| Inertial Confinement approaches to the problem of production of energy from Fusion reactions (ICF) require temperatures of 100,000,000(DEGREES)K, pressures of 10('12) atm. and densities of 10000X the liquid density to drive a small pellet, mixture of D-T, to thermonuclear conditions and burn efficiently. The problem of the hydrodynamic analysis of a hollow pellet (fuel is located between two concentric spherical or cylindrical surfaces) undergoing compression, through implosion processes driven by an energy beam, that has been focused upon its surface, is divided into the study of the phenomena of shock implosion and the collapse of an empty cavity. Using a self-similar method it is seen how the representative curve of the state of the system, for a gas with a finite number of degrees of freedom, passes analytically through all the singularities intrinsic to the conservation equations in their reduced form i.e. the point where the pressure attains a maximum, the crossing point with the parabola Z' - (V((xi))- 1)('2) and the point where the relationship C('2)((xi))-(xi)(X + (gamma)) (U-(xi)) is satisfied. Through analysis of the mentioned singularities, the idealization involved when choosing a closed form, analytical value of (gamma) for the problem of shock implosion is understood. Following a similar analysis, two singular points are found for the cavity problem i.e. a maximum in the reduced local speed of sound and the intersection point with the parabola C('2)((xi))- (U((xi))-(xi))('2). The representative curve passes through all of them for the whole spectrum of (gamma). A closed form is obtained for the self-exponent before the actual integration is performed, closing together the limits in the traditional division of self-similar problems between first and second class. Approximating the representative curve of the system in the X-Y plane, with a straight line, a closed form of the functions reduced Pressure, Density, Velocity and Speed of Sound is given. A comparison is made between the values obtained for the reduced variables from a numerical integration of the equations and those obtained from the linear case. The agreement being in the order of a few percent. |
