Time-periodic Motion Research Articles

On the stability of fully nonlinear hydraulic-fall solutions to the forced water wave problem

Two-dimensional free-surface flow over localised topography is examined, with the emphasis on the stability of hydraulic-fall solutions. A Gaussian topography profile is assumed with a positive or negative amplitude modelling a bump or a dip, respectively. Steady hydraulic-fall solutions to the full incompressible, irrotational Euler equations are computed, and their linear and nonlinear stability is analysed by computing eigenspectra of the pertinent linearised operator and by solving an initial value problem. The computations are carried out numerically using a specially developed computational framework based on the finite-element method. The Hamiltonian structure of the problem is demonstrated, and stability is determined by computing eigenspectra of the pertinent linearised operator. It is found that a hydraulic-fall flow over a bump is spectrally stable. The corresponding flow over a dip is found to be linearly unstable. In the latter case, time-dependent simulations show that ultimately, the flow settles into a time-periodic motion that corresponds to an invariant solution in an appropriately defined phase space. Physically, the solution consists of a localised large-amplitude wave that pulsates above the dip while simultaneously emitting nonlinear cnoidal waves in the upstream direction and multi-harmonic linear waves in the downstream direction.

Navier–Stokes Flow Past a Rigid Body That Moves by Time-Periodic Motion

We study existence, uniqueness and asymptotic spatial behavior of time-periodic strong solutions to the Navier-Stokes equations in the exterior of a rigid body, $\mathscr B$, moving by time-periodic motion of given period $T$, when the data are sufficiently regular and small. Our contribution improves all previous ones in several directions. For example, we allow both translational, $\bfxi$, and angular, $\bfomega$, velocities of $\mathscr B$ to depend on time, and do not impose any restriction on the period $T$ nor on the averaged velocity, $\bar{\bfxi}$, of $\mathscr B$. If $\bfxi\not\equiv\0$ we assume that $\bfxi$ and $\bfomega$ are both parallel to a constant direction, while no further assumption is needed if $\bfxi\equiv\0$. We also furnish the spatial asymptotic behavior of the velocity field, $\bfu$, associated to such solutions. In particular, if $\mathscr B$ has a net motion characterized by $\bar{\bfxi}\neq\0$, we then show that, at large distances from $\mathscr B$, $\bfu$ manifests a wake-like behavior in the direction $-\bar{\bfxi}$, entirely similar to that of the velocity field of the steady-state flow occurring when $\mathscr B$ moves with velocity $\bar{\bfxi}$.

A microfluidic viscometer: Translation of oscillatory motion of a water microdroplet in oil under electric field.

The electric field induced motion of a charged water droplet suspended in a low-dielectric oil medium is exploited to evaluate the rheological properties of the suspending medium. The time-periodic electrophoretic motion of the droplet between the electrodes decorated in a polymeric micro-well is translated into a proof-of-concept microfluidic prototype, which can measure viscosities of the unknown fluid samples. The variations in the instantaneous velocities of the migrating droplet have been measured inside silicone oil of known physical properties at different electric field intensities. Subsequently, a balance between the electric field to the viscous force has been employed to evaluate the experimental charge density on the droplet surface. Thereafter, a comprehensive scaling law has been devised to find a correlation between the charge on the droplet to the dielectric permittivity of the surrounding medium, size of the water droplet, and the applied electric field intensity. Following this, the scaling law and force balance have been employed together to evaluate the unknown viscosity of an array of suspending mediums by simply analyzing the electrophoretic motion of water droplet. The model proposed is also found to be consistent when a solid amberlite microparticle has been employed as a probe instead of the water droplet. In such a scenario, minor changes in the exponents of the scaling law are found to be necessary to reproduce the results obtained using the water droplet. The method paves the way for the making of an economical and portable microfluidic rheometer with further finetuning and translational developments.

Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

Attainability of time-periodic flow of a viscous liquid past an oscillating body

A body $$\mathscr {B}$$ is started from rest by a translational motion in an otherwise quiescent Navier–Stokes liquid filling the whole space. We show, for small data, that if after some time $$\mathscr {B}$$ reaches a spinless oscillatory motion of period $$\mathcal T$$ , the liquid will eventually execute also a time periodic motion with the same period $$\mathcal T$$ . This result is a suitable generalization of the famous Finn’s starting problem for steady states, to the case of time-periodic motions.

Viscous Flow Past a Body Translating by Time-Periodic Motion with Zero Average

We study existence, uniqueness, regularity and asymptotic spatial behavior of a Navier-Stokes flow past a body moving by a time-periodic translational motion of period $T$, and with zero average. For example, $\mathscr B$ moves in an oscillating fashion. The flow is also time-periodic with same period $T$. However, sufficiently ``far" from the body, the oscillatory component decays faster than the averaged component, so that the flow shows there a distinctive steady-state character. This provides a rigorous proof of the ``steady streaming" phenomenon.

Floquet oscillations in periodically driven Dirac systems

Electrons in a lattice exhibit time-periodic motion, known as Bloch oscillation, when subject to an additional static electric field. Here we show that a corresponding dynamics can occur upon replacing the spatially periodic potential by a time-periodic driving: Floquet oscillations of charge carriers in a spatially homogeneous system. The time lattice of the driving gives rise to Floquet bands that take on the role of the usual Bloch bands. For two different drivings (harmonic driving and periodic kicking through pulses) of systems with linear dispersion we demonstrate the existence of such oscillations, both by directly propagating wave packets and based on a complementary Floquet analysis. The Floquet oscillations feature richer oscillation patterns than their Bloch counterpart and enable the imaging of Floquet bands. Moreover, their period can be directly tuned through the driving frequency. Such oscillations should be experimentally observable in effective Dirac systems, such as graphene, when illuminated with circularly polarized light.

Analytical-Numerical Approach to Describing Time-Periodic Motion of Fronts in Singularly Perturbed Reaction–Advection–Diffusion Models

The paper presents an analytical-numerical approach to the study of moving fronts in singularly perturbed reaction–diffusion–advection models. A method for generating a dynamically adapted grid for the efficient numerical solution of problems of this class is proposed. The method is based on a priori information about the motion and properties of the front, obtained by rigorous asymptotic analysis of a singularly perturbed parabolic problem. In particular, the essential parameters taken into account when constructing the grid are estimates of the position of the transition layer, as well as its width and structure. The proposed analytical-numerical approach can significantly save computer resources, reduce the computation time, and increase the stability of the computational process in comparison with the classical approaches. An example demonstrating the main ideas and methods of application of the proposed approach is considered.

Predominantly unidirectional rotation of a solid body and a viscous liquid

Under consideration are two problems of the time-periodic motion of a hydro-mechanical system consisting of a viscous liquid and solid bodies bordering it. A new hydro-mechanical effect is discovered.

Fluid transport and mixing by an unsteady microswimmer

Swimming at low Reynolds numbers typically requires time-periodic motion that breaks time-reversal symmetry, creating an actual unsteady velocity field around swimming microorganisms. A study of the effect of ``unsteadiness'' on enhanced diffusivity and tracer displacement statistics takes a step further in considering and modeling realistic microorganisms.

Periodic solutions to a mean-field model for electrocortical activity

We consider a continuum model of electrical signals in the human cortex, which takes the form of a system of semilinear, hyperbolic partial differential equations for the inhibitory and excitatory membrane potentials and the synaptic inputs. The coupling of these components is represented by sigmoidal and quadratic nonlinearities. We consider these equations on a square domain with periodic boundary conditions, in the vicinity of the primary transition from a stable equilibrium to time-periodic motion through an equivariant Hopf bifurcation. We compute part of a family of standing wave solutions, emanating from this point.

Rigid ring-shaped particles that align in simple shear flow

AbstractMost rigid, torque-free, low-Reynolds-number, axisymmetric particles undergo a time-periodic tumbling motion in a simple shear flow, with their axes of symmetry following a set of closed Jeffery orbits. We have identified a class of rigid, ring-like particles whose axes of symmetry instead reach a permanent alignment near the velocity gradient direction with the plane of the particle aligning near the flow–vorticity plane. An asymptotic analysis for small particle aspect ratio (ratio of length parallel to the axis of symmetry to diameter perpendicular to the axis) shows that an appropriate asymmetry of the ring cross-section with a thinner outer edge and thicker inner edge leads to a tendency to rotate in a direction opposite to the vorticity; this tendency can balance the usual rotation rate associated with the finite thickness of the particle. Boundary integral computations for finite particle aspect ratios are used to determine the conditions of aspect ratio and degree of asymmetry that lead to the aligning behaviour and the final orientation of the axis of symmetry of the aligned particles. The aligning particle follows an equation of motion similar to the Leslie–Erickson equation for the director of a small-molecule nematic liquid crystal. However, whereas the alignment of the director arises from intermolecular interactions, the ring-like particle aligns solely due to its intrinsic rotational motion in a low-Reynolds-number flow.

A coupled far-field formulation for time-periodic numerical problems in fluid dynamics

Consider uniform flow past an oscillating body generating a time-periodic motion in an exterior domain, modelled by a numerical fluid dynamics solver in the near field around the body. A far-field formulation, based on the Oseen equations, is presented for coupling onto this domain thereby enabling the whole space to be modelled. In particular, examples for formulations by boundary elements and infinite elements are described.

When complexity leads to simplicity: Ocean surface mixing simplified by vertical convection

The effect of weak vertical motion on the dynamics of materials that are limited to move on the ocean surface is an unresolved problem with important environmental and ecological implications (e.g., oil spills and larvae dispersion). We investigate this effect by introducing into the classical horizontal time-periodic double-gyre model vertical motion associated with diurnal convection. The classical model produces chaotic advection on the surface. In contrast, the weak vertical motion simplifies this chaotic surface mixing pattern for a wide range of parameters. Melnikov analysis is employed to demonstrate that these conclusions are general and may be applicable to realistic cases. This counter intuitive result that the very weak nocturnal convection simplifies ocean surface mixing has significant outcomes.

Experiments on the periodic oscillation of free containers driven by liquid sloshing

AbstractExperiments on the time-periodic liquid sloshing-induced sideways motion of containers are presented. The measurements are compared with finite-depth potential theory developed from standard normal mode representations for rectangular boxes, upright cylinders, wedges and cones of$9{0}^{\ensuremath{\circ} } $apex angles, and cylindrical annuli. It is assumed that the rectilinear horizontal motion of the containers is frictionless. The study focuses on measurements of the horizontal oscillations of these containers arising solely from the liquid waves excited within. While the wedge and cone exhibit only one mode of oscillation, the boxes, cylinders and annuli have an infinite number of modes. For the boxes, cylinders and one of the annuli, we have been able to excite motion and record data for both the first and second modes of oscillation. Frequencies$\omega $were acquired as the average of three experimental determinations for every filling of mass$m$in the dry containers of mass${m}_{0} $. Measurements of the dimensionless frequencies$\omega / {\omega }_{R} $over a range of dimensionless liquid masses$M= m/ {m}_{0} $are found to be in essential agreement with theoretical predictions. The frequencies${\omega }_{R} $used for normalization arise naturally in the mathematical analysis, different for each geometry considered. Free surface waveforms for a box, a cylinder, the wedge and the cone are compared at a fixed value of$M$.

A comparative study on the rheology and wave dissipation of kaolinite and natural Hendijan Coast mud, the Persian Gulf

The objective of this paper is to investigate the rheological behavior of kaolinite and Hendijan mud, located at the northwest part of the Persian Gulf, and the dissipative role of this muddy bed on surface water waves. A series of laboratory rheological tests was conducted to investigate the rheological response of mud to rotary and cyclic shear rates. While a viscoplastic Bingham model can successfully be applied for continuous controlled shear-stress tests, the rheology of fluid mud displays complex viscoelastic behavior in time-periodic motion. The comparisons of the behavior of natural Hendijan mud with commercial kaolinite show rheological similarities. A large number of laboratory wave-flume experiments were carried out with a focus on the dissipative role of the fluid mud. Assuming four rheological models of viscous, Kelvin-Voigt viscoelastic, Bingham viscoplastic, and viscoelastic-plastic for fluid mud layer, a numerical multi-layered model was applied to analyze the effects of different parameters of surface wave and muddy bed on wave attenuation. The predicted results based on different rheological models generally agree with the obtained wave-flume data implying that the adopted rheological model does not play an important role in the accuracy of prediction.

Interaction of Turing and Hopf Modes in the Superdiffusive Brusselator Model Near a Codimension Two Bifurcation Point

Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensional superdiffusive Brusselator model are analyzed. The superdiffusive Brusselator model differs from its regular counterpart in that the Laplacian operator of the regular model is replaced by @ fi =@ j»j fi , 1 < fi < 2, an integro-differential operator that reflects the nonlocal behavior of superdiffusion. The order of the operator, fi, is a measure of the rate of superdiffusion, which, in general, can be different for each of the two components. A weakly nonlinear analysis is used to derive two coupled amplitude equations describing the slow time evolution of the Turing and Hopf modes. We seek special solutions of the amplitude equations, namely a pure Turing solution, a pure Hopf solution, and a mixed mode solution, and analyze their stability to long-wave perturbations. We find that the stability criteria of all three solutions depend greatly on the rates of superdiffusion of the two components. In addition, the stability properties of the solutions to the anomalous diffusion model are different from those of the regular diffusion model. Numerical computations in a large spatial domain, using Fourier spectral methods in space and second order Runge-Kutta in time are used to confirm the analysis and also to find solutions not predicted by the weakly nonlinear analysis, in the fully nonlinear regime. Specifically, we find a large number of steady state patterns consisting of a localized region or regions of stationary stripes in a background of time periodic cellular motion, as well as patterns with a localized region or regions of time periodic cells in a background of stationary stripes. Each such pattern lies on a branch of such solutions, is stable and

Short and long waves over a muddy seabed

The available experimental results have shown that in time-periodic motion the rheology of fluid mud displays complex viscoelastic behaviour. Based on the measured rheology of fluid mud from two field sites, we study the interaction of water waves and fluid mud by a two-layered model in which the water above is assumed to be inviscid and the mud below is viscoelastic. As the fluid-mud layer in shallow seas is usually much thinner than the water layer above, the sharp contrast of scales enables an approximate analytical theory for the interaction between fluid mud and small-amplitude waves with a narrow frequency band. It is shown that at the leading order and within a short distance of a few wavelengths, wave pressure from above forces mud motion below. Over a much longer distance, waves are modified by the accumulative dissipation in mud. At the next order, infragravity waves owing to convective inertia (or radiation stresses) are affected indirectly by mud motion through the slow modulation of the short waves. Quantitative predictions are made for mud samples of several concentrations and from two different field sites.

MEASURING AND CONTROLLING THE CHAOTIC MOTION OF PROFITS

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

Topological entropy and the controlled effect of glucose in the electrical activity of pancreatic [formula omitted]-cells

Insulin secretion from electrically coupled β-cells is governed by bursting electrical activity. In response to stimulatory concentrations of glucose, the membrane potential of pancreatic β-cells may experience a transition from bursting-spiking oscillations to continuous spiking oscillations. This transition can be chaotic but becomes more and more regular with an increase in glucose. In the presence of chaos, the inhability to predict the behavior of dynamical systems suggests the application of chaos control methods, when we are more interested in obtaining attracting time periodic motion. In this article, we focus our attention on a specific mathematical model from the literature that mimics the glucose-induced electrical activity of pancreatic β-cells (Deng, 1993 [7]). Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of the kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant allows us to quantify and to distinguish different chaotic regimes. Finally, we show that chaotic orbits of the system can be controlled, without changing their orbital properties, and be turned into desired limit cycles. The control is illustrated by an application of a feedback control technique developed by Romeiras, Grebogi, Ott and Dayawansa (1992) [13]. This work provides an illustration of how our understanding of biophysically motivated models can be directly enhanced by the theory of dynamical systems.