To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
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.
Equations of gas-phase chemical kinetics (1.85) (see Section 1.3) are valid for a constant volume (V = const) BR, while occurring a reversible chemical reactions. However, in the general case, it is desirable to allow for volume variation (V = var) in the reactor R, or in an assumed reactor of the system of reactors (SR), as well as in occurrences of irreversible reactions herein, feed and discharge of substances and surface reactions [5]. Such reactions reflect the change in gas mass and its composition in the reactor due to a number of processes (for example, evaporation, condensation, combustion of metals and coal, absorption, etc.).
The evaporation of droplets is one of the major stages of the working process that defines the combustion efficiency in the propulsion and power generation systems. Droplets of different sizes moving relative to gas flow and distributed in a complicated manner evaporate in the medium with variable gas dynamic and thermodynamic parameters. The evaporation process is very complicated, which is why whatever actual problem reduces in its theoretical analyses to an approximate model, allowing one to obtain an analytical or numerical solution. For instance, the chemical nonequilibrium model of evaporation of a single-component droplet in high-temperature flow illuminated in Chapter 5 comprises dozens of assumptions. A large number of theoretical and experimental studies are dedicated to the problems of droplets evaporation and combustion.
This paper is concerned with the problem of existence and uniqueness of weak and classical solutions for a fourth-order semilinear boundary value problem. The existence and uniqueness for weak solutions follows from standard variational methods, while similar uniqueness results for classical solutions are derived using maximum principles.
Following elucidation of the basics of thermodynamics and detailed explanation of chemical kinetics of reactive mixtures, readers are introduced to unique and effective mathematical tools for the modeling, simulation and analysis of chemical non-equilibrium phenomena in combustion and flows. The reactor approach is presented considering thermochemical reactors as the focal points. Novel equations of chemical kinetics compiling chemical thermodynamic and transport processes make reactor models universal and easily applicable to the simulation of combustion and flow in a variety of propulsion and energy generation units. Readers will find balanced coverage of both fundamental material on chemical kinetics and thermodynamics, and detailed description of mathematical models and algorithms, along with examples of their application. Researchers, practitioners, lecturers, and graduate students will all find this work valuable.
We consider a theoretical model for the flow of Newtonian fluid through a long flexible-walled channel which is formed from four compliant and rigid compartments arranged alternately in series. We drive the flow using a fixed upstream flux and derive a spatially one-dimensional model using a flow profile assumption. The compliant compartments of the channel are assumed subject to a large external pressure, so the system admits a highly collapsed steady state. Using both a global (linear) stability eigensolver and fully nonlinear simulations, we show that these highly collapsed steady states admit a primary global oscillatory instability similar to observations in a single channel. We also show that in some regions of the parameter space the system admits a secondary mode of instability which can interact with the primary mode and lead to significant changes in the structure of the neutral stability curves. Finally, we apply the predictions of this model to the flow of blood through the central retinal vein and examine the conditions required for the onset of self-excited oscillation. We show that the neutral stability curve of the primary mode of instability discussed above agrees well with canine experimental measurements of the onset of retinal venous pulsation, although there is a large discrepancy in the oscillation frequency.
The withdrawal of water with a free surface through a line sink from a two-dimensional, vertical sand column is considered using the hodograph method and a novel spectral method. Hodograph solutions are presented for slow flow and for critical, limiting steady flows, and these are compared with spectral solutions to the steady problem. The spectral method is then extended to obtain unsteady solutions and hence the evolution of the phreatic surface to the steady solutions when they exist. It is found that for each height of the interface there is a unique critical coning value of flow rate, but also that the value obtained is dependent on the flow history.
This chapter aims to apply the results of earlier chapters to solar observations, considering both historical cases and recently obtained ground- or space-based observations of the Sun’s atmosphere. Coronal loops, prominences and sunspots are used to illustrate the various theoretical results. Attention to historical contributions is also part of the treatment. The founding of coronal seismology is explored and some results are applied to coronal loops. Results for resonant absorption theory are illustrated. Prominences are also explored from the viewpoint of oscillation theory, illustrating some results of prominence seismology. Finally, sunspots are discussed in the context of slow mode propagation.
The effect of gravity is investigated in this chapter and the importance of the Klein-Gordon equation is demonstrated. The Klein-Gordon equation is solved for impulsive initial conditions and the phenomenon of an oscillating wake demonstrated. Cutoff frequency is determined. Waves in a stratified incompressible medium with a horizontal magnetic field are examined, leading to the Rayleigh-Taylor dispersion relation. The compressible case is related to the topic of magnetic helioseismology. Waves in a vertical magnetic field are also discussed. For this case, the slow mode dispersion relation is obtained and exhibits a cutoff frequency.
Connection formulas for a magnetic flux tube that describe the approximate behaviour of the perturbations across thin layers where dissipative processes (here electrical conductivity) act are derived for the Alfven singularity. The tube may be twisted or untwisted. In an appropriate limit these formulas reduce to jumps across a narrow region. Such jumps are described in terms of introduced functions $F$ and $G$ and their related functions. Jump relations are used to derive approximate dispersion relations, leading to the determination of resonant absorption decay rates. Decay rates are determined for two specific density profiles, the linear one and the sinusoidal profile. Jump conditions pertaining to the slow mode are also discussed. The equivalent jump relations holding for Cartesian geometry are obtained and illustrated for a single magnetic interface, obtaining decay rates.
The modes of oscillation of a magnetic flux tube are explored, working from the fundamental differential equations obtained in Chapter 3. Sausage modes and kink modes (as in a magnetic slab) are investigated and their dispersion relations understood. Fluting modes also occur. Dispersion relations and diagrams, each similar to those arising in a slab, are derived and displayed, for both photospheric and coronal conditions. Leaky waves are explored. Resonant absorption in a flux tube is examined, with the decay rate obtained for a $\beta = 0$ tube. Two profiles of density across a thin layer on the boundary of the tube are explored, the linear profile and the sinusoidal profile, with decay rates obtained for both.
The differential equations in Cartesian geometry are solved for the magnetoacoustic waves in a magnetic slab. The case of a field-free environment is also investigated as is the $\beta = 0$ plasma. Sausage and kink waves arise and their properties are described. The notion of surface waves and body waves is introduced. Dispersion diagrams are displayed under two sets of conditions, the photospheric medium and the coronal medium. Impulsive waves are examined. Also, waves in smoothly varying profiles are explored, especially the Epstein profile. Cutoff frequencies are obtained for a range of profiles.
Surface waves are introduced, and the surface wave dispersion relation derived. Some general properties of this relation are investigated. Surface waves in certain special cases, including when one interface is field-free or when both sides of the interface are $\beta = 0$ plasmas are discussed in detail.