A scaling approach to the derivation of hydrodynamic boundary conditions

We show hydrodynamic boundary conditions to be the inherent consequence of the Onsager principle of minimum energy dissipation, provided the relevant effects of the wall potential appear within a thin fluid layer next to the solid wall, denoted the surface layer. The condition that the effect of the surface layer on the bulk hydrodynamics must be independent of its thickness h is shown to imply a set of consistent ‘scaling relationships’ between h and the surface-layer variables/parameters. The use of the scaling relations, in conjunction with the surface-layer equations of motion derived from the Onsager principle, directly leads to the hydrodynamic boundary conditions. We demonstrate the surface-layer scaling process both physically and mathematically, and relate the parameters of the boundary conditions to those in the surface-layer equations of motion. In spatial regions outside the surface layer, equivalence between the use of surface-layer dynamics and boundary conditions is numerically demonstrated for Couette flows. As an application of the present approach, we derive the liquid-crystal hydrodynamic boundary conditions in which the rotational and translational dynamics are coupled.