Time-periodic Force Research Articles

On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams

Abstract In this paper, we study a nonlinear fluid-structure interaction problem between a ‘square-root’ viscoelastic beam and a compressible viscous fluid. The beam is immersed in the fluid which fills a two-dimensional rectangular domain with periodic boundary conditions in both directions, while both the beam and the fluid are under the effect of time-periodic forces. By using a decoupling approach, at least one time-periodic weak solution to this problem is constructed which has a bounded energy and a fixed prescribed mass. The lack of a priori energy bounds is overcome by a series of estimates based on a careful choice of parameters. The most challenging one is the pressure estimate, which is obtained by utilizing the specific periodic geometry and the Bogovskiǐ operator on a fixed domain that has a uniform constant. With uniform estimates and improved regularity of the beam as in (Muha and Schwarzacher 2023 Ann. Inst. Henri Poin. Anal. Non Lineaire 39 1369–412), the time-periodic solution is constructed by a series of limit procedures, following the finite-dimensional time-space construction from (Feireisl et al 2012 Arch. Rational Mech. Anal. 204 74586).

Exact analytical investigation of Duffing oscillator vibration spectra under time-periodic oscillatory external force

This work introduces an analytical technique for determining solutions to a highly nonlinear Duffing-harmonic oscillator model problem. The parametric solutions of Duffing oscillator vibrations for an undamped case are achieved analytically in terms of Adomian polynomials by implementing the straightforward approach of the Laplace Transformation, known as the Laplace Decomposition Procedure (LDP). Possible plots of both numerical and analytical results sketched for various parameters are also presented to support our discussion. These graphs demonstrate variations of position-time, speed-time, and speed-position for an undamped Duffing oscillator case. When analyzed in general, they can provide vibration pattern descriptions for a variety of physical and engineering system configurations. We also examine their physical structures and behavioral characteristics within the conceptual framework of chaotic formalism.

Thermal fluctuations for a three-beads swimmer

We discuss a microswimmer model made of three spheres actuated by an internal active time-periodic force, tied by an elastic potential, and submitted to hydrodynamic interactions with thermal noise. The dynamical approach we use, replacing the more common kinematic one, shows the instability of the original model and the need of a confining potential to prevent the evaporation of the swimmer. We investigate the effect of the main parameters of the model, such as the frequency and phase difference of the periodic active force, the stiffness of the confining potential, the length of the swimmer and the temperature and viscosity of the fluid. Our observables of interest are the averages of the swim velocity, the energy consumption rate, the diffusion coefficient, and the swimming precision, which is limited by the energy consumption through the celebrated thermodynamic uncertainty relations. An optimum for velocity and precision is found for an intermediate frequency. Reducing the potential stiffness, the viscosity, or the length is also beneficial for the swimming performance, but these parameters are limited by the consistency of the model. Analytical approximation for many of the relevant observables is obtained for small deformations of the swimmer. We also discuss the efficiency of the swimmer in terms of its maximum precision and of the hydrodynamic, or Lighthill, criterion, and how they are connected. Published by the American Physical Society 2024

Dynamical behaviors and invariant recurrent patterns of Kuramoto-Sivashinsky equation with time-periodic forces.

In this study, we perform an extensive numerical study of the one-dimensional Kuramoto-Sivashinsky equation under time-periodic forces. We examine the statistics of chaotic solutions and the behaviors from turbulent solutions to steady periodic solutions as the period of the forces increases. When the period is small, global turbulent characteristics associated with local oscillations are found, and the forces are considered not to influence the turbulent dynamics. Long periodic orbits are found to capture the dynamics of the turbulent solutions. On the other hand, if the strength is large enough and the period is large, only regular periodic motions are found and their spatial structures are like viscous shocks.

On periodic motions of a harmonic oscillator interacting with incompressible fluids

We consider a mass–spring system immersed in an incompressible fluid flow governed by the Navier–Stokes equations subject to a prescribed time-periodic flow rate (and possibly external time-periodic body forces on the fluid and the mass). We show that, with no restriction on the period of the flow rate (and of the external forces), when the flow rate is “small”, there exists a weak time-periodic solution to the coupled system. Under some more regularity and “smallness” conditions on the flow rate (and the external forces) we also show that these solutions are, indeed, strong solutions.

Dynamic stability of the Mindlin-Reissner plate using a time-modulated axial force

A time-periodic axial force acting on a continuous structure can be destabilized by parametric excitation. Moreover, it is capable of increasing the equivalent damping and shortening the transient vibrations. In this work, this concept is extended, and dynamic stability of a Mindlin-Reissner plate is analyzed. The FE formulation for the four-node plate element is derived by employing the variational approach including the action of an axial force. The stability of the time-periodic system is evaluated both numerically and analytically, using Floquet theory and first-order approximation based on the averaging method, respectively. The influence of plate length, thickness, and aspect ratio on dynamic stability is highlighted. It is demonstrated that the instability tongue at a parametric combination resonance frequency of the summation type is shifted toward higher axial force amplitudes by increasing the plate width and decreasing the plate length. Alternatively, the axial force acting in the vicinity of a parametric combination frequency of the difference type shows a parametric antiresonance which is maximized by exactly the opposite tuning of the plate design. This guideline is important when plate design under the action of a harmonic axial force is considered.

Effect of Gravity Modulation on the Onset of Heat Transfer by Buoyancy Induced Convection in Boussinesq- Stokes Suspension with Temperature

The current paper considers the Boussinesq-Stokes suspensions and the temperature-dependent viscosity with the influence of gravity modulation to analyze a weak non-linear stability problem of Rayleigh Benard magnetoconvection. In the study of convective instability problems, the impact of time-periodic body force also known as gravity modulation or g-gitter is essential. In the problem of gravity modulation, the gravity field has two components: one is the constant part and another an externally imposed time periodic part, which can be produced by oscillating the fluid layer. The effect of varying frequency of gravitational oscillation on convection is examined. The truncated form of the Fourier series is used in the non-linear analysis. The effects of numerous factors on the onset of convection have been discussed in this paper. The thermal Nusselt number is computed and shown for slow time periods using non-linear theory. The impacts of gravity modulation frequency and amplitude have been investigated in order to study heat transport in the system, as well as other aspects that exist in the problem.

Effect of gravity modulation on weakly nonlinear bio-thermal convection in a porous medium layer

Investigating thermal convection within porous media permeated by fluids and micro-organisms stands as a significant inquiry with broad relevance across geophysical and engineering domains. Studying convection within porous media can aid in controlling temperature and nutrient distribution for cell growth and tissue regeneration, as well as the efficiency of biofuel fermentation and production processes. Hence, the primary objective of this study is to investigate the influence of time-periodic gravitational forces on Darcy–Brinkman bio-thermal convection within a porous medium layer. This medium is saturated with a Newtonian fluid that encompasses gyrotactic micro-organisms. The gravity modulation amplitude is assumed to be very small. A weak nonlinear stability analysis is performed to analyze the stationary mode of bioconvection. The heat transport, measured by the Nusselt number, is governed by a non-autonomous Ginzburg–Landau equation. The research explores the influence of several parameters on heat transport, including the Vadaszs number, the modified bioconvective Rayleigh–Darcy number, cell eccentricity, modulation frequency, and modulation amplitude. The results are presented graphically, illustrating the impact of these parameters on heat transfer. The findings reveal that both the Vadaszs number and the modulation amplitude have a positive effect on heat transfer, enhancing the process. On the other hand, an increase in the modified bioconvection Rayleigh–Darcy number and cell eccentricity leads to a decrease in heat transfer. Furthermore, a comparison between the modulated and unmodulated systems indicates that the modulated systems have a more significant influence on the stability problem compared to the unmodulated systems. This highlights the effectiveness of external modulation in controlling heat transport within the system.

Particle orbiting constrained by elastic filament as a model cilium for fluid pumping

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.

Temperature anomalies of oscillating diffusion in ac-driven periodic systems.

We analyze the impact of temperature on the diffusion coefficient of an inertial Brownian particle moving in a symmetric periodic potential and driven by a symmetric time-periodic force. Recent studies have revealed the low-friction regime in which the diffusion coefficient shows giant damped quasiperiodic oscillations as a function of the amplitude of the time-periodic force [I. G. Marchenko et al., Chaos 32, 113106 (2022)1054-150010.1063/5.0117902]. We find out that when temperature grows the diffusion coefficient increases at its minima; however, it decreases at the maxima within a finite temperature window. This curious behavior is explained in terms of the deterministic dynamics perturbed by thermal fluctuations and mean residence time of the particle in the locked and running trajectories. We demonstrate that temperature dependence of the diffusion coefficient can be accurately reconstructed from the stationary probability to occupy the running trajectories.

Paradoxical nature of negative mobility in the weak dissipation regime.

We reinvestigate a paradigmatic model of nonequilibrium statistical physics consisting of an inertial Brownian particle in a symmetric periodic potential subjected to both a time-periodic force and a static bias. In doing so, we focus on the negative mobility phenomenon in which the average velocity of the particle is opposite to the constant force acting on it. Surprisingly, we find that in the weak dissipation regime, thermal fluctuations induce negative mobility much more frequently than it happens if dissipation is stronger. In particular, for the very first time, we report a parameter set in which thermal noise causes this effect in the nonlinear response regime. Moreover, we show that the coexistence of deterministic negative mobility and chaos is routinely encountered when approaching the overdamped limit in which chaos does not emerge rather than near the Hamiltonian regime of which chaos is one of the hallmarks. On the other hand, at non-zero temperature, the negative mobility in the weak dissipation regime is typically affected by weak ergodicity breaking. Our findings can be corroborated experimentally in a multitude of physical realizations, including, e.g., Josephson junctions and cold atoms dwelling in optical lattices.

Time periodic solutions to Hibler’s sea ice model

It is shown that the viscous-plastic Hibler sea ice model admits a unique, strong T-time periodic solution provided the given T-periodic forcing functions are small in suitable norms. This is in particular true for time periodic wind forces and time periodic ice growth rates.

Existence of a Time Periodic Solution for the Compressible Euler Equation with a Time Periodic Outer Force in a Bounded Interval

Existence of a Time Periodic Solution for the Compressible Euler Equation with a Time Periodic Outer Force in a Bounded Interval

Giant oscillations of diffusion in ac-driven periodic systems.

We revisit the problem of diffusion in a driven system consisting of an inertial Brownian particle moving in a symmetric periodic potential and subjected to a symmetric time-periodic force. We reveal parameter domains in which diffusion is normal in the long time limit and exhibits intriguing giant damped quasiperiodic oscillations as a function of the external driving amplitude. As the mechanism behind this effect, we identify the corresponding oscillations of difference in the number of locked and running trajectories that carry the leading contribution to the diffusion coefficient. Our findings can be verified experimentally in a multitude of physical systems, including colloidal particles, Josephson junction, or cold atoms dwelling in optical lattices, to name only a few.

Time periodic motion of temperature driven compressible fluids

We consider the Navier–Stokes–Fourier system describing the motion of a compressible viscous fluid in a container with impermeable boundary subject to time periodic heating and under the action of a time periodic potential force. We show the existence of a time periodic weak solution for arbitrarily large physically admissible data.

Flagellar Propulsion of Sperm Cells Against a Time-Periodic Interaction Force.

Sperm cells undergo complex interactions with external environments, such as a solid-boundary, fluid flow, as well as other cells before arriving at the fertilization site. The interaction with the oviductal epithelium, as a site of sperm storage, is one type of cell-to-cell interaction that serves as a selection mechanism. Abnormal sperm cells with poor swimming performance, the major cause of male infertility, are filtered out by this selection mechanism. In this study, collinear bundles, consisting of two sperm cells, generate propulsive thrusts along opposite directions and allow to observe the influence of cell-to-cell interaction on flagellar wave-patterns. The developed elasto-hydrodynamic model demonstrates that steric and adhesive forces lead to highly symmetrical wave-pattern and reduce the bending amplitude of the propagating wave. It is measured that the free cells exhibit a mean flagellar curvature of 6.4 ± 3.5 radmm-1 and a bending amplitude of 13.8 ± 2.8 radmm-1 . After forming the collinear bundle, the mean flagellar curvature and bending amplitude are decreased to 1.8 ± 1.1 and 9.6 ± 1.4 radmm-1 , respectively. This study presents consistent theoretical and experimental results important for understanding the adaptive behavior of sperm cells to the external time-periodic force encountered during sperm-egg interaction.

Diffusion coefficient and power spectrum of active particles with a microscopically reversible mechanism of self-propelling.

Catalytically active macromolecules are envisioned as key building blocks in the development of artificial nanomotors. However, theory and experiments report conflicting findings regarding their dynamics. The lack of consensus is mostly caused by the limited understanding of the specifics of self-propulsion mechanisms at the nanoscale. Here, we study a generic model of a self-propelled nanoparticle that does not rely on a particular mechanism. Instead, its main assumption is the fundamental symmetry of microscopic dynamics of chemical reactions: the principle of microscopic reversibility. Significant consequences of this assumption arise if we subject the particle to the action of an external time-periodic force. The particle diffusion coefficient then becomes enhanced compared to the unbiased dynamics. The enhancement can be controlled by the force amplitude and frequency. We also derive the power spectrum of particle trajectories. Among the new effects stemming from the microscopic reversibility are the enhancement of the spectrum at all frequencies and sigmoid-shaped transitions and a peak at characteristic frequencies of rotational diffusion and external forcing. Microscopic reversibility is a generic property of a broad class of chemical reactions. Therefore, we expect that the presented results will motivate new experimental studies aimed at testing our predictions. This could provide new insights into the dynamics of catalytic macromolecules.

Time-Periodic Weak Solutions for an Incompressible Newtonian Fluid Interacting with an Elastic Plate

Under the action of a time-periodic external force we prove the existence of at least one time-periodic weak solution for the interaction between a three-dimensional incompressible fluid, governed by the Navier--Stokes equation and a two-dimensional elastic plate. The challenge is that the Eulerian domain for the fluid changes in time and is a part of the solution. We introduce a two fixed-point methodology: First we construct a time-periodic solutions for a given variable time-periodic geometry. Then in a second step a (set-valued) fixed point is performed w.r.t. the geometry of the domain. The existence relies on newly developed a priori estimates applicable for both coupled and uncoupled variable geometries. Due to the expected weak regularity of the solutions such Eulerian estimates are unavoidable. Note in particular that only the fluid is assumed to be dissipative; the here-produced a priori estimates show that its possible to exploit the dissipative effects of the fluid also for the solid deformation. The existence of time-periodic solutions for a given geometry is valid for arbitrary large data. The existence of periodic coupled solutions to the fluid-structure interaction is valid for all data that exclude a self-intersection a priori.

Some periodic orbits of chaotic motions for time-periodic forced two-dimensional Navier–Stokes flows

In this study, we study the two-dimensional Navier–Stokes flows with time-periodic external forces. Invariant solutions, including periodic orbits and relative periodic orbits, are extracted with the recurrent flow analysis, while low-dimensional projections based on the dynamic mode decomposition algorithm are used to reduce the cost of searching nearly recurrences. When the period of forces gets a constant increase, the flows change from the stable time-periodic state to oscillate and even turbulent flows. In all cases, one periodic orbit is identified near the initial stage. This orbit represents the stable/unstable base state, and the trajectories of vorticity fields are trapped inside it or escape away from it leading to oscillating/turbulent motions. For the oscillating flows, periodic orbits without any symmetries play the role that the flows visit them and then move away from them to other orbits. In addition, for a moderate period of forces, a bursting phenomenon occurs and the state of oscillating flows turns to turbulent flows with the rapid increase in energy. For the turbulent motions, one unstable periodic, which qualitatively represents the shapes of a large vortex dipole that exists in the turbulent motions, is obtained. Its statistical significance is shown by the frequency of that flows visit it.

More on Convergence of Chorin’s Projection Method for Incompressible Navier–Stokes Equations

Kuroki–Soga (Numer. Math. 146:401–433, 2020) proved that Chorin’s fully discrete finite difference projection method, originally introduced by Chorin (Math. Comput. 23:341–353, 1969), is unconditionally solvable and convergent within an arbitrary fixed time interval to a Leray–Hopf weak solution of the incompressible Navier–Stokes equations on a bounded domain with an arbitrary external force. This paper is a continuation of Kuroki–Soga’s work to further exhibit mathematical aspects of the method. We show time-global solvability and convergence of the scheme; \(L^2\)-error estimates for the scheme in the class of smooth exact solutions; application of the method to the problem with a time-periodic external force to investigate time-periodic (Leray–Hopf weak) solutions, long-time behaviors, error estimates, etc.