  A vanishing viscosity method is formulated for two-dimensional transonic
steady irrotational compressible fluid flows with adiabatic constant $\gamma\in
[1,3)$. This formulation allows a family of invariant regions in the phase
plane for the corresponding viscous problem, which implies an upper bound
uniformly away from cavitation for the viscous approximate velocity fields.
Mathematical entropy pairs are constructed through the Loewner-Morawetz
relation by entropy generators governed by a generalized Tricomi equation of
mixed elliptic-hyperbolic type, and the corresponding entropy dissipation
measures are analyzed so that the viscous approximate solutions satisfy the
compensated compactness framework. Then the method of compensated compactness
is applied to show that a sequence of solutions to the artificial viscous
problem, staying uniformly away from stagnation, converges to an entropy
solution of the inviscid transonic flow problem.
