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Asymptotic expansions of the Gauss hypergeometric function with large parameters, $F(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D716}_{1}\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D716}_{2}\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D716}_{3}\unicode[STIX]{x1D70F};z)$ as $|\unicode[STIX]{x1D70F}|\rightarrow \infty$, are known for many special cases, but not for one that the author encountered in recent work on fluid mechanics: $\unicode[STIX]{x1D716}_{2}=0$ and $\unicode[STIX]{x1D716}_{3}=\unicode[STIX]{x1D716}_{1}z$. This paper gives the leading term for that case if $\unicode[STIX]{x1D6FD}$ is not a negative integer and $z$ is not on the branch cut $[1,\infty )$, and it shows how subsequent terms can be found.
The numerical entropy production (NEP) for shallow water equations (SWE) is discussed and implemented as a smoothness indicator. We consider SWE in three different dimensions, namely, one-dimensional, one-and-a-half-dimensional, and two-dimensional SWE. An existing numerical entropy scheme is reviewed and an alternative scheme is provided. We prove the properties of these two numerical entropy schemes relating to the entropy steady state and consistency with the entropy equality on smooth regions. Simulation results show that both schemes produce NEP with the same behaviour for detecting discontinuities of solutions and perform similarly as smoothness indicators. An implementation of the NEP for an adaptive numerical method is also demonstrated.
Options with extendable features have many applications in finance and these provide the motivation for this study. The pricing of extendable options when the underlying asset follows a geometric Brownian motion with constant volatility has appeared in the literature. In this paper, we consider holder-extendable call options when the underlying asset follows a mean-reverting stochastic volatility. The option price is expressed in integral forms which have known closed-form characteristic functions. We price these options using a fast Fourier transform, a finite difference method and Monte Carlo simulation, and we determine the efficiency and accuracy of the Fourier method in pricing holder-extendable call options for Heston parameters calibrated from the subprime crisis. We show that the fast Fourier transform reduces the computational time required to produce a range of holder-extendable call option prices by at least an order of magnitude. Numerical results also demonstrate that when the Heston correlation is negative, the Black–Scholes model under-prices in-the-money and over-prices out-of-the-money holder-extendable call options compared with the Heston model, which is analogous to the behaviour for vanilla calls.
In this paper, we revisit our previous work in which we derive an effective macroscale description suitable to describe the growth of biological tissue within a porous tissue-engineering scaffold. The underlying tissue dynamics is described as a multiphase mixture, thereby naturally accommodating features such as interstitial growth and active cell motion. Via a linearization of the underlying multiphase model (whose nonlinearity poses a significant challenge for such analyses), we obtain, by means of multiple-scale homogenization, a simplified macroscale model that nevertheless retains explicit dependence on both the microscale scaffold structure and the tissue dynamics, via so-called unit-cell problems that provide permeability tensors to parameterize the macroscale description. In our previous work, the cell problems retain macroscale dependence, posing significant challenges for computational implementation of the eventual macroscopic model; here, we obtain a decoupled system whereby the quasi-steady cell problems may be solved separately from the macroscale description. Moreover, we indicate how the formulation is influenced by a set of alternative microscale boundary conditions.
The two volumes which make up The Seismic Wavefield provide a comprehensive guide to the understanding of seismograms in terms of physical propagation processes within the Earth. The focus is on the observation of earthquakes and man-made sources on all scales, for both body waves and surface waves. Volume I provided a general introduction and a development of the theoretical background for seismic waves. This material is exploited in Volume II to look at the way in which observed seismograms relate to the propagation processes. Volume II discusses local and regional seismic events, global wave propagation, and the three-dimensional Earth. The combination of observation and theoretical development with a strong visual focus will greatly appeal to graduate students in seismology. The two volumes will also be valuable to researchers and professionals in academia and the petroleum industry.
Modern kinetic mechanisms are intricate and can comprise tens of substances and hundreds of reactions [46, 95, 96]. For example, paper [97] deals with low-temperature decomposition of hydrocarbons to analyze the combustion mechanism totaling 340 substances and 3400 reactions. The authors of [98] exploited the combustion mechanism including 120 substances and 721 reactions for simulation of n-decane ignition. In the calculation of the oxidation of hydrocarbons described in [99], a mechanism comprising 71 substances and 417 reactions was analyzed.
In numerical analyses aimed at developing high-efficiency combustion chambers for various engines and thermal power systems, it is necessary to have an adequate understanding of hydrodynamic and chemical processes related to flowing, mixing, and combustion of two-phase fuels and oxidizers. The occurrence of such processes is described by the availability of zones differing in type, space, and time scale of these processes in the working volume.
The model of the combustion in the flame front is commonly used for the simulation of operating parameters and emission characteristics of combustion chambers of different combustion systems as one of the main simulation fragments in models of premixed flames. The typical scheme of combustion in the flame front was described in the Section 1.1. Combustion in the flame front predetermines to a considerable extent the further afterburning processes and parameters of reacting flows in the combustion unit and combustion products emission. In accordance with the generally accepted definition, the flame front is identified as a thin layer separating an unburned fresh mixture of the reactants from the combustion products wherein maximum gradients of concentrations of the reactants and reaction products are observed (Figure 4.1). Once the fresh mixture is ignited, a resulting premixed flame propagates in the x direction, consuming the unburned mixture. The chemical interaction in the flame front under conditions of intensive self-acceleration of the processes caused by the transfer of both heat and active catalyzing centers from the products of reactions to the unburned fresh mixture.
Combustion processes (that is, conversion of chemical energy of propellant components into thermal energy of combustion products) are typical for various engineering systems. Working volumes wherein these processes can occur may be represented by combustion chambers of liquid-propellant rocket engines (LPRE), solid-propellant rocket engines (SPRE), air-breathing engines (ABE) steam-gas generators, magnetohydrodynamic generators (MHD generators), boiler furnaces of thermal electric power stations, and cylinders of internal combustion engines (ICEs) [1]. Besides, further conversion of combustion products with chemical conversions can proceed also in aircraft and rocket engine nozzles, ICE exhaust systems, LPRE gas ducts, etc.
The model of the combustion in the flame front is commonly used for the simulation of operating parameters and emission characteristics of combustion chambers of different combustion systems as one of the main simulation fragments in models of premixed flames. The typical scheme of combustion in the flame front was described in the Section 1.1. Combustion in the flame front predetermines to a considerable extent the further afterburning processes and parameters of reacting flows in the combustion unit and combustion products emission. In accordance with the generally accepted definition, the flame front is identified as a thin layer separating an unburned fresh mixture of the reactants from the combustion products wherein maximum gradients of concentrations of the reactants and reaction products are observed (Figure 4.1). Once the fresh mixture is ignited, a resulting premixed flame propagates in the x direction, consuming the unburned mixture. The chemical interaction in the flame front under conditions of intensive self-acceleration of the processes caused by the transfer of both heat and active catalyzing centers from the products of reactions to the unburned fresh mixture.
Evaporation and combustion of dispersed propellants in a high-temperature reacting flow are typical for the most diverse propulsion and power generation systems such as internal combustion engines (ICE), combustors of air breathing engines (ABE), combustion chambers of liquid-propellant rocket engines (LPRE), liquid gas generators (LGG), and steam-gas generators (SGG), combustion chambers of furnaces, etc. Liquid-propellant atomization, spray formation, and droplet evaporation processes are seen to bear strong influence on the efficiency of the combustion process and, hence, the operating and ecological parameters of these combustion systems.