Surface tension gases. The pressure inside an ideal (one surface) soap bubble can be derived from thermodynamic free energy considerations. At constant temperature and particle number, , the differential Helmholtz energy is given by where is the difference in pressure inside and outside of the bubble, and is the surface tension. In equilibrium, , and so, For a spherical bubble, the volume and surface area are given simply by and Substituting these relations into the previous expression, we find which is equivalent to the Young–Laplace equation when . For real soap bubbles, the pressure is doubled due to the presence of two
Finite volume method tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, formula_42, we take the volume integral over the total volume of the cell, formula_43, which gives, On integrating the first term to get the "volume average" and applying the "divergence theorem" to the second, this yields where formula_46 represents the total surface area of the cell and formula_47 is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (8), i.e. Again, values for the edge fluxes can be reconstructed by