On "Solid Liquid" Limit of Hydrodynamic Equations

In Ref.[1-4] we have obtained new hydrodynamic equations beyond a traditional framework of Knudsen number expansions. Approaches of paper [1 J and of papers [3-4] are markedly different. In Ref.[l], a perturbation theory similar to KAM-theory was developed for the Boltzmann equation, and it was based on Newton iterative procedures and parametrix expansions. No asymptotic expansions in powers of Knudsen number were used in Pl. On the other hand, in Ref.[2-4], a partial summation of all terms of ChapmanEnskog expansion for Grad equations was employed. However, results of both approaches have much in common. In particular, for a nonlinear case, a nontrivial threshold behavior of viscid stress tem;or at high compression of a fulx was detected (see a discussion in [1]). Namely, there exists such negative value of flow divergency, divu* at which the st ress tensor diverges as a function of divu. For divu < divu", an effective viscosity is positive, while for divu > divu· it is negative (a non-physical region). In this short communication we show that, thanks to the divergency indicated, a transition from the physical region into that nonphysical is indeed impossible. An expression for the nonlinear stress tensor [1-4J reads (we consider the one-dimensional case for sirnplisity):