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.
Read more about this topic: Capillary Surface
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