Transient free‐surface flow of a viscoelastic fluid in a narrow channel

The interplay between inertia and elasticity is examined for transient free‐surface flow inside a narrow channel. The lubrication theory is extended for the flow of viscoelastic fluids of the Oldroyd‐B type (consisting of a Newtonian solvent and a polymeric solute). While the general formulation accounts for non‐linearities stemming from inertia effects in the momentum conservation equation, and the upper‐convected terms in the constitutive equation, only the front movement contributes to non‐linear coupling for a flow inside a straight channel. In this case, it is possible to implement a spectral representation in the depthwise direction for the velocity and stress. The evolution of the flow field is obtained locally, but the front movement is captured only in the mean sense. The influence of inertia, elasticity and viscosity ratio is examined for pressure‐induced flow. The front appears to progress monotonically with time. However, the velocity and stress exhibit typically a strong overshoot upon inception, accompanied by a plug‐flow behaviour in the channel core. The flow intensity eventually diminishes with time, tending asymptotically to Poiseuille conditions. For highly elastic liquids the front movement becomes oscillatory, experiencing strong deceleration periodically. A multiple‐scale solution is obtained for fluids with no inertia and small elasticity. Comparison with the exact (numerical) solution indicates a wide range of validity for the analytical result. Copyright © 2004 John Wiley & Sons, Ltd.     

The Small Index Property for Free Groups and Relatively Free Groups

1. Some groups with the small index property In this paper we shall use the methods of the paper [6] by Hodges,Hodkinson, Lascar and Shelah to show that the free group ofcountably infinite rank and certain relatively free groups ofcountably infinite rank have the small index property.     

A two-echelon base-stock inventory model with Poisson demand and the sequential processing of orders at the upper echelon

We consider a two-echelon inventory system with a number of non-identical, independent ‘retailers’ at the lower echelon and a single ‘supplier’ at the upper echelon. Each retailer experiences Poisson demand and operates a base stock policy with backorders. The supplier manufactures to order and holds no stock. Orders are produced, in first-come first-served sequence, with a fixed production time. The supplier therefore functions as an M/D/1 queue. We are interested in the performance characteristics (average inventory, average backorder level) at each retailer. By finding the distribution of order lead time and hence the distribution of demand during order lead time, we find the steady state inventory and backorder levels based on the assumption that order lead times are independent of demand during order lead time at a retailer. We also propose two alternative approximation procedures based on assumed forms for the order lead time distribution. Finally we provide a derivation of the steady state inventory and backorder levels which will be exact as long as there is no transportation time on orders between the supplier and retailers. A numerical comparison is made between the exact and approximate measures. We conclude by recommending an approach which is intuitive and computationally straightforward.     

Soluble models for dynamics driven by a super-diffusive noise

We explicitly discuss scalar Langevin type of equations where the deterministic part is linear, but where the integrated noise source is a non-linear diffusion process exhibiting superdiffusive behavior. We calculate transient and stationary probabilities and study the possibility of noise induced transitions from a unimodal to a bimodal probability shape. Illustrations from finance and dynamical systems are given.     

Self-inhibition or the lack of it in the pyrolyses of neopentane and n-hexane

Published data show that in its early stages (up to 3% decomposition), the pyrolysis of n-hexane in the ranges 723–823 K and 10–100 Torr is not inhibited by the olefin products, in contrast with neopentane pyrolysis which is very strongly inhibited in similar conditions. Detailed consideration of the chain mechanisms for the two pyrolyses shows that the reactivity of the chain terminating radical towards hydrogen abstraction from an allylic C? H bond in product olefin is the factor which determines whether or not observable self-inhibition occurs. Thus, n-hexane pyrolysis, whose chain decomposition is terminated by recombination and disproportionation of ethyl, is not significantly self-inhibited, whereas that of neopentane which is terminated by recombination of methyl is very strongly inhibited because methyl is 14× more reactive than ethyl. The implications for other alkanes are briefly discussed.     

Stock Control and the Undershoot of the Re-order Level

Practical stock-control systems are frequently based on a policy of reviewing the stock balance and placing an order if this has fallen to or below a set level R. In order to facilitate the calculation of the control parameters, the assumption is generally made that the stock balance is exactly R when an order is placed, whereas in practice it may have fallen below R. This paper uses a steady-state analysis to evaluate some relatively simple adjustments which may be incorporated to handle this undershoot.     

Determination of myo-inositol in a flow-injection system with co-immobilized enzyme reactors and amperometric detection

A selective and sensitive flow-injection system for the determination of myo-inositol (hexahydroxycyclohexane) is described. Inositol dehydrogenase, IDH, lactate dehydrogenase, LDH, and lactate oxidase, LOD, are co-immobilized on porous glass and used in a packed-bed enzyme reactor. myo-Inositol reacts to produce an equivalent amount of hydrogen peroxide, which oxidizes hexacyanoferrate(II) to hexacyanoferrate(III) in a second reactor containing immobilized peroxidase. The hexacyanoferrate(III) is then detected amperometrically at 0 mV vs. SCE in a flow-through detector. The system responds linearly to injected samples of myo-inositol (25 μl) in the concentration range 1–300 μM. The maximum throughput was 90 samples per hour. The IDH/LDH/LOD reactor was stable for at least 5 weeks.     

The maximal solution of a restricted subadditive inequality in numerical analysis

For a fixed non-negative integerp, letU _2p = {U _2p (n)},n ≥ 0, denote the sequence that is defined by the initial conditionsU _2p (0) =U _2p (1) =U _2p (2) = =U _2p (2p) = 1 and the restricted subadditive recursion $$U_{2p} (n + 2p + 1) = \mathop {\min }\limits_{0 \leqslant l \leqslant p} (U_{2p} (n + l) + U_{2p} (n + 2p - l)),n \geqslant 0$$ U _2p is of importance in the theory of sequential search for simple real zeros of real valued continuous 2p-th derivatives In this paper, several closed form expressions forU _2p (n), n > 2p, are determined, thereby providing insight into the structure ofU _2p Two of the properties thus illuminated are (a) the existence of exactlyp + 1 limit points (1 + 1/(p + 1 +i), 0 ≤i ≤p) of the associated sequence {U _2p (n + 1)/U _2p (n)},n ≥ 0, and (b) the relevance toU _2p of the classic number theoretic function ord