Time-dependent behaviour

In a time-dependent fluid, the shear rate depends upon the time for which it has been subjected to a given shear stress. Conversely, if the shear rate is kept constant, the shear stress will change with time. However, with all time-dependent fluids an equilibrium conditipn is reached if the imposed condition (e.g. shear rate or shear stress) is maintained constant. Some fluids respond so quickly to changes that the effect of time dependence can be neglected. Others may have a much longer constant, and in changing flow situations will never be in the equilibrium state. 

In general, for shear-thinning pseudoplastic fluids the apparent viscosity will gradually decrease with time if there is a step increase in its rate of shear. This phenomenon is known as thixotropy. Similarly, with a shear-thickening fluid the apparent viscosity increases under these circumstances and the fluid exhibits rheopexy or negative-thixotropy. 

A true fluid flows when it is subjected to a shear field and motion ceases as soon as the stress is removed. In contrast, an ideal solid which has been subjected to a stress recovers its original state as soon as the stress is removed. The two extremes of behaviour are therefore represented by  

Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid tmd a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study Two common and easily observable examples of viscoelastic behaviour in a liquid are  

Viscoelasflc fluids are thus capable of exerting normal stresses. Because most materials, under appropriate circumstances, show simultaneously solid-Uke md fluid-like behaviburs in vMying proportions, the notion of an icteal elastic solid or of a purely viscous fluid represents the commonly encountered limiting condition. For instance, the viscosity of ice and the elasticity of water may both pass unnoticed The response of a material may also depend upon the type of deformation to which it is subjected. A material may behave like a highly elastic solid in one flow situation, and like a viscous fluid in another. 

We now relax our implicit approximation in which the consumption of the reactant has been ignored. The full time-dependent behaviour of the dimensionless equations will be considered. The situation is not greatly affected by use of either the exponential approximation or the full Arrhenius form, so we return to the former for simplicity. We will take the example data from Table 4.1 k = 0.05, n0 = 0.5, and e = 10 2. As we are employing the exponential approximation the value of y is not important. 

can we expect any oscillatory behaviour Instability is possible only if k e 2. This requirement is satisfied here. From the data in Table 4.4, the Hopf bifurcation points for this system occur for n = 0.207 and n = 0 058. For our example, the initial value /r0 = 0.5 exceeds the upper bifurcation point, so the system at first has a stable pseudo-stationary state to approach, with dss x 10 and ass x 4.54 x 10 4. From Fig. 4.3 we may also estimate that the approach to this state will be monotonic since the initial conditions lie outside the region of damped oscillations. 

The pre-oscillatory period, during which the temperature excess decreases and the concentration of A increases, will last at least until i has fallen from H0 to the value n, i.e. until the time tf given by 

Following this the pseudo-stationary state becomes unstable and the concentration and temperature histories are expected to move away into oscillatory 

As seen in the previous chapter, the growth of observable excursions is not immediate and may take a considerable time. If oscillations do develop, we expect them to last until the pseudo-stationary state regains stability at time t  

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To determine time dependent behaviours of the specimen up to 25 measurements in series with different time delays are possible. To prevent mistakes in application many help comments appear when inputs are necessary or differences between the calibration and the measurement are detected. All calibration conditions, a description for the specimen and results can be printed or saved by the hard disk. To reduce the input expenditure, the last configuration is made to current values when the program is stopped ore leave. 

It is important to recognize that the time-dependent behaviour of tire correlation fimction during the molecular transient time seen in figure A3.8.2 has an important origin [7, 8]. This behaviour is due to trajectories that recross the transition state and, hence, it can be proven [7] that the classical TST approximation to the rate constant is obtained from A3.8.2 in the t —> 0 limit ... 

In this book no prior knowledge of plastics is assumed. Chapter 1 provides a brief introduction to the structure of plastics and it provides an insight to the way in which their unique structure affects their performance. There is a resume of the main types of plastics which are available. Chapter 2 deals with the mechanical properties of unreinforced and reinforced plastics under the general heading of deformation. The time dependent behaviour of the materials is introduced and simple design procedures are illustrated. Chapter 3 continues the discussion on properties but concentrates on fracture as caused by creep, fatigue and impact. The concepts of fracture mechanics are also introduced for reinforced and unreinforced plastics. 

But, computational difficulties can arise due to the iterative methods used to solve recycle problems and obtain convergence. A major limitation of modular-sequential simulators is the inability to simulate the dynamic, time dependent, behaviour of a process. 

The second category, time-dependent behaviour, is common but difficult to deal with. The best known type is the thixotropic fluid, the characteristic of which is that when sheared at a constant rate (or at a constant shear stress) the apparent viscosity decreases with the duration of shearing. Figure 1.21 shows the type of flow curve that is found. The apparent viscosity continues to fall during shearing so that if measurements are made for a series of increasing shear rates and then the series is reversed, a hysteresis loop is observed. On repeating the measurements, similar behaviour is seen but at lower values of shear stress because the apparent viscosity continues to fall. 

The concentrations a and P vary in time as they approach or move away from any particular stationary state. Often it is convenient to visualize the time-dependent behaviour another way, by plotting the variation of one concentration against that of the other, in what is known as the a-/ phase plane. 

Given a particular chemical reaction, which obeys a known set of reaction rate equations, we now seek to find out how the concentrations of the various species vary in space along the reactor, i.e. we wish to find the concentrations c(x). To show how this problem is equivalent to that of determining the time-dependent behaviour in a well-stirred closed vessel, we can take a general example for which a reactant A is converted to a product B. Let the rate law appropriate to a well-stirred closed vessel be... 

So far almost all aspects of the stationary-state and even the time-dependent behaviour of this reaction-diffusion system differ only qualitatively from that found in the corresponding CSTR. In this section, however, we can consider a variation for which there can be no parallel in the well-stirred system—that of a reaction-diffusion cell set up with asymmetric boundary conditions. Thus we might consider our infinite slab with separate reservoirs on each side, with different concentrations of the autocatalyst in each reservoir. (For simplicity we will take the reactant concentration to be equal on each side.) Thus if we identify the reservoir concentration for p < — 1 as / L and on the other side (p + 1) as / R, the simple boundary conditions in eqn (9.11) are replaced by... 

It thus seems fair to conclude that explanations for C2 have to be sought in the equilibrium rather than the time dependent behaviour, although the actual magnitude of the C2 contribution may often be determined by the perhaps time dependent pre-existing order in the polymer, which is trapped by the crosslinking process. 

The magnitude of the correction to the simple (zero-order) autocatalator provided by the first-order equations, (14) and (15) for the time dependent behaviour and (17) for the stationary-states, can be seen quantitatively for a typical system from the first column of numerical results in Table 1. The corrections in ass and /3SS appear always to be an order of magnitude less than e and k, even when these parameters are of order unity. The second column, which gives the solutions of the full set of three equations (9a)-(9c), shows that the first-order equations actually overestimate the deviation from the e = k = 0 autocatalator. Returning to column I, we see that the stationary-state concentration of X is adequately given by ss = kol, even for e = k = 0.1, and is very accurately given by ss = Kass/3ss(l - e ) even for e = k = 1. Notice that for row (d) there are three stationary-state solutions at Tres = 225. 

Rheomalaxic Time-dependent behaviour in which shear rate changes cause irreversible changes in viscosity. Emulsions that invert when sheared irreversibly. An emulsion which, when sheared, inverts to a higher (or lower) viscosity emulsion, and does not re-invert when the shear is removed. 

Rheomalaxis is a special case of time-dependent behaviour in which the shear rate changes cause irreversible changes in viscosity. The change can be negative, as when structural linkages are broken, or positive, as when structural elements become entangled (like work-hardening). Example An emulsion which when... 

In the second part, we discuss possible applications of attosecond laser pulses to future studies of time-resolved electron dynamics in strongly driven systems. We discuss our current understanding of the time-dependent behaviour of non-perturbatively driven electrons in atoms, molecules and clusters. In Sect. 3.4 we discuss differences that arise when the generation of attosecond pulses is performed in different atomic media. This is followed in Sect. 3.5 by a description of the role of electron dynamics in dynamical alignment and enhanced ionization of molecules. Finally, in Sect. 3.6 the role of electron dynamics in laser heating of large clusters is discussed. 

The time dependent behaviour is therefore described by the differential equation... 

Transient photoelectrochemical behaviour of colloidal CdS The experiments described in this section are performed by recording light-on transient photocurrents from aqueous dispersions of 2-12 nm radii CdS particles (prepared as above) at a stationary optical rotating disc electrode. However, to be able to interpret the results from these experiments, it was first necessary to model the time-dependent behaviour of the mass transport limited photocurrent at the ORDE. 

Relaxation processes are universal. They are found in all branches of physics mechanical relaxation (stress and strain relaxation, creep), ultrasonic relaxation, dielectric relaxation, luminescence depolarisation, electronic relaxation (fluorescence), etc. Also the chemical reaction might be classified under the relaxation phenomena. It will be readily understood that especially in polymer science this time-dependent behaviour is of particular importance. 

We also note that the process of decay in catalytic cracking has been amply demonstrated(l)(2)(3) to be a function of the time of exposure of the catalyst to the reactants i.e. of the time on stream. Such time-dependent behaviour indicates that the kinetics of the process of catalyst decay are the same as those to be expected in a batch reaction. On reflection, it is obvious that the situation of the catalyst charge in a steady-state reactor is in fact that of a batch of reactant (the catalyst) undergoing a chemical process (catalyst decay) as a function of the time on stream i.e. the time it spends at reaction conditions - at reaction temperature and in the presence of an atmosphere of feed, products and potential poisons. 

As an alternative to the above method for eliminating the NOE an instrumental technique is available. This depends upon the realization (247) that the time-dependent behaviour of the NOE and of spin decoupling are different. Thus the NOE takes a time comparable for Tj to build up or to decay after application or removal of a rf field, whereas spin decoupling effects appear or disappear almost instantaneously. Consequently if the proton decoupler is gated off immediately prior to... 

Since the contribution om s makes the relaxation much faster than that from for the large time scale the time-dependent behaviour of IVi(t) is mostly governed by two poles of s = 0 and 5 = So which are quite close together but not quite the same leading to... 

Figure 2 shows a model/measurement comparison from a butenedial photolysis experiment in the absence of NOx. The loss of butenedial is well predicted by MCMvS.l. However, the HO2 concentration is over-estimated by MCMv3.1 by almost an order of magnitude during the early part of the experiment. The time-dependent behaviour is also not well reproduced by the simulation as in the experiment an initial fast increase in concentration is followed by a slower linear increase until the chamber closes, while the simulation shows a fast rise followed by a fall in the HO2 concentration even while the photolysis continues. The photolysis mechanism for butenedial in the absence of NOx as implemented in MCMvS.l is shown schematically in Figure 4. This indicates fliat two HO2 radicals should be formed for each molecule of maleic anhydride and glyoxal produced, and while both these product concentrations are over-estimated this is not sufficient to account for the large over-prediction ofH02.

Several successive sets of creep tests were carried out between 1996 and 2002. The tests results show that the argillites has a time-dependent behaviour even in undrained conditions. That observation confirms that the time-dependent behaviour of the argilites is thus dependent on both... 

With regard to the influence of temperature, it is shown that the time-dependent behaviour may be enhanced by an increase in temperature (Figure 7a). At low deviatoric stress, temperature effects are discreet for an rise from 20°C to 50 °C in temperature. However, from 50°C to 120°C, an acceleration of the viscoplastic strain is observed obviously, whatever the level of deviatoric stress might be. 

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