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The hydrodynamic equations describing the steady, onedimensional flow of a viscous, heat-conducting, compressible gas are treated. The equation of state for a perfect gas is assumed valid, and the coefficients of specific heat are assumed constant. A complete integral of the energy equation is found for a Prandtl Number of /4, which is a good approximation for the actual value of many gases over wide temperature ranges. A particular integral contained in this complete integral is (u/2) + cpT ~ a constant, which is a frequently used one-dimensional energy equation. The restrictions on its use are rigorously derived and stated. With the complete integral of the energy equation, three different types of solutions which depend on the boundary conditions are shown to be obtainable. The usual shock wave solution (satisfying the Rankine-Hugoniot relations) is only one of these three types. Several features of the shock wave solution are shown. For example, the entropy has a maximum value at the inflection point in the velocity distribution. The validity of the continuum hypothesis is examined, and i t is shown that the thickness of the wave as defined by Prandtl becomes infinite as the initial Mach Number of the flow becomes infinite provided that n, the exponent in the temperature-viscosity relation JU/JU0 = (T/To) , is greater than V2. Furthermore, i t is concluded tha t the continuum theory may give reasonably correct results for flows with Mach Numbers up to 1.3. The other two cases of flows contained in the equations are terminating flows—i.e., flows tha t can be valid only in a finite region. Their mathematical nature is demonstrated by two numerical examples, which indicate compression shocks followed by expansion waves. However, the physical significance, if any exists, of such solutions remains open to question.
Morduchow et al. (Tue,) studied this question.