Capillary surface
In fluid mechanics and mathematics, a capillary surface is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid interfaces. Capillary surfaces are of interest in mathematics because the problems involved are very nonlinear and have interesting properties, such as discontinuous dependence on boundary data at isolated points. In particular, static capillary surfaces with gravity absent have constant mean curvature, so that a minimal surface is a special case of static capillary surface. They are also of practical interest for fluid management in space (or other environments free of body forces), where both flow and static configuration are often dominated by capillary effects. The stress balance equation The defining equation for a capillary surface is called the stress balance equation, which can be derived by considering the forces and stresses acting on a small volume that is partly bounded by a capillary surface. For a fluid meeting another fluid (the "other" fluid notated with bars) at a surface , the equation reads where is the unit normal pointing toward the "other" fluid (the one whose quantities are notated with bars), is the stress tensor (note that on the left is a tensor-vector product), is the surface tension associated with the interface, and is the surface gradient. Note that the quantity is twice the mean curvature of the surface. In fluid mechanics, this equation serves as a boundary condition for interfacial flows, typically complementing the Navier–Stokes equations. It describes the discontinuity in stress that is balanced by forces at the surface. As a boundary condition, it is somewhat unusual in that it introduces a new variable: the surface that defines the interface. It's not too surprising then that the stress balance equation normally mandates its own boundary conditions. For best use, this vector equation is normally turned into 3 scalar equations via dot product with the unit normal and two selected unit tangents: Note that the products lacking dots are tensor products of tensors with vectors (resulting in vectors similar to a matrix-vector product), those with dots are dot products. The first equation is called the normal stress equation, or the normal stress boundary condition. The second two equations are called tangential stress equations. The stress tensor The stress tensor is related to velocity and pressure. Its actual form will depend on the specific fluid being dealt with, for the common case of incompressible Newtonian flow the stress tensor is given by where is the pressure in the fluid, is the velocity, and is the viscosity. Static interfaces In the absence of motion, the stress tensors yield only hydrostatic pressure so that , regardless of fluid type or compressibility. Considering the normal and tangential equations, The first equation establishes that curvature forces are balanced by pressure forces. The second equation implies that a static interface cannot exist in the presence of nonzero surface tension gradient. If gravity is the only body