Generalized Boltzmann Equation: Slip-No -Slip Dynamic Transition in Flows of Strongly Non-Linear Fluids

Abstract

The Navier-Stokes equations, are understood as the result of the low-order expansion in powers of dimensionless rate of strain $\eta_{ij}=\tau_{0}S_{ij}$, where $\tau_{0}$ is the microscopic relaxation time of a close-to- thermodynamic equilibrium fluid. In strongly sheared non-equilibrium fluids where $|\eta_{ij}|\geq 1$, the hydrodynamic description breaks down. According to Bogolubov's conjecture, strongly non-equlibrium systems are characterized by an hierarchy of relaxation times corresponding to various stages of the relaxation process. A "hydro-kinetic" equation with the relaxation time involving both molecular and hydrodynamic components proposed in this paper, reflects qualitative aspects of Bogolubov's hierarchy. It is shown that, applied to wall flows, this equation leads to qualitatively correct results in an extremely wide range of parameter $\eta$-variation. Among other features, it predicts the onset of slip velocity at the wall as an instability of the corresponding hydrodynamic approximation.