In the general case, for a free surface and where there is an applied "over-pressure", Δp, at the interface in equilibrium, there is a balance between the applied pressure, the hydrostatic pressure and the effects of surface tension. The Young–Laplace equation becomes:
The equation can be non-dimensionalised in terms of its characteristic length-scale, the capillary length:
and characteristic pressure
For clean water at standard temperature and pressure, the capillary length is ~2 mm.
The non-dimensional equation then becomes:
Thus, the surface shape is determined by only one parameter, the over pressure of the fluid, Δp and the scale of the surface is given by the capillary length. The solution of the equation requires an initial condition for position, and the gradient of the surface at the start point.