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The flow of a gas in a shock tube is treated in the context of kinetic theory as an initial value problem for the one-dimensional Krook equation. The only simplifying assumption made is that the gas is one-dimensional. This corresponds to a gas with an adiabatic constant γ = 3. A finite difference method is proposed, utilizing the fact that, for sufficiently small time intervals, the gas experiences essentially free molecular flow, so that the collision effects can be treated as a first-order correction. The conservation laws are not used. The computed solution agrees excellently with the classical solution, but in addition, has shock structure, diffusion of the contact discontinuity, and dispersion of the expansion wave, all incorporated. The same procedure is used to calculate steady-state shock structure, to which the shock developed in the shock tube is compared. It is seen that for the strength of the shock calculated (Pressure ratio of tube 10:1, shock Mach number 1.43, γ = 3), the shock is essentially ``fully developed'' after about 20 mean collision times of the low-pressure gas. In conclusion, an exact reduction procedure is given by which problems for a monatomic three-dimensional gas (γ=53) are reduced to the solution of two simultaneous Krook equations, manageable by the present procedure without vastly increasing the need for computing capacity.

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Just before mitral-septal contact, the protruding leaflets project at high angles relative to flow. At these high angles, flow drag, the pushing force of flow, is the dominant hydrodynamic force on the protruding leaflet and appears to be the immediate cause of obstruction. The high angle between flow direction and the protruding leaflet precludes significant Venturi effects. Even earlier in systole, at leaflet coaptation, flow drag is dominant in half of the patients, with angles relative to flow > 15 degrees. After obstruction is triggered, it appears from our data and model that the leaflet is forced against the septum by the pressure difference across the orifice. The increasing acceleration of Doppler flow is explained by a time-dependent amplifying feedback loop in which the rising pressure difference across the orifice leads to a smaller orifice and a higher pressure difference.