Browsing by Author "Battat, D."

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    A non-equilibrium kinetic description of shock wave structure
    (College of Aeronautics, 1968-06) Battat, D.
    A formulation for the shock wave structure is devised by viewing the transition as a phenomenon in which non-equilibrium effects play an important role. The essence of the method is the approximation of Boltzmann's equation by a simpler kinetic model. Initially, the distribution function in Boltzmann's collision integral is expressed in terms of a function of deviation from local equilibrium. Then, by suitably transforming the complete collision term, the molecular velocities after collision are eliminated. At this stage the formulation of the method is specialized to hard sphere molecules and the problem of deriving a model equation thus reduces to one of assigning an expression for the deviation function. In the first instance, this function is chosen to be zero and an exploratory model is obtained which, when its variable collision frequency is replaced by its mean value, reduces identically to the Bhatnagar-Gross-Krook model. However, it is found that the exploratory model provides a somewhat crude representation of Boltzmann's equation and is shown to imply a Prandtl number very nearly equal to unity. A more accurate model is then derived by choosing for the deviation function the first order term of Chapman-Enskog’s sequence, leading to the Navier-Stokes equations. Here, the specific form of Boltzmann's collision term is represented more accurately than hitherto and the model is found to possess all the known features of the Boltzmann equation. It is shown that this model contains a description of a gas in non-equilibrium state.
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    Shock wave structure in highly rarefied flows
    (College of Aeronautics, 1964-07) Battat, D.
    The Boltzmann equation is written in terms of two functions associated with the gain and loss of a certain type of molecule due to collisions. Its integral form is then applied to the problem of normal shock structure, and an iteration technique is used to determine the solution. The first approximation to the velocity distribution function of the Chapman-Enskog sequence, which leads to the Navier-Stokes equations, is used to initiate the iteration scheme. Expressions for the distribution function and the flow parameters pertinent to the first iteration are derived and show that the B-G-K model results can be obtained as a special case. This model is found to be valid in the continuum regime only, and is consequently limited to the study of strong shocks. In the present treatment the iteration is carried out on the distribution function and the analysis indicates that the method is equally valid for variations in both Mach and Knudsen numbers. Finally, the results of the first approximation are simplified, and expressed in a form suitable for numerical computation, and the range of their validity is discussed. The method should be equally suitable for other flow problems of linear or nonlinear nature.