Time-Independent Behavior

FIGU RE 17.6 Rheological classification of fluids showing (a) Newtonian, (b) non-Newtonian, 

---

It is clear that equation (3.19) and (3.20) contain the time independent behavior given by equations (3.4) and (3.11) as special cases by letting p =... 

Viscous Hquids are classified based on their rheological behavior characterized by the relationship of shear stress with shear rate. Eor Newtonian Hquids, the viscosity represented by the ratio of shear stress to shear rate is independent of shear rate, whereas non-Newtonian Hquid viscosity changes with shear rate. Non-Newtonian Hquids are further divided into three categories time-independent, time-dependent, and viscoelastic. A detailed discussion of these rheologically complex Hquids is given elsewhere (see Rheological measurements). 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

All fluids for which the viscosity varies with shear rate are non-Newtonian fluids. For uou-Newtouiau fluids the viscosity, defined as the ratio of shear stress to shear rate, is often called the apparent viscosity to emphasize the distiuc tiou from Newtonian behavior. Purely viscous, time-independent fluids, for which the apparent viscosity may be expressed as a function of shear rate, are called generalized Newtonian fluids. 

Based on a mechanical model in a time-independent flow, de Gennes derivation tries to extrapolate it to a time-dependent chain behavior. His implicit assumptions have been criticised by Bird et al. [55]. More recent calculations extending the de Gennes dumbbell to the bead-spring situation [56] tend, nevertheless, to confirm the existence of a well-characterized CS transition results with up to 100-bead chains show a critical value of the strain rate at scs = 0.5035/iz which is just 7% higher than the value predicted by de Gennes. 

These equations can be solved numerically with a computer, without making any approximations. Naturally all the involved kinetic parameters need to be either known or estimated to give a complete solution capable of describing the transient (time dependent) kinetic behavior of the reaction. However, as with any numerical solution we should anticipate that stability problems may arise and, if we are only interested in steady state situations (i.e. time independent), the complete solution is not the route to pursue. 

The concentration distributions found at different times after the start of current flow are shown in Fig. 11.4. It is a typical feature of the solution obtained that the variable parameters x and t do not appear independently but always as the ratio Like Eq. 11.15), this indicates that the diffusion front advances in proportion to the square root of time. This behavior arises because as the diffusion front advances toward the bulk solution, the concentration gradients become flatter and thus diflusion slows down. 

In reaction rate studies one s goal is a phenomenological description of a system in terms of a limited number of empirical constants. Such descriptions permit one to predict the time-dependent behavior of similar systems. In these studies the usual procedure is to try to isolate the effects of the different variables and to investigate each independently. For example, one encloses the reacting system in a thermostat in order to maintain it at a constant temperature. 

Similarly, improvement in the accuracy of the nuclear dynamics would be fruitful. While in this review we have shown that, in the absence of any approximations beyond the use of a finite basis set, the multiple spawning treatment of the nuclear dynamics can border on numerically exact for model systems with up to 24 degrees of freedom, we certainly do not claim this for the ab initio applications presented here. In principle, we can carry out sequences of calculations with larger and larger nuclear basis sets in order to demonstrate that experimentally observable quantities have converged. In the context of AIMS, the cost of the electronic structure calculations precludes systematic studies of this convergence behavior for molecules with more than a few atoms. A similar situation obtains in time-independent quantum chemistry—the only reliable way to determine the accuracy of a particular calculation is to perform a sequence of... 

Since each input of mass to a perfect plug flow unit is independent of what has been input previously, its condition as it moves along the reactor will be determined solely by its initial condition and its residence time, independently of what comes before or after. Practically, of course, some interaction will occur at the boundary between successive inputs of different compositions or temperatures. This is governed by diffusional behaviors which are beyond the scope of the present work. 

Deviations from the Forster decay (Eq. 9.29) arise from the geometrical restrictions. In the case of spheres, the restricted space results in a crossover from a three-dimensional Forster-type behavior to a time-independent limit. In an infinite cylinder, the cylindrical geometry leads to a crossover from a three-dimensional to a one-dimensional behavior. In both cases, the geometrical restriction induces a slower relaxation of the donor. 

To get the solution of Eq. 21-4 for a time-dependent rate k(t) you could use Eq. 9 of Box 12.1. Since in the above examples, k(t) is constant except for an abrupt change at t = 30 d, you can also use the solution for time-independent coefficients J and k (Box 12.1, Eq. 6) for the first 30 days and then insert the final concentration as the initial value for the second 30 days. The three concentration curves are shown in Fig. 2. Their behavior can be understood qualitatively with the following considerations ... 

Extensive measurements show that self-diffusivities in the relaxed glassy state are time independent and closely exhibit Arrhenius behavior (i.e., In Z), vs. 1/T plots appear as essentially straight lines) [8-11]. The diffusion therefore is thermally activated (in contrast to self-diffusion in the liquid above Tg as described in Section 10.1). 

Chapter HI relates to measurement of flow properties of foods that are primarily fluid in nature, unithi.i surveys the nature of viscosity and its relationship to foods. An overview of the various flow behaviors found in different fluid foods is presented. The concept of non-Newtonian foods is developed, along with methods for measurement of the complete flow curve. The quantitative or fundamental measurement of apparent shear viscosity of fluid foods with rotational viscometers or rheometers is described, unithi.2 describes two protocols for the measurement of non-Newtonian fluids. The first is for time-independent fluids, and the second is for time-dependent fluids. Both protocols use rotational rheometers, unit hi.3 describes a protocol for simple Newtonian fluids, which include aqueous solutions or oils. As rotational rheometers are new and expensive, many evaluations of fluid foods have been made with empirical methods. Such methods yield data that are not fundamental but are useful in comparing variations in consistency or texture of a food product, unit hi.4 describes a popular empirical method, the Bostwick Consistometer, which has been used to measure the consistency of tomato paste. It is a well-known method in the food industry and has also been used to evaluate other fruit pastes and juices as well. 

The behavior of a mixture is determined by a system of ordinary differential equations, while the required state, either equilibrium or stationary, is determined by a time-independent system of algebraic equations. Therefore, at first glance one would not expect any qualitative difference between the equilibrium and stationary states. Ya.B. shows that in the equilibrium case, even for an ideal system, a variational principle exists which guarantees uniqueness. Such a principle cannot be formulated for the case of an open system with influx of matter and/or energy. 

---