Reuel Shinnar

Breakup of a laminar capillary jet of a viscoelastic fluid

The stability of non-Newtonian jets was investigated. A linearized stability analysis shows that a liquid column of a viscoelastic fluid exhibits more rapid growth of axisymmetric wave disturbances than a Newtonian fluid of the same zero shear viscosity. This result is independent of the form of constitutive equation chosen. Experiments in weakly elastic fluids confirm this expectation, whereas data on fluids with more pronounced elastic properties indicate that non-linear phenomena are dominating. The disturbances appear as a series of droplets connected by random lengths of threads, which thin with distance and eventually lead to jet breakup. Even in dilute viscoelastic solutions, jet breakup does not occur by the growth of clearly defined waves.

How long do new-onset seizures in children last?

Although there are data on the duration of seizures in patients with refractory epilepsy, little is known about the duration of seizures in nonrefractory epilepsy populations. In a prospective study, seizure duration was determined in 407 children with a first unprovoked seizure using a structured interview and review of medical and ambulance records. Analysis focused on the distribution of seizure duration and on the conditional probability that a seizure would stop once it had already lasted for a specified time. Seizures lasted ≥5 minutes in 50% of cases, ≥10 minutes in 29%, ≥20 minutes in 16%, and ≥30 minutes in 12%. Seizure duration data were best fit as the sum of two exponential distributions, one with a mean of 3.6 minutes accounting for 76% of cases and the other with a mean of 31 minutes accounting for 24% of cases. The longer a seizure lasted, the less likely it was to stop within the next few minutes. In the 182 children with 2 or more seizures, the durations of the first and second seizures were highly correlated (r = 0.395, p < 0.0001). We conclude that the distribution of seizure duration in children with a first unprovoked seizure differs markedly from that observed in patients with refractory epilepsy. A subgroup of patients are predisposed to prolonged seizures. The data suggest that, once a seizure lasts for more than 5–10 minutes, it is unlikely to stop spontaneously within the next few minutes, and intervention is therefore indicated. These findings also support the continued use of the current definition of status epilepticus as a seizure lasting for 30 minutes or longer for epidemiologic studies.

Quantitative models in crime control

In this study we propose several mathematical models for describing the behavior of crime-related systems. In the first part is presented a simple model for describing and measuring crime systems with recidivism. This model may be used in estimating the quantitative effects of alternative policies on crime rates. An important factor not recognized by the model is deterrence. One derivative of the mathematical analyses is a proposed method for measuring the relative severity of sentencing in a given system. The second part of the paper deals with the criminal population as a whole, trying to bring into account the non-stationary behavior of the system. This study should be regarded as a preliminary attempt to develop elementary mathematical models fitting crime systems. Some of the assumptions do not have substantial direct support by data.

The stability of steady shear flows of some viscoelastic fluids

Abstract The stability of steady shear flows of some viscoelastic fluids relative to small perturbations in the initial data is examined. It is shown that all values of the steady shear rate where the flow curve exhibits a zero or negative slope the flow is unstable. This conclusion is corroborated by existing experimental data and bears upon the phenomenon of melt fracture.

Instability of Jets of Non‐Newtonian Fluids

The instability of jets of some non‐Newtonian fluids under the influence of externally controlled disturbances was studied. Dispersion curves were generated over a range of frequencies. Jets of a 0.1% solution of Carbopol in water, a viscoinelastic fluid, showed a breakup pattern similar to that of jets of water. The dispersion curve matched a viscosity intermediate to the zero shear viscosity of the solution and the viscosity that corresponds to the average shear rate in the tube. Subject to external disturbances, jets of Separan solutions showed a remarkably ordered and regular pattern. On jets of 0.1% Separan, sinusoidal wave formation was absent and a ligament‐droplet configuration formed from the outset. On the less elastic 0.05% Separan jets, there is an initial region in which sinusoidal waves grow exponentially; the growth is later arrested and a ligament‐droplet configuration results. In the initial region the growth rates are about equal to the growth rates that a jet of a Newtonian fluid of the...

The Extent of Correlations in a Stochastic Coalescence Process

Abstract A stochastic model of coalescence is set up and solved for the probabilities of all possible histories of particle growth. The full stochastic model is compared with the so-called kinetic model to which it reduces in the absence of correlations. A primary objective is the assessment of the extent of correlations in poorly mixed systems or in systems of small populations. The study shows that insofar as the total number of particles is concerned, regardless of their size distribution, the results from the kinetic equations match the true stochastic averages even for very small initial populations. But, when size distributors are considered, then, in systems of small population or in large systems that are poorly mixed, the results of the kinetic equations may differ substantially from the stochastic means in the long-term tail; apart from the tail, the distributions from the full stochastic and kinetic models match quite well.

CHEMICAL REACTOR MODELLING FOR PURPOSES OF CONTROLLER DESIGN

Abstract The relation between modelling and design of control systems for chemical reactors is discussed, using several practical examples. Criteria for controlability and satisfactory response are proposed. The dependence of the controller structure on the properties of the reactor model is investigated, and it is shown that the most important decisions in controller design occur during process design. The information required for controller design is different from the information required for scaleup, and depends on the design approach. Examples chosen are control of a fluid catalytic cracker, a hydrocracker, and a crystallizer. Control and reactor modelling are not only often separate activities by themselves, but they are done in separate groups and current practice is to call the control engineering in after the design is finished or the plant either operates or is scheduled to start operation. If we want to make significant advances, we have to realize that some of the most important decisions that...

Interpretation and evaluation of multiple tracer experiments

Tracer experiments for studying transport processes in multiphase systems are discussed. Such experiments are useful in in vivo physiological studies of transport processes across membranes, in engineering studies of porous packed columns, and in chromatography. In all these processes, the main forward flow is in one continuous phase from which fluid diffuses into a stagnant outer phase (and back). A nondiffusible tracer is used to characterize the flow in the main phase (inside the capillaries of an organ or between the particles of a column), and diffusible tracers are used to study the transport through the interface (membrane or film) and in the outer phase (extravascular space or inside a porous particle). From the concentration history of the different tracers at the outlet, we can reconstruct sojourn time distributions in the different phases. The statistical properties and the relations between the distributions and their moments are discussed. Methods are given for estimating interfacial transport coefficients (or permeabilities) as well as the diffusion coefficients in the outer phase from the moments of the measured distributions. It is also shown that these relations simplify considerably the mathematical modelling of such systems.

The effect of imperfect mixing on stirred combustion reactors

Mixed gas reactors, even when special pains have been taken to ensure good mixing, often depart quite noticeably from the ideal of instantaneous mixing on the micro-scale. The present paper offers a quantitative measure of the extent of this departure, and shows how it is related to the reactor performance. The intensity of the mixing is characterized by a single parameter that measures the time scale of decay of nonuniformities in composition in the reactor. This is related to the design parameters of a gas reactor by dimensional arguments drawn from the theory of isotropic turbulence. The analysis of reactor performance is carried out in terms of a coalescence model for the micro-mixing that incorporates this time scale. The mixing model is borrowed from chemical engineering studies. Calculations are presented for adiabatic reaction systems, showing the shift in the effective reaction-rate curve, and especially the reduction in blow-out limit, with decreasing mixing intensity. Such calculations permit one to see how high a mixing intensity is needed to approximate perfect mixing, and how this level depends on the reaction kineties.

A stochastic model for fluidized beds

Abstract A mathematical model for gas-fluidized beds is proposed that allows for a randomly fluctuating flow pattern. It is shown how mean first-order conversion is related to contact time distribution for arbitrary models of this type. A simplified version of the model is then studied, and it is found that the effect of fluctuating flow is similar to that of stagnancy in steady systems. This effect is inconsistent with the usual steady-state models, but it is shown that some published data on conversion in fluidized beds [6] exhibit this effect.