4. Transcriticality

The partitioning of the flow domain of fluids of instantaneous elasticity into two distinct regimes, a relatively high-speed supercritical region away from the boundaries where the viscoelastic Mach number M ≫ 1 and where the vorticity is governed by a hyperbolic equation and a low-speed region near the boundaries where it is governed by an elliptic equation, is discussed. The smallness of the Reynolds Re number is not a sufficient reason for neglecting the inertial non-linearity in the flow of viscoelastic fluids with instantaneous elasticity, those which do not have a Newtonian base viscosity. The pivotal parameter is the product of the Reynolds number and the Weissenberg number M ² = Re We, where M is the viscoelastic Mach number, the ratio of a characteristic velocity of the fluid to that of the speed of shear waves. The neglect of the inertial terms may only be justified for small values of the Mach number M.