Capillary surface
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.
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,[2] which can be derived by considering the forces and stresses acting on a small volume that is partly bounded by a capillary surface.
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),
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
It's not too surprising then that the stress balance equation normally mandates its own boundary conditions.
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
In the absence of motion, the stress tensors yield only hydrostatic pressure so that
The second equation implies that a static interface cannot exist in the presence of nonzero surface tension gradient.
This nonlinear partial differential equation when supplied with the right boundary conditions will define the static interface.
The linear solution to pressure implies that, unless the gravity term is absent, it is always possible to define the
Nondimensionalized, the Young-Laplace equation is usually studied in the form [1] where (if gravity is in the negative
This nonlinear equation has some rich properties, especially in terms of existence of unique solutions.
This is interesting because there isn't another physical equation to determine the pressure difference.
In a capillary tube, for example, implementing the contact angle boundary condition will yield a unique solution for exactly one value of
Solutions often aren't unique, this implies that there are multiple static interfaces possible; while they may all solve the same boundary value problem, the minimization of energy will normally favor one.
For example, if there are two different fluids (say liquid and gas) inside a solid container with gravity and other energy potentials absent, the energy of the system is where the subscripts
Note that inclusion of gravity would require consideration of the volume enclosed by the capillary surface and the solid walls.
This does not pose a problem; since only changes in energy are of primary interest.
is a constant, and the contact angle is known, it may be shown that (again, for two different fluids in a solid container) so that where
is the contact angle and the capital delta indicates the change from one configuration to another.
To obtain this result, it's necessary to sum (distributed) forces at the contact line (where solid, gas, and liquid meet) in a direction tangent to the solid interface and perpendicular to the contact line: where the sum is zero because of the static state.
When solutions to the Young-Laplace equation aren't unique, the most physically favorable solution is the one of minimum energy, though experiments (especially low gravity) show that metastable surfaces can be surprisingly persistent, and that the most stable configuration can become metastable through mechanical jarring without too much difficulty.
On the other hand, a metastable surface can sometimes spontaneously achieve lower energy without any input (seemingly at least) given enough time.
Boundary conditions for stress balance describe the capillary surface at the contact line: the line where a solid meets the capillary interface; also, volume constraints can serve as boundary conditions (a suspended drop, for example, has no contact line but clearly must admit a unique solution).
For static surfaces, the most common contact line boundary condition is the implementation of the contact angle, which specifies the angle that one of the fluids meets the solid wall.
For dynamic interfaces, the boundary condition showed above works well if the contact line velocity is low.
