A formal analogy is shown to exist between the one-dimensional, steady flow of an incompressible, non-Newtonian fluid, characterized by a viscosity coefficient depending on the shear rate tensor invariants, and the flow of a fictitious, inviscid, calorically perfect gas moving in a two-dimensional (2-D) space orthogonal to the viscous fluid velocity. The functional relationship between the gas density rho and the potential. velocity function phi turns out to be the same as the relationship between the viscosity eta and the velocity to of the incompressible fluid. The analogy allows the definition of a number characteristic of non-Newtonian flows with the same meaning as the Mach number, from which the elliptic or hyperbolic nature of the equation of motion can be inferred without calculating its analytical expression. The existence of the Mach number implies that some constitutive equations developed either experimentally or theoretically may not be realistic under all circumstances. The analysis is applied to three well-known models of non-Newtonian fluids (the "power law" model, the steady-state shear solution of the finite extensibility non-linear elasticity (FENE) model and the Zwanzig model), in order to evaluate their ranges of validity. An exact analytic solution of Poiseuille's problem is obtained for the Zwanzig fluid, showing that the flow cannot exist if the tube diameter exceeds a critical value; the non-existence of the solution is interpreted as a gas-dynamic tunnel cut-off effect, since the Mach number of the fictitious gas becomes equal to unity on the tube walls.