Oscillations periodic

The modification of the surface force apparatus (see Fig. VI-4) to measure viscosities between crossed mica cylinders has alleviated concerns about surface roughness. In dynamic mode, a slow, small-amplitude periodic oscillation was imposed on one of the cylinders such that the separation x varied by approximately 10% or less. In the limit of low shear rates, a simple equation defines the viscosity as a function of separation... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Johnson B R and Scott S K 1997 Complex and non-periodic oscillations in the FI2 + O2 reaction J. Chem. Soc. Faraday Trans. 93 2997-3004... 

Next suppose we consider the effect of a periodically oscillating stress on a Voigt element of modulus G and viscosity 77. Remember from the last section that for a Voigt element the appUed stress equals the sum of the elastic and viscous responses of the model. Therefore, for a stress which varies periodically, Eq. (3.64) becomes... 

Periodic oscillations have been observed as a morphological instability in several systems grown under various conditions [148]. The correspondence of the observed structures with results of theoretical modelling [139,149] is striking. 

Unit of frequency a periodic oscillation has a frequency of n hertz if in one second it goes through n cycles. 

There are several other comparable rheological experimental methods involving linear viscoelastic behavior. Among them are creep tests (constant stress), dynamic mechanical fatigue tests (forced periodic oscillation), and torsion pendulum tests (free oscillation). Viscoelastic data obtained from any of these techniques must be consistent data from the others. 

Interesting results have also been obtained with light-induced oscillations of silicon in contact with ammonium fluoride solutions. The quantum efficiency was found to oscillate complementarity with the PMC signal. The calculated surface recombination rate also oscillated comple-mentarily with the charge transfer rate.27,28 The explanation was a periodically oscillating silicon oxide surface layer. Because of a periodically changing space charge layer, the situation turned out to be nevertheless relatively complicated. 

Bhawe (14) has simulated the periodic operation of a photo-chemically induced free-radical polymerization which has both monomer and solvent transfer steps and a recombination termination reaction. An increase of 50% in the value of Dp was observed over and above the expected value of 2.0. An interesting feature of this work is that when very short period oscillations were employed, virtually time-invariant products were predicted. 

Surface force apparatus has been applied successfully over the past years for measuring normal surface forces as a function of surface gap or film thickness. The results reveal, for example, that the normal forces acting on confined liquid composed of linear-chain molecules exhibit a periodic oscillation between the attractive and repulsive interactions as one surface continuously approaches to another, which is schematically shown in Fig. 19. The period of the oscillation corresponds precisely to the thickness of a molecular chain, and the oscillation amplitude increases exponentially as the film thickness decreases. This oscillatory solvation force originates from the formation of the layering structure in thin liquid films and the change of the ordered structure with the film thickness. The result provides a convincing example that the SFA can be an effective experimental tool to detect fundamental interactions between the surfaces when the gap decreases to nanometre scale. 

Periodic oscillations of the equivalence ratio when the fuel is injected as a liquid [43-45]... 

The above set of conditions are complete in the sense that a transition from any initial state to any final state can be controlled perfectly. This idea can also be applied to multilevel problems. In the practical applications, the quadratic chirping, that is, one-period oscillation, is quite useful, as demonstrated by numerical applications given below. 

When metals have Raman active phonons, optical pump-probe techniques can be applied to study their coherent dynamics. Hase and coworkers observed a periodic oscillation in the reflectivity of Zn and Cd due to the coherent E2g phonons (Fig. 2.17) [56]. The amplitude of the coherent phonons of Zn decreased with raising temperature, in accordance with the photo-induced quasi-particle density n.p, which is proportional to the difference in the electronic temperature before and after the photoexcitation (Fig. 2.17). The result indicated the resonant nature of the ISRS generation of coherent phonons. Under intense (mJ/cm2) photoexcitation, the coherent Eg phonons of Zn exhibited a transient frequency shift similar to that of Bi (Fig. 2.9), which can be understood as the Fano interference [57], A transient frequency shift was aslo observed for the coherent transverse optical (TO) phonon in polycrystalline Zr film, in spite of much weaker photoexcitation [58],... 

Figure 3. Constraints from orbital frequencies. The 1330 Hz curve is for the highest kilohertz quasi-periodic oscillation frequency yet measured (for 4U 0614+091, by van Straaten et al. 2000). The 1500 Hz curve shows a hypothetical constraint for a higher-frequency source. Other lines are as in Figure 1. All curves are drawn for nonrotating stars the constraint wedges would be enlarged slightly for a rotating star (see Miller, Lamb, Psaltis 1998).

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

Fig. 6. Poincare maps. The section was chosen I (z) = zs = 0 and the crosses indicate no periodic oscillation. Once again, the smallest attractor corresponds to experiment E2.b. zi,Z2,zs are also dimensionless.

Sound waves provide a periodic oscillation of pressure and temperature. In water, the pressure perturbation is most important in non-aqueous solution, the temperature effect is paramount. If cu (= 2 nf, where/is the sound frequency in cps) is very much larger than t (t, relaxation time of the chemical system), then the chemical system will have no opportunity to respond to the very high frequency of the sound waves, and will remain sensibly unaffected. 

Only deterministic models for cellular rhythms have been discussed so far. Do such models remain valid when the numbers of molecules involved are small, as may occur in cellular conditions Barkai and Leibler [127] stressed that in the presence of small amounts of mRNA or protein molecules, the effect of molecular noise on circadian rhythms may become significant and may compromise the emergence of coherent periodic oscillations. The way to assess the influence of molecular noise on circadian rhythms is to resort to stochastic simulations [127-129]. Stochastic simulations of the models schematized in Fig. 3A,B show that the dynamic behavior predicted by the corresponding deterministic equations remains valid as long as the maximum numbers of mRNA and protein molecules involved in the circadian clock mechanism are of the order of a few tens and hundreds, respectively [128]. In the presence of molecular noise, the trajectory in the phase space transforms into a cloud of points surrounding the deterministic limit cycle. 

The MgS0i 7H20 crystal in an aqueous solution grew along a periodic oscillation curve of the shape factors. The variations in the curve suggest that a crystal exhibits an analogous shape almost periodically within a certain range of shape factors. 

That Is9 a growing crystal takes on various shapes while displaying a periodic oscillation curve with a constant period, a constant amplitude, and a fixed axis at the steady state of the shape factor. The amplitude is altered by the temperature (we are currently researching the amplitudes of various other crystals, of which one example follows). 

At Re = 130, a weak long-period oscillation appears in the tip of the wake (T2). Its amplitude increases with Re, but the flow behind the attached wake remains laminar to Re above 200. The amplitude of oscillation at the tip reaches 10% of the sphere diameter at Re = 270 (GIO). At about this Re, large vortices, associated with pulsations of the fluid circulating in the wake, periodically form and move downstream (S6). Vortex shedding appears to result from flow instability, originating in the free surface layer and moving downstream to affect the position of the wake tip (Rll, R12, S6). 

Near these limits there was always a "spin in the deton, which appeared on the pressure record as a periodic oscillation of the pressure for some distance behind the wave front] 14) G.B. Kistiakowsky, IEC... 

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