In addition to hydrodynamic equations in the bulk region, proper hydrodynamic boundary conditions are essential to theoretically understanding and numerically simulating the hydrodynamics of many soft matter systems, e.g. two-phase flows on solid surfaces and viscoelastic fluids on solid surfaces. In this work, we show that Onsager's variational principle, first proposed by Lars Onsager in his seminal work in 1931, provides a convenient instrument for deriving not only hydrodynamic equations in the bulk region but also thermodynamically consistent hydrodynamic boundary conditions. To demonstrate its applications, we consider one-component liquid-vapor flows and immiscible two-phase flows on solid surfaces where moving contact line is involved.
Xu, X., & Qian, T. (2017). Hydrodynamic Boundary Conditions Derived from Onsager’s Variational Principle. In Procedia IUTAM (Vol. 20, pp. 144–151). Elsevier B.V. https://doi.org/10.1016/j.piutam.2017.03.020