  It is shown here that a subset of the implicit analytical shock solutions
discovered by Becker and by Johnson can be inverted, yielding several exact
closed-form solutions of the one-dimensional compressible Navier-Stokes
equations for an ideal gas. For a constant dynamic viscosity and thermal
conductivity, and at particular values of the shock Mach number, the velocity
can be expressed in terms of a polynomial root. For a constant kinematic
viscosity, independent of Mach number, the velocity can be expressed in terms
of a hyperbolic tangent function. The remaining fluid variables are related to
the velocity through simple algebraic expressions. The solutions derived here
make excellent verification tests for numerical algorithms, since no source
terms in the evolution equations are approximated, and the closed-form
expressions are straightforward to implement. The solutions are also of some
academic interest as they may provide insight into the non-linear character of
the Navier-Stokes equations and may stimulate further analytical developments.
