Continuum Modelling of Nanoscale Hydrodynamics

Abstract

It may come as a surprise to most physicists and material scientists that up until recently, continuum hydrodynamics can not accurate model fluid(s) flow at the nanoscale. The problem lies not in the Navier Stokes (NS) equation, which expresses momentum balance and therefore must be valid, but in the boundary condition at the fluid-solid interface, required for the solution of the NS equation. Traditionally the no-slip boundary condition (NSBC) has always been used to solve hydrodynamic problems, which states that there can be no relative motion at the fluid-solid interface. The purposes of this article are to delineate the problem and to present its resolution through the application of Onsager’s principle of minimum energy dissipation, which underlies almost all the linear response phenomena in dissipative systems. Even though lacking in first principles support, NSBC was widely regarded as a cornerstone in continuum hydrodynamics, owing to its broad applicability in diverse fluid-flow problems. However, an important exception was pointed out some time ago in the so-called the moving contact line (MCL) problem in immiscible flows. Here the contact line is defined as the intersection of the two-phase immiscible fluid-fluid interface with the solid wall. In 1974, Dussan and Davis showed with rigor that under the assumptions of (1) fluid incompressibility, (2) rigid and flat solid wall, (3) impenetrable fluid-fluid interface, and (4) NSBC, there is a velocity discontinuity at the MCL, and the tangential force exerted by the fluids on the solid wall in the vicinity of the MCL is infinite. Subsequently, by employing molecular dynamic (MD) simulations, Koplik et al. (1988) and Thompson and Robbins (1988) have shown that near-complete slip occurs at the MCL, in direct contradiction to the NSBC. The failure of the NSBC and the lack of a viable alternative mean that the usual continuum hydrodynamics can not accurately model fluids flow on the micro/nanoscale. This has often been referred to as an example of the so-called “breakdown of continuum.”