer segment length near the fovea in healthy human subjects. We focus primarily on the healthy eye but our model could be the basis for future efforts designed to explore various retinal pathologies, eye-related injuries, and treatments of these conditions.

Dynamics of certain Euler-Bernoulli rods and rings from a minimal coupling quantum isomorphism.

In some parameter and solution regimes, a minimally coupled nonrelativistic quantum particle in one dimension is isomorphic to a much heavier, vibrating, very thin Euler-Bernoulli rod in three dimensions with ratio of bending modulus to linear density (ℏ/2m)^{2}. For m=m_{e}, this quantity is comparable to that of a microtubule. Axial forces and torques applied to the rod play the role of scalar and vector potentials, respectively, and rod inextensibility plays the role of normalization. We show how an uncertainty principle ΔxΔp_{x}≳ℏ governs transverse deformations propagating down the inextensible, force and torque-free rod, and how orbital angular momentum quantized in units of ℏ or ℏ/2 (depending on calculation method) emerges when the force and torque-free inextensible rod is formed into a ring. For torqued rings with large wave numbers, a "twist quantum" appears that is somewhat analogous to the magnetic flux quantum. These and other results are obtained from a purely classical treatment of the rod, i.e., without quantizing any classical fields.

On two initial-boundary-value problems for impact dynamics of metaconcrete rods

On two initial-boundary-value problems for impact dynamics of metaconcrete rods

Bonded straight and helical flagellar filaments form ultra-low-density glasses

We study how the three-dimensional shape of rigid filaments determines the microscopic dynamics and macroscopic rheology of entangled semidilute Brownian suspensions. To control the filament shape we use bacterial flagella, which are microns-long helical or straight filaments assembled from flagellin monomers. We compare the dynamics of straight rods, helical filaments, and shape-diblock copolymers composed of seamlessly joined straight and helical segments. Caged by their neighbors, straight rods preferentially diffuse along their long axis, but exhibit significantly suppressed rotational diffusion. Entangled helical filaments escape their confining tube by corkscrewing through the dense obstacles created by other filaments. By comparison, the adjoining segments of the rod-helix shape-diblocks suppress both the translation and the corkscrewing dynamics. Consequently, the shape-diblock filaments become permanently jammed at exceedingly low densities. We also measure the rheological properties of semidilute suspensions and relate their mechanical properties to the microscopic dynamics of constituent filaments. In particular, rheology shows that an entangled suspension of shape rod-helix copolymers forms a low-density glass whose elastic modulus can be estimated by accounting for how shear deformations reduce the entropic degrees of freedom of constrained filaments. Our results demonstrate that the three-dimensional shape of rigid filaments can be used to design rheological properties of semidilute fibrous suspensions.

Parameters and dynamical graphs analysis about the new dynamics traveling wave solutions of longitudinal bud equation in a magneto-electro-elastic round rod

In this paper, we investigate the nonlinear longitudinal wave equation (LWE) which involves mathematical physics in a magneto-electro-elastic (MEE) circular rod. By three different methods, we found that various forms for exact traveling wave solutions of longitudinal bud equation among an MEE round rod. These new exact and solitary wave solutions are derived in the form of rational, single periodic function, double periodic Jacobian elliptic functions (JEF), especially about Weierstrass elliptic function (WEF). Additionally, certain interesting (3D), (2D) figures represent the LWE provides the corporal information to explain the physically phenomena. By choosing different suitable values of the free parameters, we made some charts and we analyzed these charts to get a lot of valuable information to understand the MEE circular rod. Our work describes the dynamics of the MEE circular rod. Finally, we conclude with some perspectives for our future research work.

Universal flow-induced orientational ordering of colloidal rods in planar shear and extensional flows: Dilute and semidilute concentrations

The design of flow processes to build a macroscopic bulk material from rod-shaped colloidal particles has drawn considerable attention from researchers and engineers. Here, we systematically explore and show that the characteristic strain rate of the flow universally determines the orientational ordering of colloidal rods. We employed the fluctuating lattice Boltzmann method by simulating hydrodynamically interacting Brownian rods in a Newtonian liquid moving under various flow types. By modeling a rigid rod as a chain of nonoverlapping solid spheres with constraint forces and torque, we elucidate rigid rod dynamics with an aspect ratio (L/d) either 4.1 or 8.1 under various rotational Péclet number (Per) conditions. The dynamics of colloidal rods in dilute (nL3=0.05) and semidilute suspensions (nL3=1.1) were simulated for a wide range of Per (0.01<Per<1000) under shear flows including Couette and Poiseuille flows in a planar channel geometry, and an extensional and mixed-kinematics flow in a periodic four-roll mill geometry, where n is the number density, and d and L are the diameter and length of the rod, respectively. By evaluating the degree of orientational alignment of rods along the flows, we observed that there is no significant difference between flow types, and the flow-induced ordering of rods depends on the variation of Per up to moderate Per (Per<100). At a high Per (Per>100), the degree of orientational ordering is prone to diversify depending on the flow type. The spatial inhomogeneity of the strain-rate distribution leads to a substantial decrease in the orientational alignment at high Per.

Brownian dynamics simulations of hard rods in external fields and with contact interactions.

We propose a simulation method for Brownian dynamics of hard rods in one dimension for arbitrary continuous external force fields. It is an event-driven procedure based on the fragmentation and mergers of clusters formed by particles in contact. It allows one to treat particle interactions in addition to the hard-sphere exclusion as long as the corresponding interaction forces are continuous functions of the particle coordinates. We furthermore develop a treatment of sticky hard spheres as described by Baxter's contact interaction potential.

Nonlinear dynamic response on multi-fault rod fastening rotor with variable parameters

A multi-fault nonlinear model of rod fastening rotor is proposed considering the structural properties of rod fastening and the integration of multiple factors. The nonlinear analysis is performed to present the nonlinear characteristics of gas turbine rotor system with multi faults. The fault dynamics of rod fastening rotor is exposed for interaction and failure mechanism of multiple faults. The variable parameter analysis reveals the impact of system parameters variation on vibration response in multi-fault rod fastening rotor system. Analysis results reflect that multi faults enhances the nonlinearity of the system and makes dynamic characteristics more complicated. Complex nonlinear characteristics of the dynamical system are presented under different system parameters. The multi-fault nonlinear dynamic model and the nonlinear analysis of variable parameter contribute to the fault identification and stability research in the rod fastening rotor of gas turbine, which will provide the reference for the design and application of rod fastening rotor system.

A finite volume algorithm for the dynamics of filaments, rods, and beams

Filaments, rods, and beams are ubiquitous in biology and in many man-made products and structures. While a substantial amount of research has been done to understand the statics and dynamics of these long, thin objects, there remain many unanswered and unstudied problems related to the dynamics of bending and twisting filamentary objects. Simulating the general dynamics of these structures in 3D remains challenging. For example, the net force and torque on a free filament immersed in fluid at low Reynolds number must be zero. However, standard finite difference approaches will often fail to preserve the zero force and torque conditions. These numerical artifacts cause spurious rotations and translations that prohibit, or at least limit, their accuracy in simulating the dynamics of filaments, rods, and beams in these contexts (such as the free-swimming motion of a filamentary microorganism). Here we develop a finite volume discretization based on the Kirchoff equations that naturally guarantees the correct total integral of the forces and torques on filaments, rods, or beams. We then couple this discretization to resistive force theory to develop a stable, accurate dynamic algorithm of filament motion at low Reynolds number. We use a range of sample problems to highlight the utility, stability, and accuracy of this method. While our sample problems focus on low Reynolds number dynamics in the context of resistive force theory (RFT), our discretized finite volume algorithm is general and can be applied to inertial dynamics, immersed boundary methods, and boundary integral methods, as well.

Numerical and Experimental Analysis for the Dynamics of Flawed–Machining Rod–Disk Rotor with Inner Misalignment

The nonlinear dynamic effects of the misalignment between inner disks in a flawed–machining rod–disk rotor is studied. Non–uniform stress distribution due to inner misalignment is obtained based on 3D static solution. The concomitant unbalances, including constant mass eccentricity and speed–variant rotor bending, are taken into account in the nonlinear dynamics. The dynamic results show that rotor bending leads to stability reduction and vibration growth. There is a distinctive feature in that the rotor’s vibration goes up again after critical speed. The maximum allowable inner misalignment is obtained according to its stability boundaries. An uneven tightening method is also presented to reduce adverse effects when the inner misalignment exists. Moreover, an experiment is designed to measure the vibration characteristics for the rod–disk rotor bearing system with inner misalignment. The results show that the theoretical result of vibration amplitude of the flawed rod–rotor bearing system is basically consistent with the experimental value. It is also found that the precise rotor performs the periodic motion, but the flawed rod–disk rotor exhibits the period–doubling orbit. This phenomenon proves that the flawed rod–disk rotor loses stability more easily than the precise rotor due to inner misalignment. However, the amplitude of harmonic frequency components for the precise rod–disk rotor system is obviously larger than the flawed rod–disk rotor system with inner misalignment. The peak value of the vibration amplitude increases when the inner misalignment becomes larger. On the whole, this work presents numerical and experimental analysis to study the dynamics of flawed-machining rod–disk rotor with inner misalignment. It also establishes the relationship between centration precision and dynamic features.

Simulating the dynamics of Janus rods

Simulating the dynamics of Janus rods

Numerical modeling of straight and helical elastic rods under fluid flow using immersed boundary method

This paper presents a three-dimensional computational model built using immersed boundary finite volume method to explore the dynamics of straight and helical elastic rods rotating under an applied fluid flow in a channel. Numerical simulations are done for low and high rotational frequencies of a base motor attached at one fixed end of the rods. Simulations reveal that under low rotational frequency, a straight rod (bend at free end) always performs stable twirling motion and eventually attains straight state (mechanical equilibrium) under an applied fluid flow. But on the other hand, for the same conditions a helical rod always keeps its helical shape during interaction with fluid and never attains stable straight state. For the case of high rotational frequency, the straight rod executes whirling motion in which it attains helical shape during all the time. Whirling motion is also observed for helical rod under high rotational frequency. For similar conditions, the instantaneous shapes obtained by straight and helical rods are different which indicates that the initial configuration of the rod as well as rotational frequency have significant impact in deciding the dynamics of the elastic rod under fluid flow. In the biological realm, these elastic rods represent flagellum of monotrichous bacteria which helps for propulsion in fluid. Hence, the present simulation results will help to develop efficient bacteria inspired artificial microrobot for biomedical applications.

Emergent collective dynamics of pusher and puller squirmer rods: swarming, clustering, and turbulence.

We study the interplay of steric and hydrodynamic interactions in suspensions of elongated microswimmers by simulating the full hydrodynamics of squirmer rods in the quasi two-dimensional geometry of a Hele-Shaw cell. To create pusher or puller-type squirmer rods, we concentrate the surface slip-velocity field more to the back or to the front of the rod and thereby are able to tune the rod's force-dipole strength. We study a wide range of aspect ratios and area fractions and provide corresponding state diagrams. The flow field of pusher-type squirmer rods destabilizes ordered structures and favors the disordered state at small area fractions and aspect ratios. Only when steric interactions become relevant, we observe a turbulent and dynamic cluster state, while for large aspect ratios a single swarm and jammed cluster occurs. The power spectrum of the turbulent state shows two distinct energy cascades at small and large wave numbers with power-law scaling and non-universal exponents. Pullers show a strong tendency to form swarms instead of the disordered state found for neutral and pusher rods. At large area fractions a dynamic cluster is observed and at larger aspect ratio a single swarm or jammed cluster occurs.

Dynamics of Brownian Janus rods at a liquid–liquid interface

Amphiphilic Janus nanoparticles exhibit higher interfacial activity and adsorb more strongly to fluid interfaces than homogeneous nanoparticles of similar sizes. Taking advantage of both shape and chemical anisotropy on the same particle, Janus particles offer rich self-assembly possibilities for nanotechnology. By using dissipative particle dynamics simulation, the translational diffusion of Janus nanoparticles at the interface between two immiscible fluids is investigated. The particle aspect ratio affects both particle's translational thermal motion and the average orientation of the particle with respect to the interface at equilibrium. This behavior is also linked to the interfacial tension of the system. Our findings provide fundamental insights into the dynamics and self-assembly of anisotropic Brownian particles at interfaces.