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 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 ${\displaystyle S}$, the equation reads

where ${\displaystyle \scriptstyle \mathbf {\hat {n}} }$ is the unit normal pointing toward the "other" fluid (the one whose quantities are notated with bars), ${\displaystyle \scriptstyle \sigma _{ij}}$ is the stress tensor (note that on the left is a tensor-vector product), ${\displaystyle \scriptstyle \gamma }$ is the surface tension associated with the interface, and ${\displaystyle \scriptstyle \nabla _{S}}$ is the surface gradient. Note that the quantity ${\displaystyle \scriptstyle -\nabla _{\!S}\cdot \mathbf {\hat {n}} }$ is twice the mean curvature of the surface.

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