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The integrability condition for the magnetohydrodynamic equations for rotational symmetric stationary plasma with infinite conductivity is obtained. It is found that the total enthalpy is conserved on a magnetic surface. Using this condition, a geometrical representation of the flow (sub- and supersonic) is given.

The formation and structure of normal shock waves in compressible, polytropic gases is investigated using the continuum gas dynamical equations. A phenomenological model for temperature-dependent viscosity and thermal conductivity is preassumed. The fundamental resulting nonlinear differential equation is solved exactly, and the complete structure of the shock wave is determined. It is found that the flow quantities and characteristics depend upon one parameter, the upstream Prandtl number Pr1, which satisfies the inequality 3γ/(γ+1)≤2Pr1<3, where γ is the ratio of specific heats. For smaller values of Pr1, the strength and compression ratio of the shock increase. Special cases of interest cannot be obtained from this solution and are to be treated independently. Numerical results are given and discussed.

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The MHD equations are used to discuss the formation and structure of shock waves in channel flow of an ionized gas with finite viscosity, thermal and electrical conductivity. The fundamental resulting complicated nonlinear differential equations are solved exactly, and the shock structure is determined completely. A necessary compatibility condition for the evolution of shock waves is obtained in the form:where Pr and Prm are ordinary and magnetic Prandtl numbers and γ the ratio of specific heats. The flow quantities and parameters are found to depend, in a non-analytical manner, on Pr and Prm, so that special cases of interest are to be treated separately. Numerical results are given and discussed.