Influence of gravitational forces and fluid flows on the shape of surfaces of a viscous fluid of capillary size

The Navier-Stokes equations and boundary conditions for viscous fluids of capillary size are formulated in curvilinear coordinates associated with a geometry of the fluid-gas interface. As a result, the fluid dynamics of drops and menisci, which takes into account the influence of gravitational forces and flows on the surface shape, can be described. This gives a convenient basis for numerical studies. Estimations of the effects are presented for the case of an evaporating sessile drop.