Angular Frequency Of Oscillation Research Articles

Structure of the streaming flow generated by a sphere in a fluid undergoing rectilinear oscillation

A solid body in a viscous fluid undergoing oscillatory motion naturally produces a steady secondary flow due to convective inertia. This phenomenon is embodied in the streaming flow generated by a sphere in an unbounded fluid executing rectilinear oscillations. We review the considerable literature on this canonical problem and summarise exact and asymptotic formulas in the small-amplitude limit. These analytical formulas are used to explore the characteristic flow structure of this problem and clarify previously unreported features. A single, toroidal-shaped vortex exists in each hemisphere regardless of the oscillation frequency, which can drive a counter-flow away from the sphere. The vortex centre moves monotonically away from the sphere with decreasing oscillation frequency, and engulfs the entire flow domain for $\beta \equiv \omega R^2/\nu < 16.317$ , where $\omega$ is the angular oscillation frequency, $R$ the sphere radius, and $\nu$ the fluid kinematic viscosity. This seemingly abrupt change in flow structure at the critical frequency $\beta _{critical} =16.317$ , and its quantification, appear to have not been reported previously. We perform a direct numerical simulation of the Navier–Stokes equations, to (1) confirm existence of this critical frequency at finite amplitude, and (2) examine its variation with amplitude. This reveals a universal relationship between the critical frequency and oscillation amplitude, clarifying previous reports on the structure of this streaming flow. The critical frequency is shown to be identical for the streaming flow and the cycle-averaged particle paths, establishing that the critical frequency is accessible directly using standard measurements.

Universal generation of devil's staircases near Hopf bifurcations via modulated forcing of nonlinear systems

The discrete circle map is the archetypical example of a driven periodic system, showing a complex resonance structure under a change of the forcing frequency known as the devil's staircase. Adler's equation can be seen as the direct continuous equivalent of the circle map, describing locking effects in periodic systems with continuous forcing. This type of locking produces a single fundamental resonance tongue without higher-order resonances, and a devil's staircase is not observed. We show that, with harmonically modulated forcing, nonlinear oscillations close to a Hopf bifurcation generically reproduce the devil's staircase even in the continuous case. Experimental results on a semiconductor laser driven by a modulated optical signal show excellent agreement with our theoretical predictions. The locking appears as a modulation of the oscillation amplitude as well as the angular oscillation frequency. Our results show that by proper implementation of an external drive, additional regions of stable frequency locking can be introduced in systems which originally show only a single Adler-type resonance tongue. The induced resonances can be precisely controlled via the modulation parameters.

Mechanical response of industrial benchmark lipsticks under large-scale deformations

This work documents the first account of advanced mechanical properties of six commercial lipsticks, some of which serve as market leads. We systematically studied their nonlinear viscoelastic properties under large amplitude oscillatory shear deformations. At large strains, all lipsticks showed intercycle strain softening, the extent of which initially depended on the prototype in the nonlinear regime. This behavior, markedly, was absent after the crossover of the dynamic moduli. Parameters obtained from the strain amplitude sweeps, i.e., the intrinsic elastic modulus and the stress maximum, demonstrated distinct prototype dependence. The Lissajous plots and the dimensionless nonlinear indices were determined using the MITlaos software. They showed intracycle elastic strain stiffening and viscous shear thinning. The angular oscillation frequency directly influenced the linear viscoelastic measures of all the benchmark lipsticks, and the nonlinear properties of only a few benchmark ones. The current study generates standard nonlinear rheology data that can be associated with the lipstick sensory attributes and typical tribological parameters. This may serve as an effective way to examine the transition from the initial spreading to the post-application sensation.

CFD study on particle grouping under an oscillatory flow in a wavy duct

Particle grouping that arises from temporal and spatial variation of air flow is a promising method to facilitate agglomeration and removal of aerosol. We conduct a computational investigation on this phenomenon under an oscillatory flow in a wavy duct using a discrete phase method. This setup is simple and practically feasible. The grouping behavior is affected by the flow Reynolds number, duct wavelength, angular frequency of oscillation, ratio of minimum to maximum duct diameter and velocity ratio. Three different regimes are identified: no grouping, stable/strong grouping and unstable/weak grouping. It is found that a moderate delay of flow response to the imposed pulsations is critical for grouping. Among the investigated values for the periodicity parameters, the strongest grouping takes place when the duct wavelength is four times the maximum duct diameter, and the angular frequency of oscillation is five times the reciprocal of characteristic flow time. Moreover, an increase in the velocity ratio and a decrease in the diameter ratio further enhance the grouping behavior under otherwise identical conditions. Due to the flow complexity in this realistic setup, the criterion for particle grouping is different from that based on an ideal 1D standing wave flow field in the literature.

The Determination of the Parameters of a Vibration Machinef the Internal Compaction of Concrete Mixtures

The determination of the rational parameters of a vibration machine for the internal compaction of concrete mixtures. We researched the “vibration machine – concrete medium” system wherein the concrete medium is represented by a system with distributed parameters that takes into account the resilient, viscous, inertial and energy properties of the compacted concrete medium. We researched the propagation of the viscous-resilient-plastic deformation waves in the contracted concrete mixture represented in the form of a half-space. We determined the amplitudes of the oscillations and stresses occurring in the compacted mixture depending on its physical and mechanical characteristics. We determined the oscillation amplitude of the vibration machine, identified its rational parameters. We determined the regularity of the motion of the compacted mixture and the vibration machine depending on the physical and mathematical characteristics of the compacted mixture, the thickness and width of the compacted layer, the angular oscillation frequency and the amplitude of the exciting force. The presented dependences enable the substantiation of the rational parameters of the vibration machine and the modes of vibratory action.

Predator-prey system with seasonally varying additional food to predators

We have proposed a three species food chain model with additional food to predators. We have studied the dynamics of this predator-prey model with seasonally varying additional food to predators. Bifurcation analysis of the proposed model is done with respect to quality and quantity of additional food, amplitude of oscillation and angular frequency of oscillations. The bifurcation analysis of our model revealed the vital role of seasonality parameters in the controllability of the predator-prey system. Our analysis predicts that seasonality parameters are very important to determine the quality and quantity of additional food for controlling the dynamics of a food chain model.

Transport by Oscillatory Flow in Soils with Rate‐Limited Mass Transfer: 1. Theory

Oscillatory flow of fluid in a porous medium can generate a one‐way transport of heat or chemicals if there is a gradient of temperature or chemical concentration and a rate‐limited heat or mass transfer between the moving fluid and an immobile phase. For chemical transport in soils, the immobile phase can occur in stagnant porosity, by sorption, or by dissolution of a vapor in the pore water. As a function of oscillation frequency, the transport rate has a broad peak near the value ωτc = 1, where ω is the angular frequency of oscillation and τc is the characteristic equilibration time of the mobile phase. The transport rate is proportional to the gradient and to the square of the amplitude of periodic fluid displacement. A unique diffusivity derived from the analysis enables prediction of the long‐term transport by a diffusion calculation without fluid flow, thereby providing a tool for estimating the removal of contaminant vapors by passive soil vapor extraction (PSVE). We compared predictions of the analytic theory with numerical simulations of PSVE. The mobile–immobile model is also applicable to transport in other cases of oscillatory flow in porous media, including cyclic motion of water or gas due to tidal aquifers or earth tides.

Harmonic Vibrations in Thermoelasticity of Microstretch Materials

Consider a cylinder made of a microstretch thermoelastic material for which one plane end is subjected to plane boundary data varying harmonically in time. On the lateral surface and other base, we have zero body force and heat supply. By using a Toupin type measure associated with the corresponding steady-state vibration, and by assuming that the angular frequency of oscillations is lower than a certain critical frequency, we show that the amplitude of the vibrations decays exponentially with the distance to the base. This decay estimate is similar to that of the Saint-Venant type.

Coupled oscillations of double-walled carbon nanotubes

A study of the coupled axial and angular oscillations of double-walled carbon nanotubes (DWNTs) was performed using molecular dynamics simulations. In order to determine the oscillation frequencies inner and outer shells have been assumed to behave as rigid bodies, a 6–12 Lennard–Jones potential was used to model the van der Waals forces between them, and friction was neglected. Armchair (5, 5)/(10, 10) configurations, with tube lengths in the range of 25–1000 Å were investigated. The axial oscillation frequency was found to be a decreasing function of the DWNT length L (∼78 GHz for L=25 Å and ∼2 GHz for L=1000 Å). The angular oscillation frequency was found to be nearly constant at ∼58 GHz, independent of the DWNT length. In addition, sustained high-frequency angular motion can be maintained for sufficiently long DWNT (on the order of a few nanometers). For shorter DWNTs, the angular motion can be altered by interference with the axial motion, and becomes irregular.

Aerodynamic focusing in a channel with oscillating walls

We investigate analytically and numerically aerodynamic focusing of aerosol particles in the incompressible Poiseuille flow in a two-dimensional channel where the channel walls perform small oscillations perpendicular to the channel axis. The fluid–particle interaction is described by a linear drag force. Focusing efficiency is investigated for variety of flow and particle parameters, expressed in terms of dimensionless groups, namely parameter ε , characterizing the amplitude of wall oscillations; flow velocity parameter Π U = U / ω h where U is the maximal velocity of the imposed flow, h is the channel half-height, ω is the angular frequency of oscillations; and frequency parameter ω τ , where τ is Stokes relaxation time. It is shown that wall channel oscillations with frequency of about 1 kHz can focus micron-size particles on axial distances comparable to several channel heights.

Quantum Turbulence Generated and Detected by a Vibrating Quartz Fork

Flow due to a commercially available vibrating quartz fork is studied in gaseous helium, He I, He II and 3He–B, over a wide range of temperatures and pressures. On increasing the driving force, the flow changes in character from laminar (characterized by a linear velocity versus drive dependence) to turbulent (characterized by a square root velocity versus drive dependence). In classical fluids, we characterize this transition by a critical Reynolds number, Re c =U cr δ/ν, where U cr is the critical velocity, ν stands for the kinematic viscosity, $\delta=\sqrt{2\nu/\omega}$ is the viscous penetration depth and ω is the angular frequency of oscillations. U cr of order 10 cm/s observed in He II and 1 mm/s in 3He–B agree with those found with other vibrating objects such as spheres, wires and grids, as well as with available numerical simulations of vortex motion in an applied ac flow.

Transition from laminar to turbulent drag in flow due to a vibrating quartz fork

Flow due to a commercially available vibrating quartz fork is studied in gaseous helium, He I and He II, over a wide range of temperatures and pressures. On increasing the driving force applied to the fork, the drag changes in character from laminar (characterized by a linear drive vs velocity dependence) to turbulent (characterized by a quadratic drive vs velocity dependence). We characterize this transition by a critical Reynolds number Recrdelta=Ucrdelta/nu, where Ucr is the critical velocity, nu stands for the kinematic viscosity, delta=sqrt[2nu/omega] is the viscous penetration depth, and omega is the angular frequency of oscillations. We have experimentally verified that the corresponding scaling Ucr proportional, sqrt[nuomega] holds in a classical viscous fluid over two decades of nu.

Steady time-harmonic oscillations in a linear thermoelastic plate model

We examine the bending of a Mindlin-type thermoelastic plate when the source terms are time-harmonic with angular frequency ω \omega , and sufficient time has elapsed for the system to have reached a steady-state. We show that in an infinite plate the solution can be represented as the sum of five waves all but one of which exhibit damping. By formulating appropriate radiation conditions we prove uniqueness results for exterior boundary value problems subject to certain regularity assumptions and a condition on the angular frequency of oscillation.

Probability density functions for the translational itinerant oscillator

The translational itinerant oscillator model for molecular motion in fluids is developed in terms of two phenomenological parameters. These are : (a) the retarding frictional force acting on a rigid cage which undergoes translational Brownian motion, and (b) the angular frequency of oscillation of a particle hamonically bound within the cage. The equations of motion contain fewer parameters than those of Damle et al. [6] and may be solved using Laplace transforms. The underlying probability density functions in phase and configuration space may then be calculated.

Mathematical analysis of isometric cardiac muscle contraction: The effects of stimulus interval, temperature, and calcium

The time course of oscillatory and nonoscillatory isometric rested-state contraction in rat ventricle has been described by solutions of a nonhomogeneous second-order linear differential equation. The experimental conditions constitute a definition of the initial conditions of the differential equation and thus determine the form of the transient solution. The response of the contractile system can be further characterized by a damping ratio defined in terms of the constants of the homogeneous part of the differential equation. Test contractions early in the recovery of contractility were more heavily damped than the rested-state contraction. During the first phase of recovery, their damping decreased to that of the rested-state contraction. Under conditions of oscillation both low temperature and high calcium decreased the damping, the latter by a specific effect on the angular frequency of oscillation.

Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior

A theory of electron oscillations of an unbounded plasma of uniform ion density is developed, taking into account the effects of random thermal motions, but neglecting collisions.The first problem considered is that of finding the frequencies at which a plasma can undergo organized steady-state oscillations of small enough amplitude so that a linear approximation applies. It is found that long wave-length oscillations of plasmas with a Maxwell distribution of electron velocities are characterized by the steady-state dispersion relation ${\ensuremath{\omega}}^{2}=\ensuremath{\omega}_{P}^{}{}_{}{}^{2}+(\frac{3\ensuremath{\kappa}T}{m}){(\frac{2\ensuremath{\pi}}{\ensuremath{\lambda}})}^{2}$. Here ${\ensuremath{\omega}}_{P}$ is the plasma frequency, $T$ the absolute temperature of the electron gas, $\ensuremath{\lambda}$ the wave-length, and $\ensuremath{\omega}$ the angular frequency of oscillation. It is also shown that organized oscillations of wave-lengths smaller than the Debye length for the electron gas are not possible.The theory is then extended to describe the processes by which oscillations are set up. It is found that, for a given wave-length, a plasma can oscillate with arbitrary frequency, but that those frequencies not given by the steady-state dispersion relation describe motions in which, after some time, there is no contribution to macroscopic averages. These additional frequencies lead asymptotically only to microscopic fluctuations of the charge density about the organized oscillation of the plasma. In this way, one can describe the manner in which the system develops organized behavior.The treatment is then applied to large steady-state oscillations for which the equations are non-linear. One obtains solutions in which particles close to the wave velocity are trapped in the trough of the potential, oscillating back and forth about a mean velocity equal to that of the wave. One can also obtain non-linear traveling pulse solutions in which a group of particles, moving as a pulse, creates a reaction on the surrounding charge, which traps the particles and holds them together.