Different Types Of Shocks Research Articles

On the Duration of the Problem of the Points

Abstract We consider an r-player version of the famous problem of the points, which was the stimulus for the correspondence between Pascal and Fermat in the 17th century. At each play of a game, exactly one of the players wins a point, player i winning with probability pi. The game ends the first time a player has accumulated his or her required number of points—this requirement being ni for player i. A reliability application would be to suppose that a system is subject to r different types of shocks and failure occurs the first time there have been ni type i shocks for any i = 1, …, r. Our main result is to show that N, the total number of plays, is an increasing failure-rate random variable. In addition, we prove some Schur convexity results regarding P{N ≤ k} as a function of p (for ni ≡ n) and as a function of n (for pi ≡ 1/r).

One-dimensional magnetogasdynamics in oblique fields

Earlier work on the dynamics of a perfectly conductive gas in situations where all variables depend on one space co-ordinate only is extended to the case where the magnetic field has a component in the direction of variation. The theory is developed for an arbitrary gas in equilibrium, subject only to certain reasonable restrictions.The first main section studies the variation of the transverse field component in slow and fast simple waves and the tendency of compressive waves in which the transverse field does not change sign to steepen into shocks.The next section develops a symmetrical treatment of the Rayleigh, Fanno and other processes of ordinary steady one-dimensional gasdynamics, generalized to allow for electromagnetic effects. The slow, fast and Alfvén wave speeds are critical. A particular case of this analysis is a generalization of ordinary gasdynamics which allows for the effect of transverse forces such as occur in turbo-machinery.The final section is an exhaustive study of shocks in the presence of a field component normal to the shock front. From the generalized Rayleigh line it is established that there are up to six different types of shocks, all compressive and distinguishable by the relative magnitudes of the upstream and downstream normal velocities in comparison with the local slow, fast and Alfvén wave speeds.Some aspects of shock structure are discussed briefly.