Isotropic Interfaces and Mean Curvature

For the particular case of an axisymmetric surface r(z), the curvature is the sum of the radius of the osculating circle (in the plane shared by the surface normal) and the curvature of r(z) in two dimensions  

The convention that a convex interface of a solid body has positive mean curvature and a concave interface has negative mean curvature is adopted throughout this book (see Section 14.1). A table of surface formulae is provided in Table C.l. 

As will become evident below, mean interface curvature is useful when the interfacial energy is isotropic (not dependent on interface inclination). 

Weighted Mean Curvature of an Interface. The weighted mean curvature, k7, has exactly the same geometrical properties as the mean curvature except that it is weighted by the possibly orientation-dependent magnitude of the interfacial tension. It is particularly useful for addressing capillarity problems when the interfacial energy is anisotropic, that is, dependent upon the interface orientation (Section C.3). 

---