Time-periodic Motions Research Articles

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.

Numerical investigation of flow regimes in T‐shaped micromixers: Benchmark between finite volume and spectral element methods

AbstractComputational fluid dynamics (CFD) is very appealing to investigate mixing and reaction in microdevices, as it allows easily investigating different operating conditions as well as mixer geometries. This latter aspect is very important as the flow in microdevices is laminar so the mixing between reactants should be promoted by a clever mixer design, aimed at breaking the flow symmetries. Recently time periodic motions that improve mixing have been observed to take place in a T‐junction at low Reynolds numbers. In this case the numerical modelling should be based on direct numerical simulations (DNS), thus involving high computational resources. In this work, two different CFD approaches, i.e., finite volume and spectral element methods, are applied and compared for the analysis of the mixing process in the well known T‐shaped micromixer. Spectral elements methods are particularly suited for DNS; however, they have been scarcely applied to study micromixers, while plenty of works can be found with finite volume methods. The analysis is carried out using both ideal and non‐ideal liquid binary mixtures, the latter presenting a negative fluidity of mixing (i.e., the viscosity of the mixture is higher than that of the pure components). Moreover the numerical results are validated with simple flow visualization experiments.

The effect of time-periodic base angular motions upon dynamic response of asymmetric rotor systems

The effect of time-periodic base angular motions upon dynamic response of asymmetric rotor systems is investigated in this article. Both the disk and rotating shaft with asymmetric cross sections a...

Metal-Organic Framework "Swimmers" with Energy-Efficient Autonomous Motility.

Placed at a water/air interface, particles of porphyrin-based MOFs (metal-organic frameworks) cut from large-area films display efficient, multiple-use autonomous motility powered by release of solvents incorporated in the MOF matrix and directionality dictated by their shapes. The particles can be refueled multiple times and can achieve speeds of ca. 200 mm·s-1 with high kinetic energy per unit of chemical "fuel" expended (>50 μJ·g-1). Efficiency of motion depends on the nature of the fuel used as well as the microstructure and surface wettability of the MOF surface. When multiple movers are present at the interface, they organize into "open" structures that exhibit collective, time-periodic motions.

Effect of impurities on the onset and growth of gravitational instabilities in a geological CO2 storage process: Linear and nonlinear analyses

Because 75% of the total cost of carbon capture and storage (CCS) arises from the separation of CO2 from gas streams (Nsakala et al., 2001), the co-injection of CO2 and impurities such as H2S, N2, and SO2 is considered a cost-effective alternative to pure CO2 geological sequestration. Here, the effect of the impurities on the onset of gravitational instability has been analyzed theoretically and numerically. Linear stability equations have been derived and solved analytically and numerically. Double diffusive effects make the system stable or unstable depending on the values of the diffusivity ratio, δB, and buoyancy ratio, rβrC. In addition, using the Fourier spectral method, we have traced the temporal evolution of the gravitational fingers numerically. The shape and the growth history of the fingers are strongly dependent on the impurity content. The time-periodic oscillatory motions are not observed in the present linear and nonlinear analyses. For a given Rayleigh number, the dissolution of N2 and H2S impurities makes the system stable, whereas dissolved SO2 accelerates the onset of instability.

Parametric instability of flexible rotor-bearing system under time-periodic base angular motions

Parametric instability of flexible rotor-bearing system under time-periodic base angular motions is analyzed in this paper. The accurate finite element model for the flexible rotor-bearing system under time-varying base angular motions is derived based upon the energy theorem and Lagrange’s principle. Three base angular motions, including the rolling, pitching and yawing motions, are assumed to be sinusoidal perturbations superimposed upon constant terms. Considering the time-varying base movements, the second order differential equations of the system will have time-periodic gyroscopic and stiffness coefficients. The discrete state transition matrix (DSTM) method is introduced for numerically acquiring the instability regions. Based upon these, instability computations for a rotor-bearing system with one base motion alone and two base motions together are conducted, respectively. The effects of rotating speed, amplitudes of base motion and phases between two base motions on both the primary and combination instability regions are discussed in detail.

A Note on the Existence and Uniqueness of Time-Periodic Electro-rheological Flows

We prove the existence and uniqueness of time-periodic motions of an electro-rheological fluid, subject to a time-periodic body force. Further if the force vanishes in its periodicity interval and if a suitable relation between the size of the force and its period holds, then the solution periodically becomes extinct.

Dynamic behaviors of a geared rotor system under time-periodic base angular motions

The dynamic behaviors of a geared rotor system under time-periodic base motions are studied in this paper. The equations of motion for a spur gear pair system are derived based on the energy theorem and Lagrange's principle. Three base angular motions, including the rolling, pitching and yawing motions, are assumed to be sinusoidal perturbations superimposed upon constant terms. Considering the time-varying base movements and gear meshing, the second order differential equations of the system will not only have time-periodic gyroscopic and stiffness coefficients, but also the multi-frequency external excitations. Numerical method is utilized to obtain the lateral and torsional responses of the geared system under transmission error and unbalanced mass excitations. The effects of various base angular motions on both frequency response and response spectra are discussed in detail. Compared with the other two types of periodic base motions, the rolling base motion has the greatest impact on the dynamic behaviors of the geared system. The transient response of the geared rotor system is always greatly enhanced no matter which type of base angular motion is in operation.

Existence of time-periodic weak solutions to the stochastic Navier-Stokes equations around a moving body

The existence of time-periodic stochastic motions of an incompressible fluid is obtained. Here the fluid is subject to a time-periodic body force and an additional time-periodic stochastic force that is produced by a rigid body moves periodically stochastically with the same period in the fluid.

Dynamic effects associated with spatial discretization of nonlinear wave equations

A new phenomenon is detected that the attractors of a nonlinear wave equation can differ substantially from those of its finite-dimensional analogue obtained by replacing the spatial derivatives with corresponding difference operators (regardless of the discretization step). The presentation is based on a typical example, namely, on the boundary value problem for a Van-der-Pol-type telegraph equation with zero Neumann conditions at the ends of the unit interval. Under certain generic conditions, the problem is shown to admit only stable time-periodic motions, which are fairly numerous. When the problem is replaced by an approximating system of ordinary differential equations, the situation becomes fundamentally different: all the periodic motions (except for one or two) become unstable and, instead of them, stable two-dimensional invariant tori appear.

Dynamics and rheology of biaxial liquid crystal polymers in shear flows

We study nematodynamics of a mesoscopic system consisting of sheared biaxial liquid crystalline polymers using a hydrodynamical kinetic theory, in which the biaxial liquid crystal polymer is modeled as a rigid, biaxial, ellipsoidal molecule immersed in viscous solvent. The governing Smoluchowski equation in the model is solved in selected regions of the material parameter space and a range of accessible shear rates using a Wigner–Galerkin spectral method. In addition to the truly biaxial flow-aligning steady states, log-rolling steady states, and out-of-plane steady states, we report the presence of two new time-periodic motions, chaotic motion and associated phase transitions in a range of shear rates and selected material parameters. Rheological signatures of the sheared mesoscopic system are identified with predominant shear thinning in all phases and alternating signs between the normal stress differences in steady vs time-dependent motions.

Shear-induced mesostructures in biaxial liquid crystals

We study nematodynamics of a mesoscopic system consisting of sheared biaxial liquid crystalline polymers using a hydrodynamical kinetic theory. We solve the governing Smoluchowski equation using the Galerkin method in selected regions of the material parameter space and a range of accessible shear rates to investigate stable mesoscopic states and robust structures. The imposed shear flow breaks the rotational symmetry in the quiescent state to induce truly biaxial flow-aligning steady states, logrolling states, out-of-plane steady states, exotic time-periodic motions, and chaotic motions in different regimes of material parameters and shear rates.

On the linear stability of Stokes layers

Oscillatory flows occur naturally, with applications ranging across many disciplines from engineering to physiology. Transition to turbulence in such flows is a topic of practical interest and this article discusses some recent work that has furthered our understanding of the stability of a class of time-periodic fluid motions. Our study starts with an examination of the linear stability of a classical flat Stokes layer. Although experiments conducted over many years have demonstrated conclusively that this layer is unstable at a sufficiently large Reynolds number, it has only been relatively recently that rigorous theoretical confirmation of this behaviour has been obtained. The analysis and numerical calculations for the planar Stokes layer were subsequently extended to flows in channels and pipes and for the flow within a torsionally oscillating circular cylinder. We discuss why our predictions for the onset of instability in these geometries are in disappointingly poor agreement with experimental results. Finally, some suggestions for future experimental work are given and some areas for future theoretical analysis outlined.

An elastic rod model for anguilliform swimming

We develop a model for anguilliform (eel-like) swimming as an elastic rod actuated via time-dependent intrinsic curvature and subject to hydrodynamic drag forces, the latter as proposed by Taylor (in Proc Roy Proc Lond A 214:158-183, 1952). We employ a eometrically exact theory and discretize the resulting nonlinear partial differential evolution both to perform numerical simulations, and to compare with previous models consisting of chains of rigid links or masses connected by springs, dampers, and prescribed force generators representing muscles. We show that muscle activations driven by motoneuronal spike trains via calcium dynamics produce intrinsic curvatures corresponding to near-sinusoidal body shapes in longitudinally-uniform rods, but that passive elasticity causes Taylor's assumption of prescribed shape to fail, leading to time-periodic motions and lower speeds than those predicted Taylor (in Proc Roy Proc Lond A 214:158-183, 1952). We investigate the effects of bending stiffness, body geometry, and activation patterns on swimming speed, turning behavior, and acceleration to steady swimming. We show that laterally-uniform activation yields stable straight swimming and laterally differential activation levels lead to stable turns, and we argue that tapered bodies with reduced caudal (tail-end) activation (to produce uniform intrinsic curvature) swim faster than ones with uniform activation.

Existence of time-periodic solutions to the Navier–Stokes equations around a moving body

We demonstrate the existence of time-periodic motions of an incompressible Navier-Stokes fluid subject to a time-periodic body force, occupying the region exterior to a body that performs a periodic rigid motion of same period.

Separation regions in two-dimensional high-Reynolds-number flow

Two-dimensional flow inside a closed, rigid container is driven by a spatially invariant source of vorticity. At high Reynolds number an asymptotic structure with an inviscid core and boundary layers emerges. For an internal flow, the slip boundary velocity must in general decelerate in order to be periodic. Hence an adverse pressure gradient acts in the standard boundary-layer equations with the associated danger of flow separation. This problem, considered inside a nearly circular container, is found to be regularized by the shear stress and displacement conditions imposed on the boundary layer. Flow reversal can then occur within the layer and the formation of small separation regions can be studied. Both steady and time-periodic motions are considered. Asymptotic solutions in the large-Reynolds-number limit are constructed for various parameter regimes involving the shape perturbation parameter δ and the frequency of time oscillations controlled by the Womersley number α −1 .

Time-periodic quasigeostrophic motion under dissipation and forcing

The quasigeostrophic equation is a prototypical geophysical fluid model. In this paper, we consider time-periodic motions of this model under dissipation and time-dependent wind forcing. We show that when the wind forcing is time-periodic and the spatial square-integral of the wind forcing is bounded in time, the full nonlinear quasigeostrophic (QG) model has time-periodic motions, under some conditions on the β parameter, Ekman number, viscosity and the domain size.

Generation of Localization in a Discrete Chain with Periodic Boundary Conditions: Numerical and Analytical Results

We study the generation of localization in a discrete chain composed of N subsystems and periodic boundary conditions. Strongly localized motions are studied; that is, time-periodic motions where nearly all of the energy is spatially confined to a single subsystem. For varying N, numerical results indicate that the strongly localized solutions are generated through a bifurcation from an in-phase spatially extended solution. However, in the limit as $N \rightarrow \infty$, the bifurcation point tends to infinity, and a smooth transition from localization to nonlocalization occurs. We then present an analytic technique to complement the numerical results. It is based on the matching local asymptotic expansions of a solution branch using Pade approximants. This leads to global analytic representations of the considered solutions, valid over the entire range of the control parameter.

DIRECT AND INVERSE PROBLEMS ENCOUNTERED IN VIBRO-IMPACT OSCILLATIONS OF A DISCRETE SYSTEM

We study direct and inverse problems that arise in the vibro-impact oscillations of a discrete system. Specifically, we examine a class of systems with two coordinates undergoing single- or double-sided impacts; however, the presented techniques are sufficiently general to apply to systems with multiple impacts. The analytical methods employed are a nonlinear normal mode (NNM)-type analysis and a boundary value problem (BVP) formulation, and enable the computation of various branches of bifurcating periodic solutions with different impacting characteristics. Additional insight on the dynamics of these systems is obtained by direct integrations of the equations of motion and by numerical Poincaré maps. It is found that the vibro-impact systems considered possess rich nonlinear dynamics, including vibro-impact localized and nonlocalized time-periodic motions, complicated bifurcation structures giving rise to new types of single- and double-sided impacting motions, mode instabilities, and chaotic responses. We also formulate inverse vibro-impact problems, whereby we seek the class of dynamical systems that produce specified orbits in the configuration plane. The solutions of the inverse problems are generally non-unique, since they can be reduced to underdetermined sets of algebraic equations with multiple infinities of unknowns. Numerical applications are provided to demonstrate the techniques and validate the analytical results.

Water waves in a suspended container

Analysis and experiments are carried out on a horizontal rectangular wave tank which swings at the lower end of a pendulum. The walls of the tank generate waves which affect the motion of the pendulum. For small displacements of the tank, linearised shallow water equations are used to model the motion, and there exist time-periodic solutions for the system whose periods are governed by a transcendental relation. Numerical and analytic solutions of this relation show that the fundamental period is greater than both the period of the empty tank (moving like a simple pendulum) and the fundamental period of the standing wave which occurs when the tank is removed from its supports and held fixed. For a rectangular tank the theory compares well with some experimental measurements. Qualitative observations are also made of the effect of breaking waves on the tank motion: for a tank which has a mass small compared with its load the energy dissipated by breaking waves can rapidly reduce the amplitude of swing of the tank. Potential flow theory is used with linearised free-surface boundary conditions to find time periodic motions for a tank with a hyperbolic cross section.