Feeding high-pressure gas into the gas space of the tanks filled with liquids and even solids aims to maintain this gas space at a preselected pressure history bounded by tanks’ structural requirements or required propellant supply pressures, to prevent propellant pump cavitation, to avoid uncontrolled chemical reactions in gas space, etc. The pressurization process and corresponding pressurization systems are used in diverse technical devices. These include apparatus for chemical technology, oil and ore tankers, aircraft fuel tanks, and propellant tanks of LPREs. Processes related to high-pressure and often high-temperature gas feeding are extremely diversified because of the complex flow patterns of gas in the free space of the tanks, possible heat exchange with structural elements and the propellant, mass exchange caused by the evaporation of liquids, the chemical reactions in gas, and liquid phases.

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.

Modeling and numerical simulation of combustion in the cylinders of spark-ignition and compression-ignition internal combustion engines (ICEs) provide a considerable contribution in engines engineering and the optimization of engines performance, efficiency, and emissions. This chapter demonstrates the application of the reactor approach and the chemical nonequilibrium model (Chapters 1–3) to the simulation of combustion in the cylinder of the spark-ignition ICE aiming to predict the variation in ionized particle concentration as control variables. It is known that the combustion of hydrocarbon fuels with oxidizers at high pressures and temperatures is accompanied by the output of some ionized substances. Research on the ionization in flames was started in the mid-1950s for the purpose of optimization of magnetohydrodynamic generators as well as the study of ionized particle formation in combustion products of propulsion systems, particularly in the thrust chambers and exhaust plumes of rocket engines [1, 160, 215, 227, 228]. This study was later extended to the combustion in the ICE for the purpose of employing empirical and theoretical data on the ionization of combustion products for engine performance control intended for the optimization of the combustion process, the reduction of fuel consumption, the reduction of exhaust gas emission, the optimization of the exhaust gas recirculation (EGR) process, etc. [292–305].

Gas–liquid reacting flows seem to be one of the most complex and, at the same time, most prevalent fields of application for mathematical simulation of high-temperature processes. Of these processes, the phenomena are fluid atomization polydispersity and droplet secondary fragmentation, droplet heating and evaporation, turbulence, reactions in the gas phase, the difference in the velocity between the gas and droplet phases (slip velocity), and the multidimensional nature of fluid flow. Such flows make the core of processes proceeding in combustion chambers of air-breathing jet engines [216, 231, 239, 240], rocket engines [160, 215, 228, 229, 241, 242], gas generator driving turbopumps, pressurization systems of the LPRE propellant tanks [160, 215, 228, 241–243], vapor-gas generators [50, 55, 56], afterburners of air-breathing jet engines [216, 231, 239, 240], and different furnaces [58].

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 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.