Time-periodic Forcing Research Articles

Active micromixer based on artificial cilia

We propose a design for an active micromixer that is inspired by the motion of ciliated micro-organisms occurring in nature. The conceptual design consists of an array of individually addressable artificial cilia in the form of microactuators covering the channel wall. The microactuators can be set into motion by an external stimulus such as an electric or a magnetic field, inducing either a primary or secondary motion in the surrounding fluid. To validate the concept and to help to design the precise mixer configuration, we developed a computational fluid-structure model. This model is based on a fictitious domain method that couples the microactuator motion to the concomitant fluid flow, fully capturing the mutual fluid-structure interactions. The simulated flow patterns resulting from the motion of single and multiple actuated elements (in a microchannel filled with a Newtonian fluid) under the action of a time-periodic forcing function are analyzed using dynamical systems theory to quantify the mixing efficiency. The results show that with a proper actuation scheme, two microactuators placed on the same wall of a microchannel can indeed induce effective mixing by chaotic advection; their distance should be small, but collisions should be avoided, and they can be actuated in a rather broad regime around 90° out of phase. Placing actuators on opposite walls also induces exponential stretching in the fluid, but if their length is relatively small, of the order of 20% of the channel height, mixing effectiveness is higher when they are arranged on the same wall.

2D Stochastic Navier–Stokes Equations with a Time-Periodic Forcing Term

We study the long time behavior of the solution X(t, s, x) of a 2D-Navier–Stokes equation subjected to a periodic time dependent forcing term. We prove in particular that as $${t \to \infty}$$ , $${\mathbb{E}[\varphi(X(t, s, x))]}$$ approaches a periodic orbit independently of s and x for any continuous and bounded real function $${\varphi}$$ .

The amplitude equations for the first instability of electro-convection in nematic liquid crystals in the case of two unbounded space directions

Electro-convection in nematic liquid crystals is a paradigm for pattern formation in anisotropic systems. In this paper we discuss the amplitude equations obtained for this pattern forming system close to the first instability in the case of two unbounded space directions. We prove error estimates showing the validity of these formal approximations for a regularized version of the weak electrolyte model (WEM). New mathematical aspects occur due to the possible instability mechanisms of the WEM and due to the external time-periodic forcing.

Frequency locking and complex dynamics near a periodically forced robust heteroclinic cycle

Robust heteroclinic cycles occur naturally in many classes of nonlinear differential equations with invariant hyperplanes. In particular they occur frequently in models for ecological dynamics and fluid mechanical instabilities. We consider the effect of small-amplitude time-periodic forcing and describe how to reduce the dynamics to a two-dimensional map. In the limit where the heteroclinic cycle loses asymptotic stability, intervals of frequency locking appear. In the opposite limit, where the heteroclinic cycle becomes strongly stable, the dynamics remains chaotic and no frequency locking is observed.

Capturing patterns and symmetries in chaotic granular flow

Segregation patterns formed by time-periodic flow of polydisperse granular material (varying in particle size) in quasi-two-dimensional (quasi-2D) tumblers capture the symmetries of Poincaré sections, stroboscopic maps of the underlying flow, derived from a continuum model which contains no information about particle properties. We study this phenomenon experimentally by varying the concentration of small particles in a bidisperse mixture in quasi-2D tumblers with square and pentagonal cross sections. By coupling experiments with an analysis of periodic points, we explain the connection between the segregation patterns and the dynamics of the underlying flow. Analysis of the eigenvectors and unstable manifolds of hyperbolic points shows that lobes of segregated small particles stretch from hyperbolic points toward corners of the tumbler, demonstrating the connection between regions of chaotic flow and the shape of the segregation patterns. Furthermore, unstable manifolds map the shape of lobes of segregated particles. The techniques developed here can also be applied to nonpolygonal tumblers such as elliptical tumblers, as well as to circular tumblers with time-periodic forcing.

비선형 Ekman 펌핑 모델의 개발

Developed in this study is a nonlinear Ekman pumping model to be used in simulating the rotating flows with quasi-three-dimensional Navier-Stokes equations. In this model, the Ekman pumping velocity is given from the solution of the Ekman boundary-layer equations for the region adjacent to the bottom wall of the flow domain; the boundary-layer equations are solved in the momentum-integral form. The developed model is then applied to rotating flows in a rectangular container receiving a time-periodic forcing. By comparing our results with the DNS and experimental data we have validated the developed model. We also compared our results with those given from the classical Ekman pumping model. It was found that our model can predict the rotating flows more precisely than the classical linear model.

Evolving eddy structures in oscillatory Stokes flows in domains with sharp corners

Stokes flow of a viscous fluid in a cylindrical container driven by time-periodic forcing, either at the boundary or through oscillation of the cylinder about an axis parallel to its generators, is considered. The behaviour is governed by a dimensionless frequency parameter η and by the geometry of the cylinder cross-section. Various cross-sections (square, rhombus, and sector of a circle) are first treated by either finite-difference or analytic techniques and typical transitions of streamline topology during flow reversals are identified. Attention is then focused on the asymptotic behaviour near any sharp corner on the boundary. For small η, a regular perturbation expansion reveals the manner in which local flow reversal proceeds during each half-cycle of the flow. The behaviour depends on the corner angle, and different regimes are identified for both types of forcing. For example, for an oscillating square domain, eddies grow symmetrically from each corner and participate in the subsequent flow reversal in the interior. For large η, the corner eddies merge into Stokes-type boundary layers which drive the interior flow-reversal process. In general, the local corner analysis provides a key to an understanding of the global flow evolution.

Coherent control of population transfer between communicating defects

Population transfer between two identical, communicating defects in a one-dimensional tight-binding lattice can be systematically controlled by external time-periodic forcing. Employing a force with slowly changing amplitude, the time it takes to transfer a particle from one defect to the other can be altered over several orders of magnitude. An analytical expression is derived which shows how the forcing effectively changes the energy splitting between the defect states, and numerical model calculations illustrate the possibility of coherent control of the transfer.

Averaging of time - periodic systems without a small parameter

In this article, we present a new approach to averaging in non-Hamiltonian systems with periodic forcing. The results here do not depend on the existence of a small parameter. In fact, we show that our averaging method fits into an appropriate nonlinear equivalence problem, and that this problem can be solved formally by using the Lie transform framework to linearize it. According to this approach, we derive formal coordinate transformations associated with both first-order and higher-order averaging, which result in more manageable formulae than the classical ones. Using these transformations, it is possible to correct the solution of an averaged system by recovering the oscillatory components of the original non-averaged system. In this framework, the inverse transformations are also defined explicitly by formal series; they allow the estimation of appropriate initial data for each higher-order averaged system, respecting the equivalence relation. Finally, we show how these methods can be used for identifying and computing periodic solutions for a very large class of nonlinear systems with time-periodic forcing. We test the validity of our approach by analyzing both the first-order and the second-order averaged system for a problem in atmospheric chemistry.

The asymptotic behaviour of solutions of the forced Burgers equation on the circle

We describe the asymptotic behaviour of entropy solutions of the inviscid Burgers equation on the circle with time-periodic forcing. These solutions converge to periodic states, but the period of these limit states may be greater than the period of the forcing. A corollary is that the only solutions without shocks are periodic. This result can be interpreted in terms of the invariant curves of the associated dynamics.

A numerical and experimental study on advection in three-dimensional Stokes flows

Three-dimensional advection of passive tracers in non-inertial flows is studied in a finite cylinder confined by two parallel endwalls by means of numerical simulations and laboratory experiments. The fluid is set in motion through steady or time-periodic forcing by in-plane motion of the endwalls via a given forcing protocol. The numerical analysis centres on a dynamical-systems approach and concerns symmetry-based identification of coherent structures in the web of tracer paths (collectively defining the flow topology) for a number of archetypal flow configurations. The role of the flow topology in the process of tracer transport is investigated by numerical tracking of finite-size material objects released at strategic locations in the flow. Experimental validation of key aspects of the numerical results has been carried out in laboratory experiments by flow visualization with dye and flow measurement via three-dimensional particle tracking velocimetry.

Time Asymptotics of the Schrödinger Wave Function in Time-Periodic Potentials

We study the transition to the continuum of an initially bound quantum particle in $\RR^d$, $d=1,2,3$, subjected, for $t\ge 0$, to a time periodic forcing of arbitrary magnitude. The analysis is carried out for compactly supported potentials, satisfying certain auxiliary conditions. It provides complete analytic information on the time Laplace transform of the wave function. From this, comprehensive time asymptotic properties (Borel summable transseries) follow. We obtain in particular a criterion for whether the wave function gets fully delocalized (complete ionization). This criterion shows that complete ionization is generic and provides a convenient test for particular cases. When satisfied it implies absence of discrete spectrum and resonances of the associated Floquet operator. As an illustration we show that the parametric harmonic perturbation of a potential chosen to be any nonzero multiple of the characteristic function of a measurable compact set has this property.

Forced Oscillations of a Damped Korteweg-de Vries Equation in a Quarter Plane

Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in the context of a damped Korteweg-de Vries equation. Thus, consideration is given to the initial-boundary-value problem [Formula: see text] For this problem, it is shown that if the small amplitude, boundary forcing h is periodic of period T, say, then the solution u of (*) is eventually periodic of period T. More precisely, we show for each x > 0, that u(x, t + T) - u(x, t) converges to zero exponentially as t → ∞. Viewing (*) (without the initial condition) as an infinite dimensional dynamical system in the Hilbert space Hs(R+) for suitable values of s, we also demonstrate that for a given, small amplitude time-periodic boundary forcing, the system (*) admits a unique limit cycle, or forced oscillation (a solution of (*) without the initial condition that is exactly periodic of period T). Furthermore, we show that this limit cycle is globally exponentially stable. In other words, it comprises an exponentially stable attractor for the infinite-dimensional dynamical system described by (*).

Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing

Homoclinic orbits of semilinear parabolic partial differentialequations can split under time-periodic forcing as for ordinarydifferential equations. The stable and unstable manifoldmay intersect transverse at persisting homoclinic points. Thesize of the splitting is estimatedto be exponentially small of order exp$(-c/\epsilon)$ in the period $\epsilon$ of the forcingwith $\epsilon \rightarrow 0$.

Experimental investigation of propagating flow instabilities generated by super vibratory agitation

In the present experiment a multi-fin oscillator, consisting of multiple parallel diaphragms, has been employed to vertically oscillate the fluid in a rectangular tank filled with quiescent water. The fins perform a combined heaving and pitching oscillation while the whole oscillating machine is stationary. The distance between two adjacent fins is regarded as the oscillator characteristic length ( D). While applying a time-periodic forcing, flow field measurements were carried out using a two-component laser-Doppler anemometer. The measured velocity data have been phase-synchronised with the periodic excitation by sensing the position of the oscillator with a Hall-probe. Two configurations were studied: open geometry (tank dimensions ≫D, a domain practically without boundaries) and closed geometry (a domain with no-slip boundaries). The wakes behind the oscillating fins show average velocity profiles with the form of jets. The flow pattern comprises back-flow areas in-between the fins up to a streamwise distance depending on flow geometry and driving frequency. The main flow is horizontally directed for open geometry and slightly upward-directed for closed geometry, under an angle decreasing with the streamwise distance from the fins. Concentrated-vorticity regions originating from the fins extend up to a streamwise distance of ≈0.5 D. For larger distances from the fins, the flow turbulence becomes homogeneous (turbulence intensity values are low and nearly identical in streamwise and spanwise directions).

Computing global orbits of the forced spherical pendulum

Orbits of the spherical pendulum with time-periodic forcing are considered. A numerical framework is developed, which allows orbits to explore the entire “globe”: the spherical pendulum is considered as an invariant manifold in an ambient six-dimensional Euclidean space. The numerical integrator is the second-order Störmer–Verlet method coupled with the Shake–Rattle algorithm. The algorithm preserves numerically the phase space of the sphere, which is a manifold, to machine accuracy. Poincaré sections, restricted to the configuration space, are used to illustrate the transition from oscillatory behavior to chaotic solutions, as the amplitude of the pivot motion is changed. The qualitative change in the Poincaré sections from regular to chaotic behavior is in excellent qualitative agreement with corresponding computations of the Lyapunov exponents (LEs). The LEs are also computed using a novel variant of the Shake–Rattle algorithm. The results show that irregular behavior can explore the entire sphere—even at low forcing amplitudes—and therefore local methods which parameterize only part of the sphere are inadequate in general, and may lead to spurious dynamics. The numerical framework provides a tool for detailed investigation of the symmetric chaos of the forced spherical pendulum.

On the dynamics of a vertically driven damped planar pendulum

Results on the dynamics of the planar pendulum with parametric vertical timeperiodic forcing are reviewed and extended. Numerical methods are employed to study the various dynamical features of the system about its equilibrium positions. Furthermore, the dynamics of the system far from its equilibrium points is systematically investigated by using phase portraits and Poincare sections. The attractors and the associated basins of attraction are computed. We also calculate the Lyapunov exponents to show that for some parameter values the dynamics of the pendulum shows sensitivity to initial conditions.

Astrophysical significance of the anisotropic kinetic alpha effect

The generation of large scale flows by the anisotropic kinetic alpha (AKA) effect is investigated in simulations with a suitable time-dependent space- and time-periodic anisotropic forcing lacking parity invariance. The forcing pattern moves relative to the fluid, which leads to a breaking of the Galilean invariance as required for the AKA effect to exist. The AKA effect is found to produce a clear large scale flow pattern when the Reynolds number, R, is small as only a few modes are excited in linear theory. In this case the non-vanishing components of the AKA tensor are dynamically independent of the Reynolds number. For larger values of R, many more modes are excited and the components of the AKA tensor are found to decrease rapidly with increasing value of R. However, once there is a magnetic field (imposed and of sufficient strength, or dynamo-generated and saturated) the field begins to suppress the AKA effect, regardless of the value of R. It is argued that the AKA effect is unlikely to be astrophysically significant unless the magnetic field is weak and R is small.

Bounds on dissipation for Navier–Stokes flow with Kolmogorov forcing

In this paper, bounds on average viscous dissipation are derived for Kolmogorov flow in a spatially periodic domain with steady and unsteady forcing, at arbitrarily large Grashof number G. For a force of the form F 0 sin mz i or F 0 sin mz cos ωt i , we derive various bounds on the total dissipation in the flow, D u , as well as on the dissipation D m obtained from the x-velocity averaged over the x, y plane (the mean velocity of the flow). We derive upper bounds on D u and D v= D u− D m , as well as lower bounds on D m and D m/ D u , adopting constraints of the kind introduced by Howard and Busse and assuming a steady force. The background flow method introduced by Doering and Constantin is used to obtain an improved lower bound on D m/ D u of O( G −1), and a lower bound on D u , of O( G −1/2) where G≔ F 0 L 3/ ν 2 is the Grashof number. Some of these results are then generalized to time-periodic forcing. Direct numerical simulation of the flow indicates that these bounds leave substantial gaps at large Grashof number G, the calculated D m(G) and D u(G) being O( G −1/2) and O(1), respectively, as G→∞. Our theoretical bounds on D m, D u are shown to be attained by steady laminar-type flows for neighboring forcing functions, which seems to indicate that these bounds cannot be improved by adding further dynamical constraints. However, our elementary upper bound on D v can probably be improved by placing more constraints on the flows. These results serve to emphasize the difference between boundary-driven turbulence and body-force driven turbulence where the appropriate dissipation bound is believed saturated at least up to logarithms.

Time-Averaging under Fast Periodic Forcing of Parabolic Partial Differential Equations: Exponential Estimates

The phases of a large class of parabolic partial differential equations with rapid time-periodic forcing can be separated up to exponential small errors. The originally nonautonomous equation is transformed such that the nonautonomous terms are exponentially small in the period h of the forcing. This is a counterpart for partial differential equations of the theorem by A. Neishtadt (1984, J. Appl. Math. Mech.48, 134–139) for ordinary differential equations. In our case the exponential rate depends on time t and the estimates have the form hexp(−c(t)h−1/3).