Ignition of Deflagration and Detonation Ahead of the Flame due to Radiative Preheating of Suspended Micro Particles

We study a flame propagating in the gaseous combustible mixture with suspended inert particles. The gas is assumed to be transparent for the radiation emitted by the combustion products, while particles absorb and re-emit the radiation. Thermal radiation heats the particles, which in turn transfer the heat to the surrounding gaseous mixture by means of heat conduction, so that the gas temperature lags that of the particles. We consider different scenarios depending on the spatial distribution of the particles, their size and the number density. In the case of uniform distribution of the particles the radiation causes a modest increase of the temperature ahead of the flame and the corresponding increase of the flame velocity. The effects of radiation preheating is stronger for a flame with smaller normal velocity. In the case of non-uniform distribution of the particles, such that the particles number density is smaller just ahead of the flame and increases in the distant region ahead of the flame, the preheating caused by the thermal radiation may trigger additional independent source of ignition. This scenario requires the formation of a temperature gradient with the maximum temperature sufficient for ignition in the region of denser particles cloud ahead of the advancing flame. Depending on the steepness of the temperature gradient formed in the unburned mixture, either deflagration or detonation can be initiated via the Zeldovich's gradient mechanism. The ignition and the resulting combustion regimes depend on the temperature profile which is formed in effect of radiation absorption and gas-dynamic expansion. In the case of coal dust flames propagating through a layered dust cloud the effect of radiation heat transfer can result in the propagation of combustion wave with velocity up to 1000m/s and can be a plausible explanation of the origin of dust explosion in coal mines.Comment: 45 pages, 14 figures. Accepted for publication Combustion and Flame 29 June 201

Hydrogen Atom in Electric and Magnetic Fields: Dynamical Symmetries, Superintegrable and Integrable Systems, Exact Solutions

The Hamiltonian of a pure hydrogen atom possesses the SO(4) symmetry group generated by the integrals of motion: the angular momentum and the Runge-Lenz vector. The pure hydrogen atom is a supersymmetric and superintegrable system, since the Hamilton-Jacobi and the Schr\"odinger equations are separable in several different coordinate systems and has an exact analytical solution. The Schr\"odinger equation for a hydrogen atom in a uniform electric field (Stark effect) is separable in parabolic coordinates. The system has two conserved quantities: z-projections of the generalized Runge-Lenz vector and of the angular momentum. The problem is integrable and has the symmetry group SO(2)xSO(2). The ion of the hydrogen molecule (problem of two Coulomb centers) has similar symmetry group SO(2)xSO(2) generated by two conserved z-projections of the generalized Runge-Lenz and of the angular momentum on the internuclear axis. The corresponding Schr\"odinger equation is separable in the elliptical coordinates. For the hydrogen atom in a uniform magnetic field, the respective Schr\"odinger equation is not separable. The problem is non-separable and non-integrable and is considered as a representative example of quantum chaos that cannot be solved by any analytical method. Nevertheless, an exact analytical solution describing the quantum states of a hydrogen atom in a uniform magnetic field can be obtained as a convergent power series in two variables, the radius and the sine of the polar angle. The energy levels and wave functions for the ground and excited states can be calculated exactly, with any desired accuracy, for an arbitrary strength of the magnetic field. Therefore, the problem can be considered superintegrable, although it does not possess supersymmetry and additional integrals of motion.Comment: 29 pages, 3 figure