Decay oscillatory

Two open reservoirs with cross-sectional areas Ai, A2 and liquid depths hi and h2 respectively are connected by a pipeline of length L and diameter D. The resulting liquid levels form a decaying oscillatory response to final steady state. Ramirez (1976) provides a detailed model derivation for this problem and shows how the problem is solved by analogue computation. The parameters are the same as those used by Ramirez. 

So-called solvation/structural forces, or (in water) hydration forces, arise in the gap between a pair of particles or surfaces when solvent (water) molecules become ordered by the proximity of the surfaces. When such ordering occurs, there is a breakdown in the classical continuum theories of the van der Waals and electrostatic double-layer forces, with the consequence that the monotonic forces they conventionally predict are replaced (or accompanied) by exponentially decaying oscillatory forces with a periodicity roughly equal to the size of the confined species (Min et al, 2008). In practice, these confined species may be of widely variable structural and chemical types — ranging in size from small solvent molecules (like water) up to macromolecules and nanoparticles. 

Microemuisions exist for values of the parameter y [in and see Eqs. (33) and (34)] less than 1 and greater than —1, with more negative values associated with more structure. As can be seen, the correlation function is an exponentially decaying oscillatory function of the separation r. On the other hand, for values of y > 1, the Fourier transform is simply a sum of two monotonically decaying exponential functions, and the liquid is unstructured. It is this difference in bulk behavior that proves crucial to the interfacial wetting behavior. 

Difficulties arises in smooth-walled viscometers because placing a structured liquid next to the wall changes the local microstructure [1]. For a simple suspension of smooth spherical particles, the spatial concentration of particles deep in the bulk of the sample, well away from the wall, is random. However, right at the wall, the particle concentration is zero. How do these two points join The answer is that the concentration rises rapidly as one moves away from the wall shows a decaying oscillatory behaviour, then it smoothes to the bulk concentration, see figure 1. This whole process from zero to average concentration takes about five particles diameters. The result is that the material near the wall is essentially different from the bulk, however worse than this is the effective lubricating layer near the wall where the particle concentration is first zero, and is small even up to half a particle diameter. The phenomenon of lower concentration next to the wall is called wall depletion, but is popularly known as sUp, see chapter 15 section 10 for more details. 

If a confined fluid is thermodynamically open to a bulk reservoir, its exposure to a shear strain generally gives rise to an apparent multiplicity of microstates all compatible with a unique macrostate of the fluid. To illustrate the associated problem, consider the normal stress which can be computed for various substrate separations in grand canonical ensemble Monte Carlo simulations. A typical curve, plotted in Fig. 16, shows the oscillatory decay discussed in Sec. IV A 2. Suppose that instead... 

R. Evans, J. R. Henderson, D. C. Hoyle, A. O. Parry, Z. A. Sabeur. Asymptotic decay of liquid structure oscillatory liquid-vapour density profiles and the Fisher-Widom line. Mol Phys 50 755-775, 1993. 

An oscillatory force which decays exponentially with surface separation ... 

If the roots are, however, complex numbers, with one or two positive real parts, the system response will diverge with time in an oscillatory manner, since the analytical solution is then one involving sine and cosine terms. If both roots, however, have negative real parts, the sine and cosine terms still cause an oscillatory response, but the oscillation will decay with time, back to the original steady-state value, which, therefore remains a stable steady state. 

If the roots are pure imaginary numbers, the form of the response is purely oscillatory, and the magnitude will neither increase nor decay. The response, thus, remains in the neighbourhood of the steady-state solution and forms stable oscillations or limit cycles. 

If we assume that an oscillatory system response can be fitted to a second order underdamped function. With Eq. (3-29), we can calculate that with a decay ratio of 0.25, the damping ratio f is 0.215, and the maximum percent overshoot is 50%, which is not insignificant. (These values came from Revew Problem 4 back in Chapter 5.)... 

The complementary solution consists of oscillating sinusoidal terms multiplied by an exponential. Thus the solution is oscillatory or underdamped for ( < 1. Note that as long as the damping coefficient is positive (C > 0), the exponential term will decay to zero as time goes to infinity. Therefore the amplitude of the oscillations will decrease to zero. This is sketched in Fig. 6.6. 

Note For any real sample of material the resulting oscillatory deformation is one of decaying amplitude. 

Fig. 2.4. Computed concentration.histories for autocatalytic model with rate constants given exactly as in Table 2.1 (a) exponential decay of precursor (b) intermediate concentrations a(t) and 6(r), showing initial pseudo-stationary-state behaviour but subsequent development of an oscillatory period of finite duration, 1752 s < t < 3940 s.

As ku increases, the range of reactant concentrations over which the system is unstable decreases. From eqn (2.21) we can see that oscillatory behaviour will only be possible provided k2 > 8ku, otherwise there are no real solutions for p and p . This means that the uncatalysed step converting A to B must not proceed too quickly compared with the rate at which B can decay to the final product C. 

In the previous section we saw that oscillatory behaviour will be favoured if the uncatalysed reaction converting A to B is slow compared with the decay of B to C. It is interesting, therefore, to consider briefly the system in which the uncatalysed reaction is omitted, i.e. where ku is set equal to zero. This special case actually has a few problems associated with it, but also has some important features. (The problems are related to the anomalous induction periods of autocatalytic reactions discussed in 1.4 and will be discussed later in this chapter.)... 

Though reduced to the barest of essentials, the scheme shows many features observed in real examples of oscillatory reactions a pre-oscillatory period, a period of oscillatory behaviour, and then a final monotonic decay of reactant and intermediate concentrations to their equilibrium values. We can identify from the model such features as the dependence of the length of the pre-oscillatory period on the initial reactant concentration and the rate constants, an estimate for the number of oscillations, and the length of the oscillatory phase. By tuning the parameters we can obtain as many oscillations as we wish. 

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